The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.
10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
LUNCH BREAK
1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
Tuesday, May 3, 2022
9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.
Speakers Banquet
Wednesday, May 4, 2022
9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Xin Zhou (Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
May 5, 2022
9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.
10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.
LUNCH BREAK
1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.
10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
LUNCH BREAK
1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
Tuesday, May 3, 2022
9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.
Speakers Banquet
Wednesday, May 4, 2022
9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Xin Zhou (Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
May 5, 2022
9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.
10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.
LUNCH BREAK
1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.
10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
LUNCH BREAK
1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
Tuesday, May 3, 2022
9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.
Speakers Banquet
Wednesday, May 4, 2022
9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Xin Zhou (Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
May 5, 2022
9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.
10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.
LUNCH BREAK
1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Abstract: Consider a bootstrap percolation process that starts with a set of `infected’ triangles $Y \subseteq \binom{[n]}3$, and a new triangle f gets infected if there is a copy of K_4^3 (= the boundary of a tetrahedron) in which f is the only not-yet infected triangle. Suppose that every triangle is initially infected independently with probability p=p(n), what is the threshold probability for percolation — the event that all triangles get infected? How many new triangles do get infected in the subcritical regime?
This notion of percolation can be viewed as a simplification of simple-connectivity. Namely, a stacked triangulation of a triangle is obtained by repeatedly subdividing an inner face into three faces. We ask: for which $p$ does the random simplicial complex Y_2(n,p) contain, for every triple $xyz$, the faces of a stacked triangulation of $xyz$ whose internal vertices are arbitrarily labeled in [n].
We consider this problem in every dimension d>=2, and our main result identifies a sharp probability threshold for percolation, showing it is asymptotically (c_d*n)^(-1/d), where c_d is the growth rate of the Fuss–Catalan numbers of order d.
The proof hinges on a second moment argument in the supercritical regime, and on Kalai’s algebraic shifting in the subcritical regime.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.
10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
LUNCH BREAK
1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
Tuesday, May 3, 2022
9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.
Speakers Banquet
Wednesday, May 4, 2022
9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Xin Zhou (Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
May 5, 2022
9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.
10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.
LUNCH BREAK
1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Abstract: Consider a bootstrap percolation process that starts with a set of `infected’ triangles $Y \subseteq \binom{[n]}3$, and a new triangle f gets infected if there is a copy of K_4^3 (= the boundary of a tetrahedron) in which f is the only not-yet infected triangle. Suppose that every triangle is initially infected independently with probability p=p(n), what is the threshold probability for percolation — the event that all triangles get infected? How many new triangles do get infected in the subcritical regime?
This notion of percolation can be viewed as a simplification of simple-connectivity. Namely, a stacked triangulation of a triangle is obtained by repeatedly subdividing an inner face into three faces. We ask: for which $p$ does the random simplicial complex Y_2(n,p) contain, for every triple $xyz$, the faces of a stacked triangulation of $xyz$ whose internal vertices are arbitrarily labeled in [n].
We consider this problem in every dimension d>=2, and our main result identifies a sharp probability threshold for percolation, showing it is asymptotically (c_d*n)^(-1/d), where c_d is the growth rate of the Fuss–Catalan numbers of order d.
The proof hinges on a second moment argument in the supercritical regime, and on Kalai’s algebraic shifting in the subcritical regime.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.
10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
LUNCH BREAK
1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
Tuesday, May 3, 2022
9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.
Speakers Banquet
Wednesday, May 4, 2022
9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Xin Zhou (Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
May 5, 2022
9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.
10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.
LUNCH BREAK
1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.
10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
LUNCH BREAK
1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
Tuesday, May 3, 2022
9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.
Speakers Banquet
Wednesday, May 4, 2022
9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Xin Zhou (Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
May 5, 2022
9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.
10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.
LUNCH BREAK
1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.
10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
LUNCH BREAK
1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
Tuesday, May 3, 2022
9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.
Speakers Banquet
Wednesday, May 4, 2022
9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Xin Zhou (Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
May 5, 2022
9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.
10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.
LUNCH BREAK
1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: Although everyone talks about AI + healthcare, many people were unaware of the fact that there are two possible outcomes of the collaboration, due to the inherent dissimilarity between the two giant subjects. The first possibility is healthcare-leads, and AI is for building new tools to make steps in healthcare easier, better, more effective or more accurate. The other possibility is AI-leads, and therefore the protocols of healthcare can be redesigned or redefined to make sure that the whole infrastructure and pipelines are ideal for running AI algorithms.
Our system Qianfang belongs to the second category. We have designed a new kind of clinic for the doctors and patients, so that it will be able to collect high quality data for AI algorithms. Interestingly, the clinic is based on Traditional Chinese Medicine (TCM) instead of modern medicine, because we believe that TCM is more suitable for AI algorithms as the starting point.
In this talk, I will elaborate on how we convert TCM knowledge into a modern type-safe large-scale system, the mini-language that we have designed for the doctors and patients, the interpretability of AI decisions, and our feedback loop for collecting data.
Our project is still on-going, not finished yet.Bio: Yang Yuan is now an assistant professor at IIIS, Tsinghua. He finished his undergraduate study at Peking University in 2012. Afterwards, he received his PhD at Cornell University in 2018, advised by Professor Robert Kleinberg. During his PhD, he was a visiting student at MIT/Microsoft New England (2014-2015) and Princeton University (2016 Fall). Before joining Tsinghua, he spent one year at MIT Institute for Foundations of Data Science (MIFODS) as a postdoc researcher. He now works on AI+Healthcare, AI Interpretability and AI system.
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.
10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
LUNCH BREAK
1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
Tuesday, May 3, 2022
9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.
Speakers Banquet
Wednesday, May 4, 2022
9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Xin Zhou (Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
May 5, 2022
9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.
10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.
LUNCH BREAK
1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: Although everyone talks about AI + healthcare, many people were unaware of the fact that there are two possible outcomes of the collaboration, due to the inherent dissimilarity between the two giant subjects. The first possibility is healthcare-leads, and AI is for building new tools to make steps in healthcare easier, better, more effective or more accurate. The other possibility is AI-leads, and therefore the protocols of healthcare can be redesigned or redefined to make sure that the whole infrastructure and pipelines are ideal for running AI algorithms.
Our system Qianfang belongs to the second category. We have designed a new kind of clinic for the doctors and patients, so that it will be able to collect high quality data for AI algorithms. Interestingly, the clinic is based on Traditional Chinese Medicine (TCM) instead of modern medicine, because we believe that TCM is more suitable for AI algorithms as the starting point.
In this talk, I will elaborate on how we convert TCM knowledge into a modern type-safe large-scale system, the mini-language that we have designed for the doctors and patients, the interpretability of AI decisions, and our feedback loop for collecting data.
Our project is still on-going, not finished yet.Bio: Yang Yuan is now an assistant professor at IIIS, Tsinghua. He finished his undergraduate study at Peking University in 2012. Afterwards, he received his PhD at Cornell University in 2018, advised by Professor Robert Kleinberg. During his PhD, he was a visiting student at MIT/Microsoft New England (2014-2015) and Princeton University (2016 Fall). Before joining Tsinghua, he spent one year at MIT Institute for Foundations of Data Science (MIFODS) as a postdoc researcher. He now works on AI+Healthcare, AI Interpretability and AI system.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
On May 6–8, 2022, the CMSA hosted a second NSF FRG Workshop.
This project brings together a community of researchers who develop theoretical and computational models to characterize shapes. Their combined interests span Mathematics (Geometry and Topology), Computer Science (Scientific Computing and Complexity Theory), and domain sciences, from Data Sciences to Computational Biology.
Scientific research benefits from the development of an ever-growing number of sensors that are able to capture details of the world at increasingly fine resolutions. The seemingly unlimited breadth and depth of these sources provide the means to study complex systems in a more comprehensive way. At the same time, however, these sensors are generating a huge amount of data that comes with a high level of complexity and heterogeneity, providing indirect measurements of hidden processes that provide keys to the systems under study. This has led to new challenges and opportunities in data analysis. Our focus is on image data and the shapes they represent. Advances in geometry and topology have led to powerful new tools that can be applied to geometric methods for representing, searching, simulating, analyzing, and comparing shapes. These methods and tools can be applied in a wide range of fields, including computer vision, biological imaging, brain mapping, target recognition, and satellite image analysis.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Abstract: We have recently initiated a generalization of discrete exterior calculus to differential forms with values in a vector bundle. A discrete vector bundle with connection over a simplicial complex has fibers at vertices and transport maps on edges, just as in lattice gauge theory. The first part of this work involves defining and examining properties of a combinatorial exterior covariant derivative and wedge product. We find that these operators commute with pullback under simplicial maps of the base space. From these definitions emerges a combinatorial curvature. In the second part of this work we showed that the curvature behaves as one expects: it measures failure of parallel transport to be independent of the path, and is the local obstruction to a trivialization. For a bundle with metric, metric compatibility of the discrete connection is equivalent to a Leibniz rule. Vanishing curvature is indeed equivalent to an appropriately defined discrete flat connection, and curvature obstructs trivializability. In this talk I will focus on just the first part, and talk about naturality of the discrete exterior covariant derivative and discrete wedge product using simple examples. Joint work with Daniel Berwick-Evans (UIUC) and Mark Schubel (Apple, Inc.).
11:10–12:00 pm
David Gu
Title: Surface Quadrilateral Meshing Based on Abel-Jacobi Theory
Abstract: Surface quadrilateral meshing plays an important role in many fields. For example, in CAD (computer-aided design), all shapes are represented as Spline surfaces, which requires structured quad-meshing; in CAE (computer-aided engineering), the surface tessellation greatly affects the accuracy and efficiency of numerical simulations. Although the research on mesh generation has a long history, it remains a great challenge to automatically generate structured quad-meshes with high qualities. The key is to find the governing equation for the singularities of the global structured quad-meshes.
In this talk, we introduce our recent discovery: the singularities of a quad-mesh are governed by the Abel theorem. We show that each quad-mesh determines a conformal structure and a meromorphic quadratic differential, the configuration of the mesh singularities can be described as the divisor of the differential. The quad-mesh divisor minus four times of the divisor of a holomorphic one-form is principal and satisfies the Abel theorem: its image under the Jacobi map is zero in the Jacobi variety.
This leads to a rigorous and efficient algorithm for surface structured quadrilateral meshing. After determining the singularities, the metric induced by the quad-mesh can be computed using the discrete Yambe flow, and the meromorphic quartic differential can be constructed, the trajectories of the differentials give the quad-mesh. The method can be applied directly for geometric modeling and computational mechanics.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Justin Solomon
Title: Geometry Processing with Volumes
Abstract: Many algorithms in geometry processing are restricted to two-dimensional surfaces represented as triangle meshes. Drawing inspiration from simulation, medical imaging, and other application domains, however, there is a substantial demand for geometry processing algorithms targeted to volumes represented as tetrahedral meshes or grids. In this talk, I will summarize some efforts in our group to develop a geometry processing toolkit specifically for volumes. Specifically, I will cover our recent work on hexahedral remeshing via cuboid decomposition, volumetric correspondence, and minimal surface computation via geometric measure theory.
3:00–3:20 pm
Oded Stein
Title: Optimization for flip-free parametrization
Abstract: Parametrizations without flipped elements are desirable in a variety of applications such as UV mapping and surface/volume correspondence. Computing flip-free parametrizations can be challenging, and there are many different approaches to the problem. In this talk we will look at multiple strategies for flip-free parametrizations that are based on the optimization of continuous energies. Due to the nature of the problem, these energies are often nonconvex and unbounded, which is a challenge for optimization methods. We will also take a closer look at our recently developed method for computing flip-free parametrizations using the Alternating Direction Method of Multipliers (ADMM).
3:20–4:00 pm
Break
4:00–4:50 pm
John Bowers
Title: Koebe-Andre’ev-Thurston Packings via Flow
Abstract: Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles.
The full Koebe-Andre’ev-Thurston theorem generalizes the circle packing theorem to allow for neighboring circles to overlap by angles up to $\pi/2$. In this talk I will show that the Connelly-Gortler method can be extended to allow for circles to overlap to angles up to $\pi/2$. This results in a new proof of the general Koebe-Andre’ev-Thurston theorem for disk patterns on $\mathbb{S}^2$ as well as a numerical algorithm for computing them. The proof involves generalizing a notion of convexity for circle polyhedra that was recently used to prove the global rigidity of certain circle packings, which is then used to show that all convex circle polyhedra are infinitesimally rigid, a result of independent interest.
5:00–5:30 pm
Movies
“conform!” & ”Koebe polyhedra”
Saturday, May 7, 2022
9:30–10:20 am
Alexander Bobenko
Title: The Bonnet problem: Is a surface characterized by its metric and curvatures?
Abstract: We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface is characterized by a metric and mean curvature function. Generically, the answer is yes. Special cases when it is not the case are classified. In particular, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas
10:20–11:00 am
Break
11:00–11:50 am
Miri Ben Chen
Title: Surface Multigrid via Intrinsic Prolongation
Abstract: The solution of a linear system is a required ingredient in many geometry processing applications, and multigrid methods are among the most efficient solution techniques. However, due to the unstructured nature of triangle meshes, mapping functions between different multigrid levels is challenging. In this talk I will present our recent work that uses an intrinsic prolongation operator as the main building block in a multigrid solver for curved triangle meshes. Our solver can be used as a black-box in any triangle-mesh based system that requires a linear solve, and leads to order of magnitude time-efficiency improvement compared to direct solvers.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Steven Gortler
Title: Reconstructing configurations and graphs from unlabeled distance measurements
Abstract: Place a configuration of n points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence)? This question is motivated by a number of inverse problem applications. In this talk, I will review what is known about this question.
3:00–3:20 pm
Bohan Zhou
Title: Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs
Abstract: Optimal transport has profound and wide applications since its introduction in 1781 by Monge. Thanks to the Benamou-Brenier formulation, it provides a meaningful functional in the image science like image and shape registrations. However, exact computation through LP or PDE is in general not practical in large scale, while the popular entropy-regularized method introduces additional diffusion noise, deteriorating shapes and boundaries. Until the recent work [Jacobs and Leger, A Fast Approach to Optimal Transport: the back-and-forth method, Numerische Mathematik, 2020], solving OT in a both accurate and fast fashion finally becomes possible. Multiple marginal optimal transport is a natural extension from OT but has its own interest and is in general more computationally expensive. The entropy method suffers from both diffusion noise and high dimensional computational issues. In this work with Matthew Parno, we extend from two marginals to multiple marginals, on a wide class of cost functions when those marginals have a graph structure. This new method is fast and does not introduce diffusion. As a result, the new proposed method can be used in many fields those require sharp boundaries. If time allows, we will illustrate by examples the faithful joint recover via MMOT of images with sharp boundaries, with applications on sea ice prediction.
3:20–4:00 pm
Break
4:00–4:50 pm
Peter Schroeder
Title: Constrained Willmore Surfaces
Abstract: The Willmore energy of a surface is a canonical example of a squared curvature bending energy. Its minimizers are therefore of interest both in the theory of surfaces and in practical applications from physical and geometric modeling. Minimizing the bending energy alone however is insufficient. Taking a cue from univariate splines which incorporate an isometry constraint we consider Willmore minimizers subject to a conformality constraint. In this talk I will report on a numerical algorithm to find such constrained minimizers for triangle meshes.
Joint work with Yousuf Soliman (Caltech), Olga Diamanti (UGraz), Albert Chern (UCSD), Felix Knöppel (TU Berlin), Ulrich Pinkall (TU Berlin).
5:00–5:50 pm
Problems and Application discussions
Sunday, May 8, 2022
9:00–9:50 am
Tianqi Wu
Title: Convergence of discrete uniformizations
Abstract: The theory of discrete conformality, based on the notion of vertex scaling, has been implemented in computing conformal maps or uniformizations of surfaces. We will show that if a Delaunay triangle mesh approximates a smooth surface, then the related discrete uniformization will converge to the smooth uniformization, with an error bounded linearly by the size of the triangles in the mesh.
10:10–11:00 am
Yanwen Luo
Title: Recent Progress in Spaces of Geodesic Triangulations of Surfaces Abstract: Spaces of geodesic triangulations of surfaces are natural discretization of the groups of surface diffeomorphisms isotopy to the identity. It has been conjectured that these spaces have the same homotopy type as their smooth counterparts. In this talk, we will report the recent progress in this problem. The key ingredient is the idea in Tutte’s embedding theorem. We will explain how to use it to identify the homotopy types of spaces of geodesic triangulations. This is joint work with Tianqi Wu and Xiaoping Zhu.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
On May 6–8, 2022, the CMSA hosted a second NSF FRG Workshop.
This project brings together a community of researchers who develop theoretical and computational models to characterize shapes. Their combined interests span Mathematics (Geometry and Topology), Computer Science (Scientific Computing and Complexity Theory), and domain sciences, from Data Sciences to Computational Biology.
Scientific research benefits from the development of an ever-growing number of sensors that are able to capture details of the world at increasingly fine resolutions. The seemingly unlimited breadth and depth of these sources provide the means to study complex systems in a more comprehensive way. At the same time, however, these sensors are generating a huge amount of data that comes with a high level of complexity and heterogeneity, providing indirect measurements of hidden processes that provide keys to the systems under study. This has led to new challenges and opportunities in data analysis. Our focus is on image data and the shapes they represent. Advances in geometry and topology have led to powerful new tools that can be applied to geometric methods for representing, searching, simulating, analyzing, and comparing shapes. These methods and tools can be applied in a wide range of fields, including computer vision, biological imaging, brain mapping, target recognition, and satellite image analysis.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Abstract: We have recently initiated a generalization of discrete exterior calculus to differential forms with values in a vector bundle. A discrete vector bundle with connection over a simplicial complex has fibers at vertices and transport maps on edges, just as in lattice gauge theory. The first part of this work involves defining and examining properties of a combinatorial exterior covariant derivative and wedge product. We find that these operators commute with pullback under simplicial maps of the base space. From these definitions emerges a combinatorial curvature. In the second part of this work we showed that the curvature behaves as one expects: it measures failure of parallel transport to be independent of the path, and is the local obstruction to a trivialization. For a bundle with metric, metric compatibility of the discrete connection is equivalent to a Leibniz rule. Vanishing curvature is indeed equivalent to an appropriately defined discrete flat connection, and curvature obstructs trivializability. In this talk I will focus on just the first part, and talk about naturality of the discrete exterior covariant derivative and discrete wedge product using simple examples. Joint work with Daniel Berwick-Evans (UIUC) and Mark Schubel (Apple, Inc.).
11:10–12:00 pm
David Gu
Title: Surface Quadrilateral Meshing Based on Abel-Jacobi Theory
Abstract: Surface quadrilateral meshing plays an important role in many fields. For example, in CAD (computer-aided design), all shapes are represented as Spline surfaces, which requires structured quad-meshing; in CAE (computer-aided engineering), the surface tessellation greatly affects the accuracy and efficiency of numerical simulations. Although the research on mesh generation has a long history, it remains a great challenge to automatically generate structured quad-meshes with high qualities. The key is to find the governing equation for the singularities of the global structured quad-meshes.
In this talk, we introduce our recent discovery: the singularities of a quad-mesh are governed by the Abel theorem. We show that each quad-mesh determines a conformal structure and a meromorphic quadratic differential, the configuration of the mesh singularities can be described as the divisor of the differential. The quad-mesh divisor minus four times of the divisor of a holomorphic one-form is principal and satisfies the Abel theorem: its image under the Jacobi map is zero in the Jacobi variety.
This leads to a rigorous and efficient algorithm for surface structured quadrilateral meshing. After determining the singularities, the metric induced by the quad-mesh can be computed using the discrete Yambe flow, and the meromorphic quartic differential can be constructed, the trajectories of the differentials give the quad-mesh. The method can be applied directly for geometric modeling and computational mechanics.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Justin Solomon
Title: Geometry Processing with Volumes
Abstract: Many algorithms in geometry processing are restricted to two-dimensional surfaces represented as triangle meshes. Drawing inspiration from simulation, medical imaging, and other application domains, however, there is a substantial demand for geometry processing algorithms targeted to volumes represented as tetrahedral meshes or grids. In this talk, I will summarize some efforts in our group to develop a geometry processing toolkit specifically for volumes. Specifically, I will cover our recent work on hexahedral remeshing via cuboid decomposition, volumetric correspondence, and minimal surface computation via geometric measure theory.
3:00–3:20 pm
Oded Stein
Title: Optimization for flip-free parametrization
Abstract: Parametrizations without flipped elements are desirable in a variety of applications such as UV mapping and surface/volume correspondence. Computing flip-free parametrizations can be challenging, and there are many different approaches to the problem. In this talk we will look at multiple strategies for flip-free parametrizations that are based on the optimization of continuous energies. Due to the nature of the problem, these energies are often nonconvex and unbounded, which is a challenge for optimization methods. We will also take a closer look at our recently developed method for computing flip-free parametrizations using the Alternating Direction Method of Multipliers (ADMM).
3:20–4:00 pm
Break
4:00–4:50 pm
John Bowers
Title: Koebe-Andre’ev-Thurston Packings via Flow
Abstract: Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles.
The full Koebe-Andre’ev-Thurston theorem generalizes the circle packing theorem to allow for neighboring circles to overlap by angles up to $\pi/2$. In this talk I will show that the Connelly-Gortler method can be extended to allow for circles to overlap to angles up to $\pi/2$. This results in a new proof of the general Koebe-Andre’ev-Thurston theorem for disk patterns on $\mathbb{S}^2$ as well as a numerical algorithm for computing them. The proof involves generalizing a notion of convexity for circle polyhedra that was recently used to prove the global rigidity of certain circle packings, which is then used to show that all convex circle polyhedra are infinitesimally rigid, a result of independent interest.
5:00–5:30 pm
Movies
“conform!” & ”Koebe polyhedra”
Saturday, May 7, 2022
9:30–10:20 am
Alexander Bobenko
Title: The Bonnet problem: Is a surface characterized by its metric and curvatures?
Abstract: We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface is characterized by a metric and mean curvature function. Generically, the answer is yes. Special cases when it is not the case are classified. In particular, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas
10:20–11:00 am
Break
11:00–11:50 am
Miri Ben Chen
Title: Surface Multigrid via Intrinsic Prolongation
Abstract: The solution of a linear system is a required ingredient in many geometry processing applications, and multigrid methods are among the most efficient solution techniques. However, due to the unstructured nature of triangle meshes, mapping functions between different multigrid levels is challenging. In this talk I will present our recent work that uses an intrinsic prolongation operator as the main building block in a multigrid solver for curved triangle meshes. Our solver can be used as a black-box in any triangle-mesh based system that requires a linear solve, and leads to order of magnitude time-efficiency improvement compared to direct solvers.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Steven Gortler
Title: Reconstructing configurations and graphs from unlabeled distance measurements
Abstract: Place a configuration of n points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence)? This question is motivated by a number of inverse problem applications. In this talk, I will review what is known about this question.
3:00–3:20 pm
Bohan Zhou
Title: Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs
Abstract: Optimal transport has profound and wide applications since its introduction in 1781 by Monge. Thanks to the Benamou-Brenier formulation, it provides a meaningful functional in the image science like image and shape registrations. However, exact computation through LP or PDE is in general not practical in large scale, while the popular entropy-regularized method introduces additional diffusion noise, deteriorating shapes and boundaries. Until the recent work [Jacobs and Leger, A Fast Approach to Optimal Transport: the back-and-forth method, Numerische Mathematik, 2020], solving OT in a both accurate and fast fashion finally becomes possible. Multiple marginal optimal transport is a natural extension from OT but has its own interest and is in general more computationally expensive. The entropy method suffers from both diffusion noise and high dimensional computational issues. In this work with Matthew Parno, we extend from two marginals to multiple marginals, on a wide class of cost functions when those marginals have a graph structure. This new method is fast and does not introduce diffusion. As a result, the new proposed method can be used in many fields those require sharp boundaries. If time allows, we will illustrate by examples the faithful joint recover via MMOT of images with sharp boundaries, with applications on sea ice prediction.
3:20–4:00 pm
Break
4:00–4:50 pm
Peter Schroeder
Title: Constrained Willmore Surfaces
Abstract: The Willmore energy of a surface is a canonical example of a squared curvature bending energy. Its minimizers are therefore of interest both in the theory of surfaces and in practical applications from physical and geometric modeling. Minimizing the bending energy alone however is insufficient. Taking a cue from univariate splines which incorporate an isometry constraint we consider Willmore minimizers subject to a conformality constraint. In this talk I will report on a numerical algorithm to find such constrained minimizers for triangle meshes.
Joint work with Yousuf Soliman (Caltech), Olga Diamanti (UGraz), Albert Chern (UCSD), Felix Knöppel (TU Berlin), Ulrich Pinkall (TU Berlin).
5:00–5:50 pm
Problems and Application discussions
Sunday, May 8, 2022
9:00–9:50 am
Tianqi Wu
Title: Convergence of discrete uniformizations
Abstract: The theory of discrete conformality, based on the notion of vertex scaling, has been implemented in computing conformal maps or uniformizations of surfaces. We will show that if a Delaunay triangle mesh approximates a smooth surface, then the related discrete uniformization will converge to the smooth uniformization, with an error bounded linearly by the size of the triangles in the mesh.
10:10–11:00 am
Yanwen Luo
Title: Recent Progress in Spaces of Geodesic Triangulations of Surfaces Abstract: Spaces of geodesic triangulations of surfaces are natural discretization of the groups of surface diffeomorphisms isotopy to the identity. It has been conjectured that these spaces have the same homotopy type as their smooth counterparts. In this talk, we will report the recent progress in this problem. The key ingredient is the idea in Tutte’s embedding theorem. We will explain how to use it to identify the homotopy types of spaces of geodesic triangulations. This is joint work with Tianqi Wu and Xiaoping Zhu.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
On May 6–8, 2022, the CMSA hosted a second NSF FRG Workshop.
This project brings together a community of researchers who develop theoretical and computational models to characterize shapes. Their combined interests span Mathematics (Geometry and Topology), Computer Science (Scientific Computing and Complexity Theory), and domain sciences, from Data Sciences to Computational Biology.
Scientific research benefits from the development of an ever-growing number of sensors that are able to capture details of the world at increasingly fine resolutions. The seemingly unlimited breadth and depth of these sources provide the means to study complex systems in a more comprehensive way. At the same time, however, these sensors are generating a huge amount of data that comes with a high level of complexity and heterogeneity, providing indirect measurements of hidden processes that provide keys to the systems under study. This has led to new challenges and opportunities in data analysis. Our focus is on image data and the shapes they represent. Advances in geometry and topology have led to powerful new tools that can be applied to geometric methods for representing, searching, simulating, analyzing, and comparing shapes. These methods and tools can be applied in a wide range of fields, including computer vision, biological imaging, brain mapping, target recognition, and satellite image analysis.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Abstract: We have recently initiated a generalization of discrete exterior calculus to differential forms with values in a vector bundle. A discrete vector bundle with connection over a simplicial complex has fibers at vertices and transport maps on edges, just as in lattice gauge theory. The first part of this work involves defining and examining properties of a combinatorial exterior covariant derivative and wedge product. We find that these operators commute with pullback under simplicial maps of the base space. From these definitions emerges a combinatorial curvature. In the second part of this work we showed that the curvature behaves as one expects: it measures failure of parallel transport to be independent of the path, and is the local obstruction to a trivialization. For a bundle with metric, metric compatibility of the discrete connection is equivalent to a Leibniz rule. Vanishing curvature is indeed equivalent to an appropriately defined discrete flat connection, and curvature obstructs trivializability. In this talk I will focus on just the first part, and talk about naturality of the discrete exterior covariant derivative and discrete wedge product using simple examples. Joint work with Daniel Berwick-Evans (UIUC) and Mark Schubel (Apple, Inc.).
11:10–12:00 pm
David Gu
Title: Surface Quadrilateral Meshing Based on Abel-Jacobi Theory
Abstract: Surface quadrilateral meshing plays an important role in many fields. For example, in CAD (computer-aided design), all shapes are represented as Spline surfaces, which requires structured quad-meshing; in CAE (computer-aided engineering), the surface tessellation greatly affects the accuracy and efficiency of numerical simulations. Although the research on mesh generation has a long history, it remains a great challenge to automatically generate structured quad-meshes with high qualities. The key is to find the governing equation for the singularities of the global structured quad-meshes.
In this talk, we introduce our recent discovery: the singularities of a quad-mesh are governed by the Abel theorem. We show that each quad-mesh determines a conformal structure and a meromorphic quadratic differential, the configuration of the mesh singularities can be described as the divisor of the differential. The quad-mesh divisor minus four times of the divisor of a holomorphic one-form is principal and satisfies the Abel theorem: its image under the Jacobi map is zero in the Jacobi variety.
This leads to a rigorous and efficient algorithm for surface structured quadrilateral meshing. After determining the singularities, the metric induced by the quad-mesh can be computed using the discrete Yambe flow, and the meromorphic quartic differential can be constructed, the trajectories of the differentials give the quad-mesh. The method can be applied directly for geometric modeling and computational mechanics.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Justin Solomon
Title: Geometry Processing with Volumes
Abstract: Many algorithms in geometry processing are restricted to two-dimensional surfaces represented as triangle meshes. Drawing inspiration from simulation, medical imaging, and other application domains, however, there is a substantial demand for geometry processing algorithms targeted to volumes represented as tetrahedral meshes or grids. In this talk, I will summarize some efforts in our group to develop a geometry processing toolkit specifically for volumes. Specifically, I will cover our recent work on hexahedral remeshing via cuboid decomposition, volumetric correspondence, and minimal surface computation via geometric measure theory.
3:00–3:20 pm
Oded Stein
Title: Optimization for flip-free parametrization
Abstract: Parametrizations without flipped elements are desirable in a variety of applications such as UV mapping and surface/volume correspondence. Computing flip-free parametrizations can be challenging, and there are many different approaches to the problem. In this talk we will look at multiple strategies for flip-free parametrizations that are based on the optimization of continuous energies. Due to the nature of the problem, these energies are often nonconvex and unbounded, which is a challenge for optimization methods. We will also take a closer look at our recently developed method for computing flip-free parametrizations using the Alternating Direction Method of Multipliers (ADMM).
3:20–4:00 pm
Break
4:00–4:50 pm
John Bowers
Title: Koebe-Andre’ev-Thurston Packings via Flow
Abstract: Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles.
The full Koebe-Andre’ev-Thurston theorem generalizes the circle packing theorem to allow for neighboring circles to overlap by angles up to $\pi/2$. In this talk I will show that the Connelly-Gortler method can be extended to allow for circles to overlap to angles up to $\pi/2$. This results in a new proof of the general Koebe-Andre’ev-Thurston theorem for disk patterns on $\mathbb{S}^2$ as well as a numerical algorithm for computing them. The proof involves generalizing a notion of convexity for circle polyhedra that was recently used to prove the global rigidity of certain circle packings, which is then used to show that all convex circle polyhedra are infinitesimally rigid, a result of independent interest.
5:00–5:30 pm
Movies
“conform!” & ”Koebe polyhedra”
Saturday, May 7, 2022
9:30–10:20 am
Alexander Bobenko
Title: The Bonnet problem: Is a surface characterized by its metric and curvatures?
Abstract: We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface is characterized by a metric and mean curvature function. Generically, the answer is yes. Special cases when it is not the case are classified. In particular, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas
10:20–11:00 am
Break
11:00–11:50 am
Miri Ben Chen
Title: Surface Multigrid via Intrinsic Prolongation
Abstract: The solution of a linear system is a required ingredient in many geometry processing applications, and multigrid methods are among the most efficient solution techniques. However, due to the unstructured nature of triangle meshes, mapping functions between different multigrid levels is challenging. In this talk I will present our recent work that uses an intrinsic prolongation operator as the main building block in a multigrid solver for curved triangle meshes. Our solver can be used as a black-box in any triangle-mesh based system that requires a linear solve, and leads to order of magnitude time-efficiency improvement compared to direct solvers.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Steven Gortler
Title: Reconstructing configurations and graphs from unlabeled distance measurements
Abstract: Place a configuration of n points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence)? This question is motivated by a number of inverse problem applications. In this talk, I will review what is known about this question.
3:00–3:20 pm
Bohan Zhou
Title: Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs
Abstract: Optimal transport has profound and wide applications since its introduction in 1781 by Monge. Thanks to the Benamou-Brenier formulation, it provides a meaningful functional in the image science like image and shape registrations. However, exact computation through LP or PDE is in general not practical in large scale, while the popular entropy-regularized method introduces additional diffusion noise, deteriorating shapes and boundaries. Until the recent work [Jacobs and Leger, A Fast Approach to Optimal Transport: the back-and-forth method, Numerische Mathematik, 2020], solving OT in a both accurate and fast fashion finally becomes possible. Multiple marginal optimal transport is a natural extension from OT but has its own interest and is in general more computationally expensive. The entropy method suffers from both diffusion noise and high dimensional computational issues. In this work with Matthew Parno, we extend from two marginals to multiple marginals, on a wide class of cost functions when those marginals have a graph structure. This new method is fast and does not introduce diffusion. As a result, the new proposed method can be used in many fields those require sharp boundaries. If time allows, we will illustrate by examples the faithful joint recover via MMOT of images with sharp boundaries, with applications on sea ice prediction.
3:20–4:00 pm
Break
4:00–4:50 pm
Peter Schroeder
Title: Constrained Willmore Surfaces
Abstract: The Willmore energy of a surface is a canonical example of a squared curvature bending energy. Its minimizers are therefore of interest both in the theory of surfaces and in practical applications from physical and geometric modeling. Minimizing the bending energy alone however is insufficient. Taking a cue from univariate splines which incorporate an isometry constraint we consider Willmore minimizers subject to a conformality constraint. In this talk I will report on a numerical algorithm to find such constrained minimizers for triangle meshes.
Joint work with Yousuf Soliman (Caltech), Olga Diamanti (UGraz), Albert Chern (UCSD), Felix Knöppel (TU Berlin), Ulrich Pinkall (TU Berlin).
5:00–5:50 pm
Problems and Application discussions
Sunday, May 8, 2022
9:00–9:50 am
Tianqi Wu
Title: Convergence of discrete uniformizations
Abstract: The theory of discrete conformality, based on the notion of vertex scaling, has been implemented in computing conformal maps or uniformizations of surfaces. We will show that if a Delaunay triangle mesh approximates a smooth surface, then the related discrete uniformization will converge to the smooth uniformization, with an error bounded linearly by the size of the triangles in the mesh.
10:10–11:00 am
Yanwen Luo
Title: Recent Progress in Spaces of Geodesic Triangulations of Surfaces Abstract: Spaces of geodesic triangulations of surfaces are natural discretization of the groups of surface diffeomorphisms isotopy to the identity. It has been conjectured that these spaces have the same homotopy type as their smooth counterparts. In this talk, we will report the recent progress in this problem. The key ingredient is the idea in Tutte’s embedding theorem. We will explain how to use it to identify the homotopy types of spaces of geodesic triangulations. This is joint work with Tianqi Wu and Xiaoping Zhu.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
On May 6–8, 2022, the CMSA hosted a second NSF FRG Workshop.
This project brings together a community of researchers who develop theoretical and computational models to characterize shapes. Their combined interests span Mathematics (Geometry and Topology), Computer Science (Scientific Computing and Complexity Theory), and domain sciences, from Data Sciences to Computational Biology.
Scientific research benefits from the development of an ever-growing number of sensors that are able to capture details of the world at increasingly fine resolutions. The seemingly unlimited breadth and depth of these sources provide the means to study complex systems in a more comprehensive way. At the same time, however, these sensors are generating a huge amount of data that comes with a high level of complexity and heterogeneity, providing indirect measurements of hidden processes that provide keys to the systems under study. This has led to new challenges and opportunities in data analysis. Our focus is on image data and the shapes they represent. Advances in geometry and topology have led to powerful new tools that can be applied to geometric methods for representing, searching, simulating, analyzing, and comparing shapes. These methods and tools can be applied in a wide range of fields, including computer vision, biological imaging, brain mapping, target recognition, and satellite image analysis.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Abstract: We have recently initiated a generalization of discrete exterior calculus to differential forms with values in a vector bundle. A discrete vector bundle with connection over a simplicial complex has fibers at vertices and transport maps on edges, just as in lattice gauge theory. The first part of this work involves defining and examining properties of a combinatorial exterior covariant derivative and wedge product. We find that these operators commute with pullback under simplicial maps of the base space. From these definitions emerges a combinatorial curvature. In the second part of this work we showed that the curvature behaves as one expects: it measures failure of parallel transport to be independent of the path, and is the local obstruction to a trivialization. For a bundle with metric, metric compatibility of the discrete connection is equivalent to a Leibniz rule. Vanishing curvature is indeed equivalent to an appropriately defined discrete flat connection, and curvature obstructs trivializability. In this talk I will focus on just the first part, and talk about naturality of the discrete exterior covariant derivative and discrete wedge product using simple examples. Joint work with Daniel Berwick-Evans (UIUC) and Mark Schubel (Apple, Inc.).
11:10–12:00 pm
David Gu
Title: Surface Quadrilateral Meshing Based on Abel-Jacobi Theory
Abstract: Surface quadrilateral meshing plays an important role in many fields. For example, in CAD (computer-aided design), all shapes are represented as Spline surfaces, which requires structured quad-meshing; in CAE (computer-aided engineering), the surface tessellation greatly affects the accuracy and efficiency of numerical simulations. Although the research on mesh generation has a long history, it remains a great challenge to automatically generate structured quad-meshes with high qualities. The key is to find the governing equation for the singularities of the global structured quad-meshes.
In this talk, we introduce our recent discovery: the singularities of a quad-mesh are governed by the Abel theorem. We show that each quad-mesh determines a conformal structure and a meromorphic quadratic differential, the configuration of the mesh singularities can be described as the divisor of the differential. The quad-mesh divisor minus four times of the divisor of a holomorphic one-form is principal and satisfies the Abel theorem: its image under the Jacobi map is zero in the Jacobi variety.
This leads to a rigorous and efficient algorithm for surface structured quadrilateral meshing. After determining the singularities, the metric induced by the quad-mesh can be computed using the discrete Yambe flow, and the meromorphic quartic differential can be constructed, the trajectories of the differentials give the quad-mesh. The method can be applied directly for geometric modeling and computational mechanics.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Justin Solomon
Title: Geometry Processing with Volumes
Abstract: Many algorithms in geometry processing are restricted to two-dimensional surfaces represented as triangle meshes. Drawing inspiration from simulation, medical imaging, and other application domains, however, there is a substantial demand for geometry processing algorithms targeted to volumes represented as tetrahedral meshes or grids. In this talk, I will summarize some efforts in our group to develop a geometry processing toolkit specifically for volumes. Specifically, I will cover our recent work on hexahedral remeshing via cuboid decomposition, volumetric correspondence, and minimal surface computation via geometric measure theory.
3:00–3:20 pm
Oded Stein
Title: Optimization for flip-free parametrization
Abstract: Parametrizations without flipped elements are desirable in a variety of applications such as UV mapping and surface/volume correspondence. Computing flip-free parametrizations can be challenging, and there are many different approaches to the problem. In this talk we will look at multiple strategies for flip-free parametrizations that are based on the optimization of continuous energies. Due to the nature of the problem, these energies are often nonconvex and unbounded, which is a challenge for optimization methods. We will also take a closer look at our recently developed method for computing flip-free parametrizations using the Alternating Direction Method of Multipliers (ADMM).
3:20–4:00 pm
Break
4:00–4:50 pm
John Bowers
Title: Koebe-Andre’ev-Thurston Packings via Flow
Abstract: Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles.
The full Koebe-Andre’ev-Thurston theorem generalizes the circle packing theorem to allow for neighboring circles to overlap by angles up to $\pi/2$. In this talk I will show that the Connelly-Gortler method can be extended to allow for circles to overlap to angles up to $\pi/2$. This results in a new proof of the general Koebe-Andre’ev-Thurston theorem for disk patterns on $\mathbb{S}^2$ as well as a numerical algorithm for computing them. The proof involves generalizing a notion of convexity for circle polyhedra that was recently used to prove the global rigidity of certain circle packings, which is then used to show that all convex circle polyhedra are infinitesimally rigid, a result of independent interest.
5:00–5:30 pm
Movies
“conform!” & ”Koebe polyhedra”
Saturday, May 7, 2022
9:30–10:20 am
Alexander Bobenko
Title: The Bonnet problem: Is a surface characterized by its metric and curvatures?
Abstract: We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface is characterized by a metric and mean curvature function. Generically, the answer is yes. Special cases when it is not the case are classified. In particular, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas
10:20–11:00 am
Break
11:00–11:50 am
Miri Ben Chen
Title: Surface Multigrid via Intrinsic Prolongation
Abstract: The solution of a linear system is a required ingredient in many geometry processing applications, and multigrid methods are among the most efficient solution techniques. However, due to the unstructured nature of triangle meshes, mapping functions between different multigrid levels is challenging. In this talk I will present our recent work that uses an intrinsic prolongation operator as the main building block in a multigrid solver for curved triangle meshes. Our solver can be used as a black-box in any triangle-mesh based system that requires a linear solve, and leads to order of magnitude time-efficiency improvement compared to direct solvers.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Steven Gortler
Title: Reconstructing configurations and graphs from unlabeled distance measurements
Abstract: Place a configuration of n points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence)? This question is motivated by a number of inverse problem applications. In this talk, I will review what is known about this question.
3:00–3:20 pm
Bohan Zhou
Title: Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs
Abstract: Optimal transport has profound and wide applications since its introduction in 1781 by Monge. Thanks to the Benamou-Brenier formulation, it provides a meaningful functional in the image science like image and shape registrations. However, exact computation through LP or PDE is in general not practical in large scale, while the popular entropy-regularized method introduces additional diffusion noise, deteriorating shapes and boundaries. Until the recent work [Jacobs and Leger, A Fast Approach to Optimal Transport: the back-and-forth method, Numerische Mathematik, 2020], solving OT in a both accurate and fast fashion finally becomes possible. Multiple marginal optimal transport is a natural extension from OT but has its own interest and is in general more computationally expensive. The entropy method suffers from both diffusion noise and high dimensional computational issues. In this work with Matthew Parno, we extend from two marginals to multiple marginals, on a wide class of cost functions when those marginals have a graph structure. This new method is fast and does not introduce diffusion. As a result, the new proposed method can be used in many fields those require sharp boundaries. If time allows, we will illustrate by examples the faithful joint recover via MMOT of images with sharp boundaries, with applications on sea ice prediction.
3:20–4:00 pm
Break
4:00–4:50 pm
Peter Schroeder
Title: Constrained Willmore Surfaces
Abstract: The Willmore energy of a surface is a canonical example of a squared curvature bending energy. Its minimizers are therefore of interest both in the theory of surfaces and in practical applications from physical and geometric modeling. Minimizing the bending energy alone however is insufficient. Taking a cue from univariate splines which incorporate an isometry constraint we consider Willmore minimizers subject to a conformality constraint. In this talk I will report on a numerical algorithm to find such constrained minimizers for triangle meshes.
Joint work with Yousuf Soliman (Caltech), Olga Diamanti (UGraz), Albert Chern (UCSD), Felix Knöppel (TU Berlin), Ulrich Pinkall (TU Berlin).
5:00–5:50 pm
Problems and Application discussions
Sunday, May 8, 2022
9:00–9:50 am
Tianqi Wu
Title: Convergence of discrete uniformizations
Abstract: The theory of discrete conformality, based on the notion of vertex scaling, has been implemented in computing conformal maps or uniformizations of surfaces. We will show that if a Delaunay triangle mesh approximates a smooth surface, then the related discrete uniformization will converge to the smooth uniformization, with an error bounded linearly by the size of the triangles in the mesh.
10:10–11:00 am
Yanwen Luo
Title: Recent Progress in Spaces of Geodesic Triangulations of Surfaces Abstract: Spaces of geodesic triangulations of surfaces are natural discretization of the groups of surface diffeomorphisms isotopy to the identity. It has been conjectured that these spaces have the same homotopy type as their smooth counterparts. In this talk, we will report the recent progress in this problem. The key ingredient is the idea in Tutte’s embedding theorem. We will explain how to use it to identify the homotopy types of spaces of geodesic triangulations. This is joint work with Tianqi Wu and Xiaoping Zhu.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
On May 6–8, 2022, the CMSA hosted a second NSF FRG Workshop.
This project brings together a community of researchers who develop theoretical and computational models to characterize shapes. Their combined interests span Mathematics (Geometry and Topology), Computer Science (Scientific Computing and Complexity Theory), and domain sciences, from Data Sciences to Computational Biology.
Scientific research benefits from the development of an ever-growing number of sensors that are able to capture details of the world at increasingly fine resolutions. The seemingly unlimited breadth and depth of these sources provide the means to study complex systems in a more comprehensive way. At the same time, however, these sensors are generating a huge amount of data that comes with a high level of complexity and heterogeneity, providing indirect measurements of hidden processes that provide keys to the systems under study. This has led to new challenges and opportunities in data analysis. Our focus is on image data and the shapes they represent. Advances in geometry and topology have led to powerful new tools that can be applied to geometric methods for representing, searching, simulating, analyzing, and comparing shapes. These methods and tools can be applied in a wide range of fields, including computer vision, biological imaging, brain mapping, target recognition, and satellite image analysis.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Abstract: We have recently initiated a generalization of discrete exterior calculus to differential forms with values in a vector bundle. A discrete vector bundle with connection over a simplicial complex has fibers at vertices and transport maps on edges, just as in lattice gauge theory. The first part of this work involves defining and examining properties of a combinatorial exterior covariant derivative and wedge product. We find that these operators commute with pullback under simplicial maps of the base space. From these definitions emerges a combinatorial curvature. In the second part of this work we showed that the curvature behaves as one expects: it measures failure of parallel transport to be independent of the path, and is the local obstruction to a trivialization. For a bundle with metric, metric compatibility of the discrete connection is equivalent to a Leibniz rule. Vanishing curvature is indeed equivalent to an appropriately defined discrete flat connection, and curvature obstructs trivializability. In this talk I will focus on just the first part, and talk about naturality of the discrete exterior covariant derivative and discrete wedge product using simple examples. Joint work with Daniel Berwick-Evans (UIUC) and Mark Schubel (Apple, Inc.).
11:10–12:00 pm
David Gu
Title: Surface Quadrilateral Meshing Based on Abel-Jacobi Theory
Abstract: Surface quadrilateral meshing plays an important role in many fields. For example, in CAD (computer-aided design), all shapes are represented as Spline surfaces, which requires structured quad-meshing; in CAE (computer-aided engineering), the surface tessellation greatly affects the accuracy and efficiency of numerical simulations. Although the research on mesh generation has a long history, it remains a great challenge to automatically generate structured quad-meshes with high qualities. The key is to find the governing equation for the singularities of the global structured quad-meshes.
In this talk, we introduce our recent discovery: the singularities of a quad-mesh are governed by the Abel theorem. We show that each quad-mesh determines a conformal structure and a meromorphic quadratic differential, the configuration of the mesh singularities can be described as the divisor of the differential. The quad-mesh divisor minus four times of the divisor of a holomorphic one-form is principal and satisfies the Abel theorem: its image under the Jacobi map is zero in the Jacobi variety.
This leads to a rigorous and efficient algorithm for surface structured quadrilateral meshing. After determining the singularities, the metric induced by the quad-mesh can be computed using the discrete Yambe flow, and the meromorphic quartic differential can be constructed, the trajectories of the differentials give the quad-mesh. The method can be applied directly for geometric modeling and computational mechanics.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Justin Solomon
Title: Geometry Processing with Volumes
Abstract: Many algorithms in geometry processing are restricted to two-dimensional surfaces represented as triangle meshes. Drawing inspiration from simulation, medical imaging, and other application domains, however, there is a substantial demand for geometry processing algorithms targeted to volumes represented as tetrahedral meshes or grids. In this talk, I will summarize some efforts in our group to develop a geometry processing toolkit specifically for volumes. Specifically, I will cover our recent work on hexahedral remeshing via cuboid decomposition, volumetric correspondence, and minimal surface computation via geometric measure theory.
3:00–3:20 pm
Oded Stein
Title: Optimization for flip-free parametrization
Abstract: Parametrizations without flipped elements are desirable in a variety of applications such as UV mapping and surface/volume correspondence. Computing flip-free parametrizations can be challenging, and there are many different approaches to the problem. In this talk we will look at multiple strategies for flip-free parametrizations that are based on the optimization of continuous energies. Due to the nature of the problem, these energies are often nonconvex and unbounded, which is a challenge for optimization methods. We will also take a closer look at our recently developed method for computing flip-free parametrizations using the Alternating Direction Method of Multipliers (ADMM).
3:20–4:00 pm
Break
4:00–4:50 pm
John Bowers
Title: Koebe-Andre’ev-Thurston Packings via Flow
Abstract: Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles.
The full Koebe-Andre’ev-Thurston theorem generalizes the circle packing theorem to allow for neighboring circles to overlap by angles up to $\pi/2$. In this talk I will show that the Connelly-Gortler method can be extended to allow for circles to overlap to angles up to $\pi/2$. This results in a new proof of the general Koebe-Andre’ev-Thurston theorem for disk patterns on $\mathbb{S}^2$ as well as a numerical algorithm for computing them. The proof involves generalizing a notion of convexity for circle polyhedra that was recently used to prove the global rigidity of certain circle packings, which is then used to show that all convex circle polyhedra are infinitesimally rigid, a result of independent interest.
5:00–5:30 pm
Movies
“conform!” & ”Koebe polyhedra”
Saturday, May 7, 2022
9:30–10:20 am
Alexander Bobenko
Title: The Bonnet problem: Is a surface characterized by its metric and curvatures?
Abstract: We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface is characterized by a metric and mean curvature function. Generically, the answer is yes. Special cases when it is not the case are classified. In particular, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas
10:20–11:00 am
Break
11:00–11:50 am
Miri Ben Chen
Title: Surface Multigrid via Intrinsic Prolongation
Abstract: The solution of a linear system is a required ingredient in many geometry processing applications, and multigrid methods are among the most efficient solution techniques. However, due to the unstructured nature of triangle meshes, mapping functions between different multigrid levels is challenging. In this talk I will present our recent work that uses an intrinsic prolongation operator as the main building block in a multigrid solver for curved triangle meshes. Our solver can be used as a black-box in any triangle-mesh based system that requires a linear solve, and leads to order of magnitude time-efficiency improvement compared to direct solvers.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Steven Gortler
Title: Reconstructing configurations and graphs from unlabeled distance measurements
Abstract: Place a configuration of n points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence)? This question is motivated by a number of inverse problem applications. In this talk, I will review what is known about this question.
3:00–3:20 pm
Bohan Zhou
Title: Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs
Abstract: Optimal transport has profound and wide applications since its introduction in 1781 by Monge. Thanks to the Benamou-Brenier formulation, it provides a meaningful functional in the image science like image and shape registrations. However, exact computation through LP or PDE is in general not practical in large scale, while the popular entropy-regularized method introduces additional diffusion noise, deteriorating shapes and boundaries. Until the recent work [Jacobs and Leger, A Fast Approach to Optimal Transport: the back-and-forth method, Numerische Mathematik, 2020], solving OT in a both accurate and fast fashion finally becomes possible. Multiple marginal optimal transport is a natural extension from OT but has its own interest and is in general more computationally expensive. The entropy method suffers from both diffusion noise and high dimensional computational issues. In this work with Matthew Parno, we extend from two marginals to multiple marginals, on a wide class of cost functions when those marginals have a graph structure. This new method is fast and does not introduce diffusion. As a result, the new proposed method can be used in many fields those require sharp boundaries. If time allows, we will illustrate by examples the faithful joint recover via MMOT of images with sharp boundaries, with applications on sea ice prediction.
3:20–4:00 pm
Break
4:00–4:50 pm
Peter Schroeder
Title: Constrained Willmore Surfaces
Abstract: The Willmore energy of a surface is a canonical example of a squared curvature bending energy. Its minimizers are therefore of interest both in the theory of surfaces and in practical applications from physical and geometric modeling. Minimizing the bending energy alone however is insufficient. Taking a cue from univariate splines which incorporate an isometry constraint we consider Willmore minimizers subject to a conformality constraint. In this talk I will report on a numerical algorithm to find such constrained minimizers for triangle meshes.
Joint work with Yousuf Soliman (Caltech), Olga Diamanti (UGraz), Albert Chern (UCSD), Felix Knöppel (TU Berlin), Ulrich Pinkall (TU Berlin).
5:00–5:50 pm
Problems and Application discussions
Sunday, May 8, 2022
9:00–9:50 am
Tianqi Wu
Title: Convergence of discrete uniformizations
Abstract: The theory of discrete conformality, based on the notion of vertex scaling, has been implemented in computing conformal maps or uniformizations of surfaces. We will show that if a Delaunay triangle mesh approximates a smooth surface, then the related discrete uniformization will converge to the smooth uniformization, with an error bounded linearly by the size of the triangles in the mesh.
10:10–11:00 am
Yanwen Luo
Title: Recent Progress in Spaces of Geodesic Triangulations of Surfaces Abstract: Spaces of geodesic triangulations of surfaces are natural discretization of the groups of surface diffeomorphisms isotopy to the identity. It has been conjectured that these spaces have the same homotopy type as their smooth counterparts. In this talk, we will report the recent progress in this problem. The key ingredient is the idea in Tutte’s embedding theorem. We will explain how to use it to identify the homotopy types of spaces of geodesic triangulations. This is joint work with Tianqi Wu and Xiaoping Zhu.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
On May 6–8, 2022, the CMSA hosted a second NSF FRG Workshop.
This project brings together a community of researchers who develop theoretical and computational models to characterize shapes. Their combined interests span Mathematics (Geometry and Topology), Computer Science (Scientific Computing and Complexity Theory), and domain sciences, from Data Sciences to Computational Biology.
Scientific research benefits from the development of an ever-growing number of sensors that are able to capture details of the world at increasingly fine resolutions. The seemingly unlimited breadth and depth of these sources provide the means to study complex systems in a more comprehensive way. At the same time, however, these sensors are generating a huge amount of data that comes with a high level of complexity and heterogeneity, providing indirect measurements of hidden processes that provide keys to the systems under study. This has led to new challenges and opportunities in data analysis. Our focus is on image data and the shapes they represent. Advances in geometry and topology have led to powerful new tools that can be applied to geometric methods for representing, searching, simulating, analyzing, and comparing shapes. These methods and tools can be applied in a wide range of fields, including computer vision, biological imaging, brain mapping, target recognition, and satellite image analysis.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Abstract: We have recently initiated a generalization of discrete exterior calculus to differential forms with values in a vector bundle. A discrete vector bundle with connection over a simplicial complex has fibers at vertices and transport maps on edges, just as in lattice gauge theory. The first part of this work involves defining and examining properties of a combinatorial exterior covariant derivative and wedge product. We find that these operators commute with pullback under simplicial maps of the base space. From these definitions emerges a combinatorial curvature. In the second part of this work we showed that the curvature behaves as one expects: it measures failure of parallel transport to be independent of the path, and is the local obstruction to a trivialization. For a bundle with metric, metric compatibility of the discrete connection is equivalent to a Leibniz rule. Vanishing curvature is indeed equivalent to an appropriately defined discrete flat connection, and curvature obstructs trivializability. In this talk I will focus on just the first part, and talk about naturality of the discrete exterior covariant derivative and discrete wedge product using simple examples. Joint work with Daniel Berwick-Evans (UIUC) and Mark Schubel (Apple, Inc.).
11:10–12:00 pm
David Gu
Title: Surface Quadrilateral Meshing Based on Abel-Jacobi Theory
Abstract: Surface quadrilateral meshing plays an important role in many fields. For example, in CAD (computer-aided design), all shapes are represented as Spline surfaces, which requires structured quad-meshing; in CAE (computer-aided engineering), the surface tessellation greatly affects the accuracy and efficiency of numerical simulations. Although the research on mesh generation has a long history, it remains a great challenge to automatically generate structured quad-meshes with high qualities. The key is to find the governing equation for the singularities of the global structured quad-meshes.
In this talk, we introduce our recent discovery: the singularities of a quad-mesh are governed by the Abel theorem. We show that each quad-mesh determines a conformal structure and a meromorphic quadratic differential, the configuration of the mesh singularities can be described as the divisor of the differential. The quad-mesh divisor minus four times of the divisor of a holomorphic one-form is principal and satisfies the Abel theorem: its image under the Jacobi map is zero in the Jacobi variety.
This leads to a rigorous and efficient algorithm for surface structured quadrilateral meshing. After determining the singularities, the metric induced by the quad-mesh can be computed using the discrete Yambe flow, and the meromorphic quartic differential can be constructed, the trajectories of the differentials give the quad-mesh. The method can be applied directly for geometric modeling and computational mechanics.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Justin Solomon
Title: Geometry Processing with Volumes
Abstract: Many algorithms in geometry processing are restricted to two-dimensional surfaces represented as triangle meshes. Drawing inspiration from simulation, medical imaging, and other application domains, however, there is a substantial demand for geometry processing algorithms targeted to volumes represented as tetrahedral meshes or grids. In this talk, I will summarize some efforts in our group to develop a geometry processing toolkit specifically for volumes. Specifically, I will cover our recent work on hexahedral remeshing via cuboid decomposition, volumetric correspondence, and minimal surface computation via geometric measure theory.
3:00–3:20 pm
Oded Stein
Title: Optimization for flip-free parametrization
Abstract: Parametrizations without flipped elements are desirable in a variety of applications such as UV mapping and surface/volume correspondence. Computing flip-free parametrizations can be challenging, and there are many different approaches to the problem. In this talk we will look at multiple strategies for flip-free parametrizations that are based on the optimization of continuous energies. Due to the nature of the problem, these energies are often nonconvex and unbounded, which is a challenge for optimization methods. We will also take a closer look at our recently developed method for computing flip-free parametrizations using the Alternating Direction Method of Multipliers (ADMM).
3:20–4:00 pm
Break
4:00–4:50 pm
John Bowers
Title: Koebe-Andre’ev-Thurston Packings via Flow
Abstract: Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles.
The full Koebe-Andre’ev-Thurston theorem generalizes the circle packing theorem to allow for neighboring circles to overlap by angles up to $\pi/2$. In this talk I will show that the Connelly-Gortler method can be extended to allow for circles to overlap to angles up to $\pi/2$. This results in a new proof of the general Koebe-Andre’ev-Thurston theorem for disk patterns on $\mathbb{S}^2$ as well as a numerical algorithm for computing them. The proof involves generalizing a notion of convexity for circle polyhedra that was recently used to prove the global rigidity of certain circle packings, which is then used to show that all convex circle polyhedra are infinitesimally rigid, a result of independent interest.
5:00–5:30 pm
Movies
“conform!” & ”Koebe polyhedra”
Saturday, May 7, 2022
9:30–10:20 am
Alexander Bobenko
Title: The Bonnet problem: Is a surface characterized by its metric and curvatures?
Abstract: We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface is characterized by a metric and mean curvature function. Generically, the answer is yes. Special cases when it is not the case are classified. In particular, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas
10:20–11:00 am
Break
11:00–11:50 am
Miri Ben Chen
Title: Surface Multigrid via Intrinsic Prolongation
Abstract: The solution of a linear system is a required ingredient in many geometry processing applications, and multigrid methods are among the most efficient solution techniques. However, due to the unstructured nature of triangle meshes, mapping functions between different multigrid levels is challenging. In this talk I will present our recent work that uses an intrinsic prolongation operator as the main building block in a multigrid solver for curved triangle meshes. Our solver can be used as a black-box in any triangle-mesh based system that requires a linear solve, and leads to order of magnitude time-efficiency improvement compared to direct solvers.
12:00–2:00 pm
Lunch Break
2:00–2:50 pm
Steven Gortler
Title: Reconstructing configurations and graphs from unlabeled distance measurements
Abstract: Place a configuration of n points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence)? This question is motivated by a number of inverse problem applications. In this talk, I will review what is known about this question.
3:00–3:20 pm
Bohan Zhou
Title: Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs
Abstract: Optimal transport has profound and wide applications since its introduction in 1781 by Monge. Thanks to the Benamou-Brenier formulation, it provides a meaningful functional in the image science like image and shape registrations. However, exact computation through LP or PDE is in general not practical in large scale, while the popular entropy-regularized method introduces additional diffusion noise, deteriorating shapes and boundaries. Until the recent work [Jacobs and Leger, A Fast Approach to Optimal Transport: the back-and-forth method, Numerische Mathematik, 2020], solving OT in a both accurate and fast fashion finally becomes possible. Multiple marginal optimal transport is a natural extension from OT but has its own interest and is in general more computationally expensive. The entropy method suffers from both diffusion noise and high dimensional computational issues. In this work with Matthew Parno, we extend from two marginals to multiple marginals, on a wide class of cost functions when those marginals have a graph structure. This new method is fast and does not introduce diffusion. As a result, the new proposed method can be used in many fields those require sharp boundaries. If time allows, we will illustrate by examples the faithful joint recover via MMOT of images with sharp boundaries, with applications on sea ice prediction.
3:20–4:00 pm
Break
4:00–4:50 pm
Peter Schroeder
Title: Constrained Willmore Surfaces
Abstract: The Willmore energy of a surface is a canonical example of a squared curvature bending energy. Its minimizers are therefore of interest both in the theory of surfaces and in practical applications from physical and geometric modeling. Minimizing the bending energy alone however is insufficient. Taking a cue from univariate splines which incorporate an isometry constraint we consider Willmore minimizers subject to a conformality constraint. In this talk I will report on a numerical algorithm to find such constrained minimizers for triangle meshes.
Joint work with Yousuf Soliman (Caltech), Olga Diamanti (UGraz), Albert Chern (UCSD), Felix Knöppel (TU Berlin), Ulrich Pinkall (TU Berlin).
5:00–5:50 pm
Problems and Application discussions
Sunday, May 8, 2022
9:00–9:50 am
Tianqi Wu
Title: Convergence of discrete uniformizations
Abstract: The theory of discrete conformality, based on the notion of vertex scaling, has been implemented in computing conformal maps or uniformizations of surfaces. We will show that if a Delaunay triangle mesh approximates a smooth surface, then the related discrete uniformization will converge to the smooth uniformization, with an error bounded linearly by the size of the triangles in the mesh.
10:10–11:00 am
Yanwen Luo
Title: Recent Progress in Spaces of Geodesic Triangulations of Surfaces Abstract: Spaces of geodesic triangulations of surfaces are natural discretization of the groups of surface diffeomorphisms isotopy to the identity. It has been conjectured that these spaces have the same homotopy type as their smooth counterparts. In this talk, we will report the recent progress in this problem. The key ingredient is the idea in Tutte’s embedding theorem. We will explain how to use it to identify the homotopy types of spaces of geodesic triangulations. This is joint work with Tianqi Wu and Xiaoping Zhu.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Title:Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Title:Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Title:Singularities along the Lagrangian mean curvature flow.
Abstract:We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Title: Symplectic deformations and the Type IIA flow
Abstract:The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Title:Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Title:Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video
10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: I will explore the interplay between Swampland conjectures and models of inflation and light Dark Matter. To that end, I will briefly review the weak gravity conjecture (WGC) and the related Festina Lente (FL) bound. These have implications for light darkly and milli-charged particles and can disfavor a large portion of parameter space. The FL bound also implies strong restrictions on the field content of our universe during inflation and presents an opportunity for inflationary model building. At the same time, it rules out some popular models like chromo-natural inflation and gauge-flation. Finally, I will review another Swampland conjecture related to Stückelberg photon masses and discuss its implications for astro-particle physics.
On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Title:Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Title:Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Title:Singularities along the Lagrangian mean curvature flow.
Abstract:We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Title: Symplectic deformations and the Type IIA flow
Abstract:The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Title:Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Title:Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video
10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: I will explore the interplay between Swampland conjectures and models of inflation and light Dark Matter. To that end, I will briefly review the weak gravity conjecture (WGC) and the related Festina Lente (FL) bound. These have implications for light darkly and milli-charged particles and can disfavor a large portion of parameter space. The FL bound also implies strong restrictions on the field content of our universe during inflation and presents an opportunity for inflationary model building. At the same time, it rules out some popular models like chromo-natural inflation and gauge-flation. Finally, I will review another Swampland conjecture related to Stückelberg photon masses and discuss its implications for astro-particle physics.
On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Title:Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Title:Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Title:Singularities along the Lagrangian mean curvature flow.
Abstract:We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Title: Symplectic deformations and the Type IIA flow
Abstract:The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Title:Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Title:Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video
10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Title:Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Title:Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Title:Singularities along the Lagrangian mean curvature flow.
Abstract:We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Title: Symplectic deformations and the Type IIA flow
Abstract:The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Title:Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Title:Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video
10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Title:Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Title:Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Title:Singularities along the Lagrangian mean curvature flow.
Abstract:We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Title: Symplectic deformations and the Type IIA flow
Abstract:The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Title:Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Title:Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video
10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: We are familiar with the idea that quantum gravity in AdS can holographically emerge from complex patterns of entanglement, but can the physics of big bang cosmology emerge from a quantum many-body system? In this talk I will argue that standard tools of holography can be used to describe fully non-perturbative microscopic models of cosmology in which a period of accelerated expansion may result from the positive potential energy of time-dependent scalar fields evolving towards a region with negative potential. In these models, the fundamental cosmological constant is negative, and the universe eventually recollapses in a time-reversal symmetric way. The microscopic description naturally selects a special state for the cosmology. In this framework, physics in the cosmological spacetime is dual to the vacuum physics in a static planar asymptotically AdS Lorentzian wormhole spacetime, in the sense that the background spacetimes and observables are related by analytic continuation. The dual spacetime is weakly curved everywhere, so any cosmological observables can be computed in the dual picture via effective field theory without detailed knowledge of the UV completion or the physics near the big bang. Based on 2203.11220 with S. Antonini, P. Simidzija, and M. Van Raamsdonk.
On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Title:Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Title:Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Title:Singularities along the Lagrangian mean curvature flow.
Abstract:We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Title: Symplectic deformations and the Type IIA flow
Abstract:The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Title:Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Title:Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video
10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: We are familiar with the idea that quantum gravity in AdS can holographically emerge from complex patterns of entanglement, but can the physics of big bang cosmology emerge from a quantum many-body system? In this talk I will argue that standard tools of holography can be used to describe fully non-perturbative microscopic models of cosmology in which a period of accelerated expansion may result from the positive potential energy of time-dependent scalar fields evolving towards a region with negative potential. In these models, the fundamental cosmological constant is negative, and the universe eventually recollapses in a time-reversal symmetric way. The microscopic description naturally selects a special state for the cosmology. In this framework, physics in the cosmological spacetime is dual to the vacuum physics in a static planar asymptotically AdS Lorentzian wormhole spacetime, in the sense that the background spacetimes and observables are related by analytic continuation. The dual spacetime is weakly curved everywhere, so any cosmological observables can be computed in the dual picture via effective field theory without detailed knowledge of the UV completion or the physics near the big bang. Based on 2203.11220 with S. Antonini, P. Simidzija, and M. Van Raamsdonk.
On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Title:Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Title:Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Title:Singularities along the Lagrangian mean curvature flow.
Abstract:We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Title: Symplectic deformations and the Type IIA flow
Abstract:The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Title:Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Title:Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video
10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: The CDF collaboration recently reported a new precise measurement of the W boson mass MW with a central value significantly larger than the SM prediction. We explore the effects of including this new measurement on a fit of the Standard Model (SM) to electroweak precision data. We characterize the tension of this new measurement with the SM and explore potential beyond the SM phenomena within the electroweak sector in terms of the oblique parameters S, T and U. We show that the large MW value can be accommodated in the fit by a large, nonzero value of U, which is difficult to construct in explicit models. Assuming U = 0, the electroweak fit strongly prefers large, positive values of T. Finally, we study how the preferred values of the oblique parameters may be generated in the context of models affecting the electroweak sector at tree- and loop-level. In particular, we demonstrate that the preferred values of T and S can be generated with a real SU(2)L triplet scalar, the humble swino, which can be heavy enough to evade current collider constraints, or by (multiple) species of a singlet-doublet fermion pair. We highlight challenges in constructing other simple models, such as a dark photon, for explaining a large MW value, and several directions for further study.
Abstract: Many statistical and computational tasks boil down to comparing probability measures expressed as density functions, clouds of data points, or generative models. In this setting, we often are unable to match individual data points but rather need to deduce relationships between entire weighted and unweighted point sets. In this talk, I will summarize our team’s recent efforts to apply geometric techniques to problems in this space, using tools from optimal transport and spectral geometry. Motivated by applications in dataset comparison, time series analysis, and robust learning, our work reveals how to apply geometric reasoning to data expressed as probability measures without sacrificing computational efficiency.
On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Title:Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Title:Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Title:Singularities along the Lagrangian mean curvature flow.
Abstract:We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Title: Symplectic deformations and the Type IIA flow
Abstract:The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Title:Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Title:Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video
10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: The CDF collaboration recently reported a new precise measurement of the W boson mass MW with a central value significantly larger than the SM prediction. We explore the effects of including this new measurement on a fit of the Standard Model (SM) to electroweak precision data. We characterize the tension of this new measurement with the SM and explore potential beyond the SM phenomena within the electroweak sector in terms of the oblique parameters S, T and U. We show that the large MW value can be accommodated in the fit by a large, nonzero value of U, which is difficult to construct in explicit models. Assuming U = 0, the electroweak fit strongly prefers large, positive values of T. Finally, we study how the preferred values of the oblique parameters may be generated in the context of models affecting the electroweak sector at tree- and loop-level. In particular, we demonstrate that the preferred values of T and S can be generated with a real SU(2)L triplet scalar, the humble swino, which can be heavy enough to evade current collider constraints, or by (multiple) species of a singlet-doublet fermion pair. We highlight challenges in constructing other simple models, such as a dark photon, for explaining a large MW value, and several directions for further study.
Abstract: Many statistical and computational tasks boil down to comparing probability measures expressed as density functions, clouds of data points, or generative models. In this setting, we often are unable to match individual data points but rather need to deduce relationships between entire weighted and unweighted point sets. In this talk, I will summarize our team’s recent efforts to apply geometric techniques to problems in this space, using tools from optimal transport and spectral geometry. Motivated by applications in dataset comparison, time series analysis, and robust learning, our work reveals how to apply geometric reasoning to data expressed as probability measures without sacrificing computational efficiency.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Title: Cobordism and Deformation Class of the Standard Model and Beyond: Proton Stability and Neutrino Mass
Abstract: ‘t Hooft anomalies of quantum field theories (QFTs) with an invertible global symmetry G (including spacetime and internal symmetries) in a d-dim spacetime are known to be classified by a d+1-dim cobordism group TPd+1(G), whose group generator is a d+1-dim cobordism invariant written as a d+1-dim invertible topological field theory. Deformation class of QFT is recently proposed to be specified by its symmetry G and a d+1-dim invertible topological field theory. Seemly different QFTs of the same deformation class can be deformed to each other via quantum phase transitions. We ask which deformation class controls the 4d ungauged or gauged (SU(3)×SU(2)×U(1))/Zq Standard Model (SM) for q=1,2,3,6 with a continuous or discrete (B−L) symmetry and with also a compatible discrete baryon plus lepton Z_{2Nf} B+L symmetry. (The Z_{2Nf} B+L is discrete due to the ABJ anomaly under the BPST instanton.) We explore a systematic classification of candidate perturbative local and nonperturbative global anomalies of the 4d SM, including all these gauge and gravitational backgrounds, via a cobordism theory, which controls the SM’s deformation class. While many Grand Unified Theories violating the discrete B+L symmetry suffer from the proton decay, the SM and some versions of Ultra Unification (constrained by Z_{16} class global anomaly that replaces sterile neutrinos with new exotic gapped/gapless topological or conformal sectors) can have a stable proton. Dictated by a Z_2 class global mixed gauge-gravitational anomaly, there can be a gapless deconfined quantum critical region between Georgi-Glashow and Pati-Salam models — the Standard Model and beyond occur as neighbor phases. We will also comment on a new mechanism to give the neutrino mass via topological field theories and topological defects. Work based on arXiv:2112.14765, arXiv:2204.08393, arXiv:2202.13498 and references therein.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Title: Cobordism and Deformation Class of the Standard Model and Beyond: Proton Stability and Neutrino Mass
Abstract: ‘t Hooft anomalies of quantum field theories (QFTs) with an invertible global symmetry G (including spacetime and internal symmetries) in a d-dim spacetime are known to be classified by a d+1-dim cobordism group TPd+1(G), whose group generator is a d+1-dim cobordism invariant written as a d+1-dim invertible topological field theory. Deformation class of QFT is recently proposed to be specified by its symmetry G and a d+1-dim invertible topological field theory. Seemly different QFTs of the same deformation class can be deformed to each other via quantum phase transitions. We ask which deformation class controls the 4d ungauged or gauged (SU(3)×SU(2)×U(1))/Zq Standard Model (SM) for q=1,2,3,6 with a continuous or discrete (B−L) symmetry and with also a compatible discrete baryon plus lepton Z_{2Nf} B+L symmetry. (The Z_{2Nf} B+L is discrete due to the ABJ anomaly under the BPST instanton.) We explore a systematic classification of candidate perturbative local and nonperturbative global anomalies of the 4d SM, including all these gauge and gravitational backgrounds, via a cobordism theory, which controls the SM’s deformation class. While many Grand Unified Theories violating the discrete B+L symmetry suffer from the proton decay, the SM and some versions of Ultra Unification (constrained by Z_{16} class global anomaly that replaces sterile neutrinos with new exotic gapped/gapless topological or conformal sectors) can have a stable proton. Dictated by a Z_2 class global mixed gauge-gravitational anomaly, there can be a gapless deconfined quantum critical region between Georgi-Glashow and Pati-Salam models — the Standard Model and beyond occur as neighbor phases. We will also comment on a new mechanism to give the neutrino mass via topological field theories and topological defects. Work based on arXiv:2112.14765, arXiv:2204.08393, arXiv:2202.13498 and references therein.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
SMaSH: Symposium for Mathematical Sciences at Harvard
On Tuesday, May 17, 2022, from 9:00 am – 5:30 pm, the Harvard John A Paulson School of Engineering and Applied Sciences (SEAS) and the Harvard Center of Mathematical Sciences and Applications (CMSA) held a Symposium for Mathematical Sciences for the mathematical sciences community at Harvard.
Organizing Committee
Michael Brenner, Applied Mathematics (SEAS)
Michael Desai, Organismic and Evolutionary Biology (FAS)
Sam Gershman, Psychology (FAS)
Michael Hopkins, Mathematics (FAS)
Gary King, Government (FAS)
Peter Koellner, Philosophy (FAS)
Scott Kominers, Economics (FAS) & Entrepreneurial Management (HBS)
Coffee and Breakfast West Atrium (ground floor of the SEC)
9:30–10:30 am
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumKosuke Imai, Government & Statistics (FAS): Use of Simulation Algorithms for Legislative Redistricting Analysis and EvaluationYannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS): The Sample Complexity of Up-to-ε Multi-Dimensional Revenue Maximization
10:30–11:00 am
Coffee Break West Atrium (ground floor of the SEC)
11:00–12:00 pm
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumSeth Neel, Technology & Operations Management (HBS): “Machine (Un)Learning” or Why Your Deployed Model Might Violate Existing Privacy LawDemba Ba, Electrical Engineering & Bioengineering (SEAS): Geometry, AI, and the Brain
12:00–1:00 pm
Lunch Break Engineering Yard Tent
1:00–2:30 pm
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumMelanie Matchett Wood, Mathematics (FAS): Understanding distributions of algebraic structures through their momentsMorgane Austern, Statistics (FAS): Limit theorems for structured random objectsAnurag Anshu, Computer Science (SEAS): Operator-valued polynomial approximations and their use.
2:30–3:00 pm
Coffee Break West Atrium (ground floor of the SEC)
3:00–4:30 pm
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumMichael Brenner, Applied Mathematics (SEAS): Towards living synthetic materialsRui Duan, Biostatistics (HSPH): Federated and transfer learning for healthcare data integrationSham M. Kakade, Computer Science (SEAS) & Statistics (FAS): What is the Statistical Complexity of Reinforcement Learning?
4:30–5:30 pm
Reception with Jazz musicians & Poster Session Engineering Yard Tent
Faculty Talks
Speaker
Title / Abstract / Bio
Anurag Anshu, Computer Science (SEAS)
Title: Operator-valued polynomial approximations and their use.
Abstract: Approximation of complicated functions with low degree polynomials is an indispensable tool in mathematics. This becomes particularly relevant in computer science, where the complexity of interesting functions is often captured by the degree of the approximating polynomials. This talk concerns the approximation of operator-valued functions (such as the exponential of a hermitian matrix, or the intersection of two projectors) with low-degree operator-valued polynomials. We will highlight the challenges that arise in achieving similarly good approximations as real-valued functions, as well as recent methods to overcome them. We will discuss applications to the ground states in physics and quantum complexity theory: correlation lengths, area laws and concentration bounds.
Bio: Anurag Anshu is an Assistant Professor of computer science at Harvard University. He spends a lot of time exploring the rich structure of quantum many-body systems from the viewpoint of quantum complexity theory, quantum learning theory and quantum information theory. He held postdoctoral positions at University of California, Berkeley and University of Waterloo and received his PhD from National University of Singapore, focusing on quantum communication complexity.
Morgane Austern, Statistics (FAS)
Title: Limit theorems for structured random objects
Abstract: Statistical inference relies on numerous tools from probability theory to study the properties of estimators. Some of the most central ones are the central limit theorem and the free central limit theorem. However, these same tools are often inadequate to study modern machine problems that frequently involve structured data (e.g networks) or complicated dependence structures (e.g dependent random matrices). In this talk, we extend universal limit theorems beyond the classical setting. We consider distributionally “structured’ and dependent random object i.e random objects whose distribution is invariant under the action of an amenable group. We show, under mild moment and mixing conditions, a series of universal second and third order limit theorems: central-limit theorems, concentration inequalities, Wigner semi-circular law and Berry-Esseen bounds. The utility of these will be illustrated by a series of examples in machine learning, network and information theory.
Bio: Morgane Austern is an assistant professor in the Statistics Department of Harvard University. Broadly, she is interested in developing probability tools for modern machine learning and in establishing the properties of learning algorithms in structured and dependent data contexts. She graduated with a PhD in statistics from Columbia University in 2019 where she worked in collaboration with Peter Orbanz and Arian Maleki on limit theorems for dependent and structured data. She was a postdoctoral researcher at Microsoft Research New England from 2019 to 2021.
Abstract: A large body of experiments suggests that neural computations reflect, in some sense, the geometry of “the world”. How do artificial and neural systems learn representations of “the world” that reflect its geometry? How, for instance, do we, as humans, learn representations of objects, e.g. fruits, that reflect the geometry of object space? Developing artificial systems that can capture/understand the geometry of the data they process may enable them to learn representations useful in many different contexts and tasks. My talk will describe an artificial neural-network architecture that, starting from a simple union-of-manifold model of data comprising objects from different categories, mimics some aspects of how primates learn, organize, and retrieve concepts, in a manner that respects the geometry of object space.
Bio: Demba Ba serves as an Associate Professor of electrical engineering and bioengineering in Harvard University’s School of Engineering and Applied Sciences, where he directs the CRISP group. Recently, he has taken a keen interest in the connection between artificial neural networks and sparse signal processing. His group leverages this connection to solve data-driven unsupervised learning problems in neuroscience, to understand the principles of hierarchical representations of sensory signals in the brain, and to develop explainable AI. In 2016, he received a Research Fellowship in Neuroscience from the Alfred P. Sloan Foundation. In 2021, Harvard’s Faculty of Arts and Sciences awarded him the Roslyn Abramson award for outstanding undergraduate teaching.
Michael Brenner, Applied Mathematics (SEAS)
Title: Towards living synthetic materials
Abstract: Biological materials are much more complicated and functional than synthetic ones. Over the past several years we have been trying to figure out why. A sensible hypothesis is that biological materials are programmable. But we are very far from being able to program materials we create with this level of sophistication. I will discuss our (largely unsuccessful) efforts to bridge this gap, though as of today I’m somewhat optimistic that we are arriving at a set of theoretical models that is rich enough to produce relevant emergent behavior.
Bio: I’ve been at Harvard for a long time. My favorite part of Harvard is the students.
Rui Duan, Biostatistics (HSPH)
Title: Federated and transfer learning for healthcare data integration
Abstract: The growth of availability and variety of healthcare data sources has provided unique opportunities for data integration and evidence synthesis, which can potentially accelerate knowledge discovery and improve clinical decision-making. However, many practical and technical challenges, such as data privacy, high dimensionality, and heterogeneity across different datasets, remain to be addressed. In this talk, I will introduce several methods for the effective and efficient integration of multiple healthcare datasets in order to train statistical or machine learning models with improved generalizability and transferability. Specifically, we develop communication-efficient federated learning algorithms for jointly analyzing multiple datasets without the need of sharing patient-level data, as well as transfer learning approaches that leverage shared knowledge learned across multiple datasets to improve the performance of statistical models in target populations of interest. We will discuss both the theoretical properties and examples of implementation of our methods in real-world research networks and data consortia.
Bio: Rui Duan is an Assistant Professor of Biostatistics at the Harvard T.H. Chan School of Public Health. She received her Ph.D. in Biostatistics in May 2020 from the University of Pennsylvania. Her research interests focus on developing statistical, machine learning, and informatics tools for (1) efficient data integration in biomedical research, (2) understanding and accounting for the heterogeneity of biomedical data, and improving the generalizability and transferability of models across populations (3) advancing precision medicine research on rare diseases and underrepresented populations.
Yannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS)
Title: The Sample Complexity of Up-to-ε Multi-Dimensional Revenue Maximization
Abstract: We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of n additive bidders whose values for m heterogeneous items are drawn independently. For any such instance and any ε > 0, we show that it is possible to learn an ε-Bayesian Incentive Compatible auction whose expected revenue is within ε of the optimal ε-BIC auction from only polynomially many samples.
Our fully nonparametric approach is based on ideas that hold quite generally, and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive, and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well-understood, our corollary for this case extends slightly the state-of-the-art.
Joint work with S. Matthew Weinberg
Bio: Yannai A. Gonczarowski is an Assistant Professor of Economics and of Computer Science at Harvard University—the first faculty member at Harvard to have been appointed to both of these departments. Interested in both economic theory and theoretical computer science, Yannai explores computer-science-inspired economics: he harnesses approaches, aesthetics, and techniques traditionally originating in computer science to derive economically meaningful insights. Yannai received his PhD from the Departments of Math and CS, and the Center for the Study of Rationality, at the Hebrew University of Jerusalem, where he was advised by Sergiu Hart and Noam Nisan. Yannai is also a professionally-trained opera singer, having acquired a bachelor’s degree and a master’s degree in Classical Singing at the Jerusalem Academy of Music and Dance. Yannai’s doctoral dissertation was recognized with several awards, including the 2018 Michael B. Maschler Prize of the Israeli Chapter of the Game Theory Society, and the ACM SIGecom Doctoral Dissertation Award for 2018. For the design and implementation of the National Matching System for Gap-Year Programs in Israel, he was awarded the Best Paper Award at MATCH-UP’19 and the inaugural INFORMS AMD Michael H. Rothkopf Junior Researcher Paper Prize (first place) for 2020. Yannai is also the recipient of the inaugural ACM SIGecom Award for Best Presentation by a Student or Postdoctoral Researcher at EC’18. His first textbook, “Mathematical Logic through Python” (Gonczarowski and Nisan), which introduces a new approach to teaching the material of a basic Logic course to Computer Science students, tailored to the unique intuitions and strengths of this cohort of students, is forthcoming in Cambridge University Press.
Kosuke Imai, Government & Statistics (FAS)
Title: Use of Simulation Algorithms for Legislative Redistricting Analysis and Evaluation
Abstract: After the 2020 Census, many states have been redrawing the boundaries of Congressional and state legislative districts. To evaluate the partisan and racial bias of redistricting plans, scholars have developed Monte Carlo simulation algorithms. The idea is to generate a representative sample of redistricting plans under a specified set of criteria and conduct a statistical hypothesis test by comparing a proposed plan with these simulated plans. I will give a brief overview of these redistricting simulation algorithms and discuss how they are used in real-world court cases.
Bio: Kosuke Imai is Professor in the Department of Government and Department of Statistics at Harvard University. Before moving to Harvard in 2018, Imai taught at Princeton University for 15 years where he was the founding director of the Program in Statistics and Machine Learning. Imai specializes in the development of statistical methods and machine learning algorithms and their applications to social science research. His areas of expertise include causal inference, computational social science, program evaluation, and survey methodology.
Sham M. Kakade, Computer Science (SEAS) & Statistics (FAS)
Title: What is the Statistical Complexity of Reinforcement Learning?
Abstract: This talk will highlight much of the recent progress on the following fundamental question in the theory of reinforcement learning: what (representational or structural) conditions govern our ability to generalize and avoid the curse of dimensionality? With regards to supervised learning, these questions are reasonably well understood, both practically and theoretically: practically, we have overwhelming evidence on the value of representational learning (say through modern deep networks) as a means for sample efficient learning, and, theoretically, there are well-known complexity measures (e.g. the VC dimension and Rademacher complexity) that govern the statistical complexity of learning. Providing an analogous theory for reinforcement learning is far more challenging, where even characterizing structural conditions which support sample efficient generalization has been far less well understood, until recently.
This talk will survey recent advances towards characterizing when generalization is possible in RL, focusing on both necessary and sufficient conditions. In particular, we will introduce a new complexity measure, the Decision-Estimation Coefficient, that is proven to be necessary (and, essentially, sufficient) for sample-efficient interactive learning.
Bio: Sham Kakade is a professor at Harvard University and a co-director of the Kempner Institute for the Study of Artificial and Natural Intelligence. He works on the mathematical foundations of machine learning and AI. Sham’s thesis helped lay the statistical foundations of reinforcement learning. With his collaborators, his additional contributions include foundational results on: policy gradient methods in reinforcement learning; regret bounds for linear bandit and Gaussian process bandit models; the tensor and spectral methods for latent variable models; and a number of convergence analyses for convex and non-convex algorithms. He is the recipient of the ICML Test of Time Award, the IBM Pat Goldberg best paper award, and INFORMS Revenue Management and Pricing Prize. He has been program chair for COLT 2011.
Sham was an undergraduate at Caltech, where he studied physics and worked under the guidance of John Preskill in quantum computing. He completed his Ph.D. with Peter Dayan in computational neuroscience at the Gatsby Computational Neuroscience Unit. He was a postdoc with Michael Kearns at the University of Pennsylvania.
Title: “Machine (Un)Learning” or Why Your Deployed Model Might Violate Existing Privacy Law
Abstract: Businesses like Facebook and Google depend on training sophisticated models on user data. Increasingly—in part because of regulations like the European Union’s General Data Protection Act and the California Consumer Privacy Act—these organizations are receiving requests to delete the data of particular users. But what should that mean? It is straightforward to delete a customer’s data from a database and stop using it to train future models. But what about models that have already been trained using an individual’s data? These are not necessarily safe; it is known that individual training data can be exfiltrated from models trained in standard ways via model inversion attacks. In a series of papers we help formalize a rigorous notion of data-deletion and propose algorithms to efficiently delete user data from trained models with provable guarantees in both convex and non-convex settings.
Bio: Seth Neel is a first-year Assistant Professor in the TOM Unit at Harvard Business School, and Co-PI of the SAFR ML Lab in the D3 Institute, which develops methodology to incorporate privacy and fairness guarantees into techniques for machine learning and data analysis, while balancing other critical considerations like accuracy, efficiency, and interpretability. He obtained his Ph.D. from the University of Pennsylvania in 2020 where he was an NSF graduate fellow. His work has focused primarily on differential privacy, notions of fairness in a variety of machine learning settings, and adaptive data analysis.
Melanie Matchett Wood, Mathematics (FAS)
Title: Understanding distributions of algebraic structures through their moments
Abstract: A classical tool of probability and analysis is to use the moments (mean, variance, etc.) of a distribution to recognize an unknown distribution of real numbers. In recent work, we are interested in distributions of algebraic structures that can’t be captured in a single number. We will explain one example, the fundamental group, that captures something about the shapes of possibly complicated or high dimensional spaces. We are developing a new theory of the moment problem for random algebraic structures which helps to to identify distributions of such, such as fundamental groups of random three dimensional spaces. This talk is based partly on joint work with Will Sawin.
Bio: Melanie Matchett Wood is a professor of mathematics at Harvard University and a Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study. Her work spans number theory, algebraic geometry, algebraic topology, additive combinatorics, and probability. Wood has been awarded a CAREER grant, a Sloan Research Fellowship, a Packard Fellowship for Science and Engineering, and the AWM-Microsoft Research Prize in Algebra and Number Theory, and she is a Fellow of the American Mathematical Society. In 2021, Wood received the National Science Foundation’s Alan T. Waterman Award, the nation’s highest honor for early-career scientists and engineers.
Abstract: In 1973, Erdős conjectured the existence of high girth (n,3,2)-Steiner systems. Recently, Glock, Kühn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős’ conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erdős’ conjecture. As for Steiner systems with more general parameters, Glock, Kühn, Lo, and Osthus conjectured the existence of high girth (n,q,r)-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of submatchings (which we view as a hypergraph H where V(H)=E(G)). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Komlós, Pintz, Spencer, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when H is a hypergraph with small codegrees (for which high girth designs is a specific instance). Indeed, the coloring version of our result even yields an almost partition of K_n^r into approximate high girth (n,q,r)-Steiner systems. If time permits, I will explain some of the other applications of our main results such as to rainbow matchings. This is joint work with Luke Postle.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
SMaSH: Symposium for Mathematical Sciences at Harvard
On Tuesday, May 17, 2022, from 9:00 am – 5:30 pm, the Harvard John A Paulson School of Engineering and Applied Sciences (SEAS) and the Harvard Center of Mathematical Sciences and Applications (CMSA) held a Symposium for Mathematical Sciences for the mathematical sciences community at Harvard.
Organizing Committee
Michael Brenner, Applied Mathematics (SEAS)
Michael Desai, Organismic and Evolutionary Biology (FAS)
Sam Gershman, Psychology (FAS)
Michael Hopkins, Mathematics (FAS)
Gary King, Government (FAS)
Peter Koellner, Philosophy (FAS)
Scott Kominers, Economics (FAS) & Entrepreneurial Management (HBS)
Coffee and Breakfast West Atrium (ground floor of the SEC)
9:30–10:30 am
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumKosuke Imai, Government & Statistics (FAS): Use of Simulation Algorithms for Legislative Redistricting Analysis and EvaluationYannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS): The Sample Complexity of Up-to-ε Multi-Dimensional Revenue Maximization
10:30–11:00 am
Coffee Break West Atrium (ground floor of the SEC)
11:00–12:00 pm
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumSeth Neel, Technology & Operations Management (HBS): “Machine (Un)Learning” or Why Your Deployed Model Might Violate Existing Privacy LawDemba Ba, Electrical Engineering & Bioengineering (SEAS): Geometry, AI, and the Brain
12:00–1:00 pm
Lunch Break Engineering Yard Tent
1:00–2:30 pm
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumMelanie Matchett Wood, Mathematics (FAS): Understanding distributions of algebraic structures through their momentsMorgane Austern, Statistics (FAS): Limit theorems for structured random objectsAnurag Anshu, Computer Science (SEAS): Operator-valued polynomial approximations and their use.
2:30–3:00 pm
Coffee Break West Atrium (ground floor of the SEC)
3:00–4:30 pm
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumMichael Brenner, Applied Mathematics (SEAS): Towards living synthetic materialsRui Duan, Biostatistics (HSPH): Federated and transfer learning for healthcare data integrationSham M. Kakade, Computer Science (SEAS) & Statistics (FAS): What is the Statistical Complexity of Reinforcement Learning?
4:30–5:30 pm
Reception with Jazz musicians & Poster Session Engineering Yard Tent
Faculty Talks
Speaker
Title / Abstract / Bio
Anurag Anshu, Computer Science (SEAS)
Title: Operator-valued polynomial approximations and their use.
Abstract: Approximation of complicated functions with low degree polynomials is an indispensable tool in mathematics. This becomes particularly relevant in computer science, where the complexity of interesting functions is often captured by the degree of the approximating polynomials. This talk concerns the approximation of operator-valued functions (such as the exponential of a hermitian matrix, or the intersection of two projectors) with low-degree operator-valued polynomials. We will highlight the challenges that arise in achieving similarly good approximations as real-valued functions, as well as recent methods to overcome them. We will discuss applications to the ground states in physics and quantum complexity theory: correlation lengths, area laws and concentration bounds.
Bio: Anurag Anshu is an Assistant Professor of computer science at Harvard University. He spends a lot of time exploring the rich structure of quantum many-body systems from the viewpoint of quantum complexity theory, quantum learning theory and quantum information theory. He held postdoctoral positions at University of California, Berkeley and University of Waterloo and received his PhD from National University of Singapore, focusing on quantum communication complexity.
Morgane Austern, Statistics (FAS)
Title: Limit theorems for structured random objects
Abstract: Statistical inference relies on numerous tools from probability theory to study the properties of estimators. Some of the most central ones are the central limit theorem and the free central limit theorem. However, these same tools are often inadequate to study modern machine problems that frequently involve structured data (e.g networks) or complicated dependence structures (e.g dependent random matrices). In this talk, we extend universal limit theorems beyond the classical setting. We consider distributionally “structured’ and dependent random object i.e random objects whose distribution is invariant under the action of an amenable group. We show, under mild moment and mixing conditions, a series of universal second and third order limit theorems: central-limit theorems, concentration inequalities, Wigner semi-circular law and Berry-Esseen bounds. The utility of these will be illustrated by a series of examples in machine learning, network and information theory.
Bio: Morgane Austern is an assistant professor in the Statistics Department of Harvard University. Broadly, she is interested in developing probability tools for modern machine learning and in establishing the properties of learning algorithms in structured and dependent data contexts. She graduated with a PhD in statistics from Columbia University in 2019 where she worked in collaboration with Peter Orbanz and Arian Maleki on limit theorems for dependent and structured data. She was a postdoctoral researcher at Microsoft Research New England from 2019 to 2021.
Abstract: A large body of experiments suggests that neural computations reflect, in some sense, the geometry of “the world”. How do artificial and neural systems learn representations of “the world” that reflect its geometry? How, for instance, do we, as humans, learn representations of objects, e.g. fruits, that reflect the geometry of object space? Developing artificial systems that can capture/understand the geometry of the data they process may enable them to learn representations useful in many different contexts and tasks. My talk will describe an artificial neural-network architecture that, starting from a simple union-of-manifold model of data comprising objects from different categories, mimics some aspects of how primates learn, organize, and retrieve concepts, in a manner that respects the geometry of object space.
Bio: Demba Ba serves as an Associate Professor of electrical engineering and bioengineering in Harvard University’s School of Engineering and Applied Sciences, where he directs the CRISP group. Recently, he has taken a keen interest in the connection between artificial neural networks and sparse signal processing. His group leverages this connection to solve data-driven unsupervised learning problems in neuroscience, to understand the principles of hierarchical representations of sensory signals in the brain, and to develop explainable AI. In 2016, he received a Research Fellowship in Neuroscience from the Alfred P. Sloan Foundation. In 2021, Harvard’s Faculty of Arts and Sciences awarded him the Roslyn Abramson award for outstanding undergraduate teaching.
Michael Brenner, Applied Mathematics (SEAS)
Title: Towards living synthetic materials
Abstract: Biological materials are much more complicated and functional than synthetic ones. Over the past several years we have been trying to figure out why. A sensible hypothesis is that biological materials are programmable. But we are very far from being able to program materials we create with this level of sophistication. I will discuss our (largely unsuccessful) efforts to bridge this gap, though as of today I’m somewhat optimistic that we are arriving at a set of theoretical models that is rich enough to produce relevant emergent behavior.
Bio: I’ve been at Harvard for a long time. My favorite part of Harvard is the students.
Rui Duan, Biostatistics (HSPH)
Title: Federated and transfer learning for healthcare data integration
Abstract: The growth of availability and variety of healthcare data sources has provided unique opportunities for data integration and evidence synthesis, which can potentially accelerate knowledge discovery and improve clinical decision-making. However, many practical and technical challenges, such as data privacy, high dimensionality, and heterogeneity across different datasets, remain to be addressed. In this talk, I will introduce several methods for the effective and efficient integration of multiple healthcare datasets in order to train statistical or machine learning models with improved generalizability and transferability. Specifically, we develop communication-efficient federated learning algorithms for jointly analyzing multiple datasets without the need of sharing patient-level data, as well as transfer learning approaches that leverage shared knowledge learned across multiple datasets to improve the performance of statistical models in target populations of interest. We will discuss both the theoretical properties and examples of implementation of our methods in real-world research networks and data consortia.
Bio: Rui Duan is an Assistant Professor of Biostatistics at the Harvard T.H. Chan School of Public Health. She received her Ph.D. in Biostatistics in May 2020 from the University of Pennsylvania. Her research interests focus on developing statistical, machine learning, and informatics tools for (1) efficient data integration in biomedical research, (2) understanding and accounting for the heterogeneity of biomedical data, and improving the generalizability and transferability of models across populations (3) advancing precision medicine research on rare diseases and underrepresented populations.
Yannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS)
Title: The Sample Complexity of Up-to-ε Multi-Dimensional Revenue Maximization
Abstract: We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of n additive bidders whose values for m heterogeneous items are drawn independently. For any such instance and any ε > 0, we show that it is possible to learn an ε-Bayesian Incentive Compatible auction whose expected revenue is within ε of the optimal ε-BIC auction from only polynomially many samples.
Our fully nonparametric approach is based on ideas that hold quite generally, and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive, and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well-understood, our corollary for this case extends slightly the state-of-the-art.
Joint work with S. Matthew Weinberg
Bio: Yannai A. Gonczarowski is an Assistant Professor of Economics and of Computer Science at Harvard University—the first faculty member at Harvard to have been appointed to both of these departments. Interested in both economic theory and theoretical computer science, Yannai explores computer-science-inspired economics: he harnesses approaches, aesthetics, and techniques traditionally originating in computer science to derive economically meaningful insights. Yannai received his PhD from the Departments of Math and CS, and the Center for the Study of Rationality, at the Hebrew University of Jerusalem, where he was advised by Sergiu Hart and Noam Nisan. Yannai is also a professionally-trained opera singer, having acquired a bachelor’s degree and a master’s degree in Classical Singing at the Jerusalem Academy of Music and Dance. Yannai’s doctoral dissertation was recognized with several awards, including the 2018 Michael B. Maschler Prize of the Israeli Chapter of the Game Theory Society, and the ACM SIGecom Doctoral Dissertation Award for 2018. For the design and implementation of the National Matching System for Gap-Year Programs in Israel, he was awarded the Best Paper Award at MATCH-UP’19 and the inaugural INFORMS AMD Michael H. Rothkopf Junior Researcher Paper Prize (first place) for 2020. Yannai is also the recipient of the inaugural ACM SIGecom Award for Best Presentation by a Student or Postdoctoral Researcher at EC’18. His first textbook, “Mathematical Logic through Python” (Gonczarowski and Nisan), which introduces a new approach to teaching the material of a basic Logic course to Computer Science students, tailored to the unique intuitions and strengths of this cohort of students, is forthcoming in Cambridge University Press.
Kosuke Imai, Government & Statistics (FAS)
Title: Use of Simulation Algorithms for Legislative Redistricting Analysis and Evaluation
Abstract: After the 2020 Census, many states have been redrawing the boundaries of Congressional and state legislative districts. To evaluate the partisan and racial bias of redistricting plans, scholars have developed Monte Carlo simulation algorithms. The idea is to generate a representative sample of redistricting plans under a specified set of criteria and conduct a statistical hypothesis test by comparing a proposed plan with these simulated plans. I will give a brief overview of these redistricting simulation algorithms and discuss how they are used in real-world court cases.
Bio: Kosuke Imai is Professor in the Department of Government and Department of Statistics at Harvard University. Before moving to Harvard in 2018, Imai taught at Princeton University for 15 years where he was the founding director of the Program in Statistics and Machine Learning. Imai specializes in the development of statistical methods and machine learning algorithms and their applications to social science research. His areas of expertise include causal inference, computational social science, program evaluation, and survey methodology.
Sham M. Kakade, Computer Science (SEAS) & Statistics (FAS)
Title: What is the Statistical Complexity of Reinforcement Learning?
Abstract: This talk will highlight much of the recent progress on the following fundamental question in the theory of reinforcement learning: what (representational or structural) conditions govern our ability to generalize and avoid the curse of dimensionality? With regards to supervised learning, these questions are reasonably well understood, both practically and theoretically: practically, we have overwhelming evidence on the value of representational learning (say through modern deep networks) as a means for sample efficient learning, and, theoretically, there are well-known complexity measures (e.g. the VC dimension and Rademacher complexity) that govern the statistical complexity of learning. Providing an analogous theory for reinforcement learning is far more challenging, where even characterizing structural conditions which support sample efficient generalization has been far less well understood, until recently.
This talk will survey recent advances towards characterizing when generalization is possible in RL, focusing on both necessary and sufficient conditions. In particular, we will introduce a new complexity measure, the Decision-Estimation Coefficient, that is proven to be necessary (and, essentially, sufficient) for sample-efficient interactive learning.
Bio: Sham Kakade is a professor at Harvard University and a co-director of the Kempner Institute for the Study of Artificial and Natural Intelligence. He works on the mathematical foundations of machine learning and AI. Sham’s thesis helped lay the statistical foundations of reinforcement learning. With his collaborators, his additional contributions include foundational results on: policy gradient methods in reinforcement learning; regret bounds for linear bandit and Gaussian process bandit models; the tensor and spectral methods for latent variable models; and a number of convergence analyses for convex and non-convex algorithms. He is the recipient of the ICML Test of Time Award, the IBM Pat Goldberg best paper award, and INFORMS Revenue Management and Pricing Prize. He has been program chair for COLT 2011.
Sham was an undergraduate at Caltech, where he studied physics and worked under the guidance of John Preskill in quantum computing. He completed his Ph.D. with Peter Dayan in computational neuroscience at the Gatsby Computational Neuroscience Unit. He was a postdoc with Michael Kearns at the University of Pennsylvania.
Title: “Machine (Un)Learning” or Why Your Deployed Model Might Violate Existing Privacy Law
Abstract: Businesses like Facebook and Google depend on training sophisticated models on user data. Increasingly—in part because of regulations like the European Union’s General Data Protection Act and the California Consumer Privacy Act—these organizations are receiving requests to delete the data of particular users. But what should that mean? It is straightforward to delete a customer’s data from a database and stop using it to train future models. But what about models that have already been trained using an individual’s data? These are not necessarily safe; it is known that individual training data can be exfiltrated from models trained in standard ways via model inversion attacks. In a series of papers we help formalize a rigorous notion of data-deletion and propose algorithms to efficiently delete user data from trained models with provable guarantees in both convex and non-convex settings.
Bio: Seth Neel is a first-year Assistant Professor in the TOM Unit at Harvard Business School, and Co-PI of the SAFR ML Lab in the D3 Institute, which develops methodology to incorporate privacy and fairness guarantees into techniques for machine learning and data analysis, while balancing other critical considerations like accuracy, efficiency, and interpretability. He obtained his Ph.D. from the University of Pennsylvania in 2020 where he was an NSF graduate fellow. His work has focused primarily on differential privacy, notions of fairness in a variety of machine learning settings, and adaptive data analysis.
Melanie Matchett Wood, Mathematics (FAS)
Title: Understanding distributions of algebraic structures through their moments
Abstract: A classical tool of probability and analysis is to use the moments (mean, variance, etc.) of a distribution to recognize an unknown distribution of real numbers. In recent work, we are interested in distributions of algebraic structures that can’t be captured in a single number. We will explain one example, the fundamental group, that captures something about the shapes of possibly complicated or high dimensional spaces. We are developing a new theory of the moment problem for random algebraic structures which helps to to identify distributions of such, such as fundamental groups of random three dimensional spaces. This talk is based partly on joint work with Will Sawin.
Bio: Melanie Matchett Wood is a professor of mathematics at Harvard University and a Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study. Her work spans number theory, algebraic geometry, algebraic topology, additive combinatorics, and probability. Wood has been awarded a CAREER grant, a Sloan Research Fellowship, a Packard Fellowship for Science and Engineering, and the AWM-Microsoft Research Prize in Algebra and Number Theory, and she is a Fellow of the American Mathematical Society. In 2021, Wood received the National Science Foundation’s Alan T. Waterman Award, the nation’s highest honor for early-career scientists and engineers.
Abstract: In 1973, Erdős conjectured the existence of high girth (n,3,2)-Steiner systems. Recently, Glock, Kühn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős’ conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erdős’ conjecture. As for Steiner systems with more general parameters, Glock, Kühn, Lo, and Osthus conjectured the existence of high girth (n,q,r)-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of submatchings (which we view as a hypergraph H where V(H)=E(G)). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Komlós, Pintz, Spencer, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when H is a hypergraph with small codegrees (for which high girth designs is a specific instance). Indeed, the coloring version of our result even yields an almost partition of K_n^r into approximate high girth (n,q,r)-Steiner systems. If time permits, I will explain some of the other applications of our main results such as to rainbow matchings. This is joint work with Luke Postle.
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.
Schedule
Speaker
Title
Abstract
March 1 – 3, 2022 10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).
March 22 – 25, 2022 22nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.
April 19 & 21, 2022 10 am – 12 pm ET, each dayZoom only
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, a precursor of general relativity characterized by a dimensionless coupling constant (the “Foeppl-von Karman number”) that can easily reach vK = 10^7 in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics, as exemplified by experiments that twist and bend individual atomically-thin free-standing graphene sheets (with vK = 10^13!) A crumpling transition out of the flat phase for thermalized elastic membranes has been predicted when kT is large compared to the microscopic bending stiffness, which could have interesting consequences for Dirac cones of electrons embedded in graphene. It may be possible to lower the crumpling temperature for graphene to more readily accessible range by inserting a regular lattice of laser-cut perforations, an expectation an confirmed by extensive molecular dynamics simulations. We then move on to analyze the physics of sheets mutilated with puckers and stitches. Puckers and stitches lead to Ising-like phase transitions riding on a background of flexural phonons, as well as an anomalous coefficient of thermal expansion. Finally, we argue that thin membranes with a background curvature lead to thermalized spherical shells that must collapse beyond a critical size at room temperature, even in the absence of an external pressure.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: Recent advances in our understanding of symmetry in quantum many-body systems offer the possibility of a generalized Landau paradigm that encompasses all equilibrium phases of matter. This talk will be an elementary review of some of these developments, based on: https://arxiv.org/abs/2204.03045
Abstract: Conditional independence (CI) is an important tool instatistical modeling, as, for example, it gives a statistical interpretation to graphical models. In general, given a list of dependencies among random variables, it is difficult to say which constraints are implied by them. Moreover, it is important to know what constraints on the random variables are caused by hidden variables. On the other hand, such constraints are corresponding to some determinantal conditions on the tensor of joint probabilities of the observed random variables. Hence, the inference question in statistics relates to understanding the algebraic and geometric properties of determinantal varieties such as their irreducible decompositions or determining their defining equations. I will explain some recent progress that arises by uncovering the link to point configurations in matroid theory and incidence geometry. This connection, in particular, leads to effective computational approaches for (1) giving a decomposition for each CI variety; (2) identifying each component in the decomposition as a matroid variety; (3) determining whether the variety has a real point or equivalently there is a statistical model satisfying a given collection of dependencies. The talk is based on joint works with Oliver Clarke, Kevin Grace, and Harshit Motwani.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
Title: Baryon Minus Lepton Number BF Theory for the Cosmological Lithium Problem
Abstract: The cosmological lithium problem—that the observed primordial abundance is lower than theoretical expectations by order one—is perhaps the most statistically significant anomaly of SM+ ΛCDM, and has resisted decades of attempts by cosmologists, nuclear physicists, and astronomers alike to root out systematics. We upgrade a discrete subgroup of the anomaly-free global symmetry of the SM to an infrared gauge symmetry, and UV complete this at a scale Λ as the familiar U(1)_{B-N_cL} Abelian Higgs theory. The early universe phase transition forms cosmic strings which are charged under the emergent higher-form symmetry of the baryon minus lepton BF theory. These topological defects catalyze interactions which turn N_g baryons into N_g leptons at strong scale rates in an analogue of the Callan-Rubakov effect, where N_g=3 is the number of SM generations. We write down a model in which baryon minus lepton strings superconduct bosonic global baryon plus lepton number currents and catalyze solely 3p^+ → 3e^+. We suggest that such cosmic strings have disintegrated O(1) of the lithium nuclei formed during Big Bang Nucleosynthesis and estimate the rate, with our benchmark model finding Λ ~ 10^8 GeV gives the right number density of strings.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Abstract: In spatial population genetics, it is important to understand the probability of extinction in multi-species interactions such as growing bacterial colonies, cancer tumor evolution and human migration. This is because extinction probabilities are instrumental in determining the probability of coexistence and the genealogies of populations. A key challenge is the complication due to spatial effect and different sources of stochasticity. In this talk, I will discuss about methods to compute the probability of extinction and other long-time behaviors for stochastic reaction-diffusion equations on metric graphs that flexibly parametrizes the underlying space. Based on recent joint work with Adrian Gonzalez-Casanova and Yifan (Johnny) Yang.
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1 October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
On August 24, 2021, the CMSA hosted our seventh annual Conference on Big Data. The Conference features many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The 2021 Big Data Conference took place virtually on Zoom.
Organizers:
Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University
Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business
Horng-Tzer Yau, Professor of Mathematics, Harvard University
Title: Robustness and stability for multidimensional persistent homology
Abstract: A basic principle in topological data analysis is to study the shape of data by looking at multiscale homological invariants. The idea is to filter the data using a scale parameter that reflects feature size. However, for many data sets, it is very natural to consider multiple filtrations, for example coming from feature scale and density. A key question that arises is how such invariants behave with respect to noise and outliers. This talk will describe a framework for understanding those questions and explore open problems in the area.
Abstract: Many data analysis pipelines are adaptive: the choice of which analysis to run next depends on the outcome of previous analyses. Common examples include variable selection for regression problems and hyper-parameter optimization in large-scale machine learning problems: in both cases, common practice involves repeatedly evaluating a series of models on the same dataset. Unfortunately, this kind of adaptive re-use of data invalidates many traditional methods of avoiding overfitting and false discovery, and has been blamed in part for the recent flood of non-reproducible findings in the empirical sciences. An exciting line of work beginning with Dwork et al. in 2015 establishes the first formal model and first algorithmic results providing a general approach to mitigating the harms of adaptivity, via a connection to the notion of differential privacy. In this talk, we’ll explore the notion of differential privacy and gain some understanding of how and why it provides protection against adaptivity-driven overfitting. Many interesting questions in this space remain open.
Joint work with: Christopher Jung (UPenn), Seth Neel (Harvard), Aaron Roth (UPenn), Saeed Sharifi-Malvajerdi (UPenn), and Moshe Shenfeld (HUJI). This talk will draw on work that appeared at NeurIPS 2019 and ITCS 2020
Title: Towards Reliable and Robust Model Explanations
Abstract: As machine learning black boxes are increasingly being deployed in domains such as healthcare and criminal justice, there is growing emphasis on building tools and techniques for explaining these black boxes in an interpretable manner. Such explanations are being leveraged by domain experts to diagnose systematic errors and underlying biases of black boxes. In this talk, I will present some of our recent research that sheds light on the vulnerabilities of popular post hoc explanation techniques such as LIME and SHAP, and also introduce novel methods to address some of these vulnerabilities. More specifically, I will first demonstrate that these methods are brittle, unstable, and are vulnerable to a variety of adversarial attacks. Then, I will discuss two solutions to address some of the vulnerabilities of these methods – (i) a framework based on adversarial training that is designed to make post hoc explanations more stable and robust to shifts in the underlying data; (ii) a Bayesian framework that captures the uncertainty associated with post hoc explanations and in turn allows us to generate explanations with user specified levels of confidences. I will conclude the talk by discussing results from real world datasets to both demonstrate the vulnerabilities in post hoc explanation techniques as well as the efficacy of our aforementioned solutions.
Abstract: Many selection processes contain a “gatekeeper”. The gatekeeper’s goal is to examine an applicant’s suitability to a proposed position before both parties endure substantial costs. Intuitively, the introduction of a gatekeeper should reduce selection costs as unlikely applicants are sifted out. However, we show that this is not always the case as the gatekeeper’s introduction inadvertently reduces the applicant’s expected costs and thus interferes with her self-selection. We study the conditions under which the gatekeeper’s presence improves the system’s efficiency and those conditions under which the gatekeeper’s presence induces inefficiency. Additionally, we show that the gatekeeper can sometimes improve selection correctness by behaving strategically (i.e., ignore her private information with some probability).
Abstract: In geometry and physics it has proved useful to relate G2 and Calabi-Yau geometry via circle bundles. Contact Calabi-Yau 7-manifolds are, in the simplest cases, such circle bundles over Calabi-Yau 3-orbifolds. These 7-manifolds provide testing grounds for the study of geometric flows which seek to find torsion-free G2-structures (and thus Ricci flat metrics with exceptional holonomy). They also give useful backgrounds to examine the heterotic G2 system (also known as the G2-Hull-Strominger system), which is a coupled set of PDEs arising from physics that involves the G2-structure and gauge theory on the 7-manifold. I will report on recent progress on both of these directions in the study of contact Calabi-Yau 7-manifolds, which is joint work with H. Sá Earp and J. Saavedra.
Title: Classical and quantum integrable systems in enumerative geometry
Abstract: For more than a quarter of a century, thanks to the ideas and questions originating in modern high-energy physics, there has been a very fruitful interplay between enumerative geometry and integrable system, both classical and quantum. While it is impossible to summarize even the most important aspects of this interplay in one talk, I will try to highlight a few logical points with the goal to explain the place and the role of certain more recent developments.
The Center of Mathematical Sciences and Applications will be hosting a workshop on Quantum Information on April 23-24, 2018. In the days leading up to the conference, the American Mathematical Society will also be hosting a sectional meeting on quantum information on April 21-22. You can find more information here.
Abstract: Kodaira’s motivation was to generalize the theory of Riemann surfaces in Weyl’s book to higher dimensions. After quickly recalling the chronology of Kodaira, I will review some of Kodaira’s works in three sections on topics of harmonic analysis, deformation theory and compact complex surfaces. Each topic corresponds to a volume of Kodaira’s collected works in three volumes, of which I will cover only tiny parts.
During the Spring 2021 Semester Artan Sheshmani (CMSA/ I.M. A.U.) will be teaching a CMSA special lecture series on Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants for algebraic surfaces.
Title: Is relativity compatible with quantum theory?
Abstract: We review the background, mathematical progress, and open questions in the effort to determine whether one can combine quantum mechanics, special relativity, and interaction together into one mathematical theory. This field of mathematics is known as “constructive quantum field theory.” Physicists believe that such a theory describes experimental measurements made over a 70 year period and now refined to 13-decimal-point precision—the most accurate experiments ever performed.
Abstract: In mid-career, as an internationally renowned mathematician, Michael Atiyah discovered that some problems in physics responded to current work in algebraic geometry and this set him on a path to develop an active interface between mathematics and physics which was formative in the links which are so active today. The talk will focus, in a fairly basic fashion, on some examples of this interaction, which involved both applying physical ideas to solve mathematical problems and introducing mathematical ideas to physicists.
Abstract: Fano and Calabi-Yau varieties play a fundamental role in algebraic geometry, differential geometry, arithmetic geometry, mathematical physics, etc. The notion of log Calabi-Yau fibration unifies Fano and Calabi-Yau varieties, their fibrations, as well as their local birational counterparts such as flips and singularities. Such fibrations can be examined from many different perspectives. The purpose of this talk is to introduce the theory of log Calabi-Yau fibrations, to remind some known results, and to state some open problems.
Title: Why do some universities have separate departments of statistics? And are they all anachronisms, destined to follow the path of other dinosaurs?
Abstract: For a long stretch of time in the history of mathematics, Number Theory and Topology formed vast, but disjoint domains of mathematical knowledge. Origins of number theory can be traced back to the Babylonian clay tablet Plimpton 322 (about 1800 BC) that contained a list of integer solutions of the “Diophantine” equation $a^2+b^2=c^2$: archetypal theme of number theory, named after Diophantus of Alexandria (about 250 BC). Topology was born much later, but arguably, its cousin — modern measure theory, — goes back to Archimedes, author of Psammites (“Sand Reckoner”), who was approximately a contemporary of Diophantus. In modern language, Archimedes measures the volume of observable universe by counting the number of small grains of sand necessary to fill this volume. Of course, many qualitative geometric models and quantitative estimates of the relevant distances precede his calculations. Moreover, since the estimated numbers of grains of sand are quite large (about $10^{64}$), Archimedes had to invent and describe a system of notation for large numbers going far outside the possibilities of any of the standard ancient systems. The construction of the first bridge between number theory and topology was accomplished only about fifty years ago: it is the theory of spectra in stable homotopy theory. In particular, it connects $Z$, the initial object in the theory of commutative rings, with the sphere spectrum $S$. This connection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory. In this talk based upon the authors’ (Yu. Manin and M. Marcolli) joint research project, I suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds.
Title: Knot Invariants From Gauge Theory in Three, Four, and Five Dimensions
Abstract: I will explain connections between a sequence of theories in two, three, four, and five dimensions and describe how these theories are related to the Jones polynomial of a knot and its categorification.
Title: Noncommutative Geometry, the Spectral Aspect
Abstract: This talk will be a survey of the spectral side of noncommutative geometry, presenting the new paradigm of spectral triples and showing its relevance for the fine structure of space-time, its large scale structure and also in number theory in connection with the zeros of the Riemann zeta function.
On August 23-24, 2018 the CMSA will be hosting our fourth annual Conference on Big Data. The Conference will feature many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The talks will take place in Science Center Hall B, 1 Oxford Street.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants.
The Center of Mathematical Sciences and Applications will be hosting a workshop on General Relativity from May 23 – 24, 2016. The workshop will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138. The workshop will start on Monday, May 23 at 9am and end on Tuesday, May 24 at 4pm.
Speakers:
Po-Ning Chen, Columbia University
Piotr T. Chruściel, University of Vienna
Justin Corvino, Lafayette College
Greg Galloway, University of Miami
James Guillochon, Harvard University
Lan-Hsuan Huang, University of Connecticut
Dan Kapec, Harvard University
Dan Lee, CUNY
Alex Lupsasca, Harvard University
Pengzi Miao, University of Miami
Prahar Mitra, Harvard University
Lorenzo Sironi, Harvard University
Jared Speck, MIT
Mu-Tao Wang, Columbia University
Please click Workshop Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
The CMSA will be hosting an F-Theory workshop September 29-30, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
On August 19-20, 2019 the CMSA will be hosting our fifth annual Conference on Big Data. The Conference will feature many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The talks will take place in Science Center Hall D, 1 Oxford Street.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants.
Videos can be found in this Youtube playlist or in the schedule below.
Due to inclement weather on Sunday, the second half of the workshop has been moved forward one day. Sunday and Monday’s talks will now take place on Monday and Tuesday.
On January 18-21, 2019 the Center of Mathematical Sciences and Applications will be hosting a workshop on the Geometric Analysis Approach to AI.
This workshop will focus on the theoretic foundations of AI, especially various methods in Deep Learning. The topics will cover the relationship between deep learning and optimal transportation theory, DL and information geometry, DL Learning and information bottle neck and renormalization theory, DL and manifold embedding and so on. Furthermore, the recent advancements, novel methods, and real world applications of Deep Learning will also be reported and discussed.
The workshop will take place from January 18th to January 23rd, 2019. In the first four days, from January 18th to January 21, the speakers will give short courses; On the 22nd and 23rd, the speakers will give conference representations. This workshop is organized by Xianfeng Gu and Shing-Tung Yau.
On August 18 and 20, 2018, the Center of Mathematic Sciences and Applications and the Harvard University Mathematics Department hosted a conference on From Algebraic Geometry to Vision and AI: A Symposium Celebrating the Mathematical Work of David Mumford. The talks took place in Science Center, Hall B.
Saturday, August 18th: A day of talks on Vision, AI and brain sciences
Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960′, due to the discovery of “supertranslation ambiguity”: the angular momentums recorded by two distant observers of the same system may not be the same. In this talk, I shall show how the mathematical theory of optimal isometric embedding and quasilocal angular momentum identifies a correction term, and leads to a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau.
Just over a century ago, the biologist, mathematician and philologist D’Arcy Thompson wrote “On growth and form”. The book – a literary masterpiece – is a visionary synthesis of the geometric biology of form. It also served as a call for mathematical and physical approaches to understanding the evolution and development of shape. In the century since its publication, we have seen a revolution in biology following the discovery of the genetic code, which has uncovered the molecular and cellular basis for life, combined with the ability to probe the chemical, structural, and dynamical nature of molecules, cells, tissues and organs across scales. In parallel, we have seen a blossoming of our understanding of spatiotemporal patterning in physical systems, and a gradual unveiling of the complexity of physical form. So, how far are we from realizing the century-old vision that “Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed” ?
To address this requires an appreciation of the enormous ‘morphospace’ in terms of the potential shapes and sizes that living forms take, using the language of mathematics. In parallel, we need to consider the biological processes that determine form in mathematical terms is based on understanding how instabilities and patterns in physical systems might be harnessed by evolution.
In Fall 2018, CMSA will focus on a program that aims at recent mathematical advances in describing shape using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems. The first workshop will focus on the interface between Morphometrics and Mathematics, while the second will focus on the interface between Morphogenesis and Physics.The workshop is organized by L. Mahadevan (Harvard), O. Pourquie (Harvard), A. Srivastava (Florida).
As part of the program on Mathematical Biology a workshop on Morphogenesis: Geometry and Physics will take place on December 3-5, 2018. The workshop will be held inroom G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
In Fall 2018, the CMSA will host a Program on Mathematical Biology, which aims to describe recent mathematical advances in using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems.
The plethora of natural shapes that surround us at every scale is both bewildering and astounding – from the electron micrograph of a polyhedral virus, to the branching pattern of a gnarled tree to the convolutions in the brain. Even at the human scale, the shapes seen in a garden at the scale of a pollen grain, a seed, a sapling, a root, a flower or leaf are so numerous that “it is enough to drive the sanest man mad,” wrote Darwin. Can we classify these shapes and understand their origins quantitatively?
In biology, there is growing interest in and ability to quantify growth and form in the context of the size and shape of bacteria and other protists, to understand how polymeric assemblies grow and shrink (in the cytoskeleton), and how cells divide, change size and shape, and move to organize tissues, change their topology and geometry, and link multiple scales and connect biochemical to mechanical aspects of these problems, all in a self-regulated setting.
To understand these questions, we need to describe shape (biomathematics), predict shape (biophysics), and design shape (bioengineering).
For example, in mathematics there are some beautiful links to Nash’s embedding theorem, connections to quasi-conformal geometry, Ricci flows and geometric PDE, to Gromov’s h principle, to geometrical singularities and singular geometries, discrete and computational differential geometry, to stochastic geometry and shape characterization (a la Grenander, Mumford etc.). A nice question here is to use the large datasets (in 4D) and analyze them using ideas from statistical geometry (a la Taylor, Adler) to look for similarities and differences across species during development, and across evolution.
In physics, there are questions of generalizing classical theories to include activity, break the usual Galilean invariance, as well as isotropy, frame indifference, homogeneity, and create both agent (cell)-based and continuum theories for ordered, active machines, linking statistical to continuum mechanics, and understanding the instabilities and patterns that arise. Active generalizations of liquid crystals, polar materials, polymers etc. are only just beginning to be explored and there are some nice physical analogs of biological growth/form that are yet to be studied.
The CMSA will be hosting a Workshop on Morphometrics, Morphogenesis and Mathematics from October 22-24 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
The Center of Mathematical Sciences and Applications will be having a conference on Big Data August 24-26, 2015, in Science Center Hall B at Harvard University. This conference will feature many speakers from the Harvard Community as well as many scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
For more info, please contact Sarah LaBauve at slabauve@math.harvard.edu.
Registration for the conference is now closed.
Please click here for a downloadable version of this schedule.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found here.
Monday, August 24
Time
Speaker
Title
8:45am
Meet and Greet
9:00am
Sendhil Mullainathan
Prediction Problems in Social Science: Applications of Machine Learning to Policy and Behavioral Economics
9:45am
Mike Luca
Designing Disclosure for the Digital Age
10:30
Break
10:45
Jianqing Fan
Big Data Big Assumption: Spurious discoveries and endogeneity
Title: Synthetic Regression Discontinuity: Estimating Treatment Effects using Machine Learning
Abstract: In the standard regression discontinuity setting, treatment assignment is based on whether a unit’s observable score (running variable) crosses a known threshold. We propose a two-stage method to estimate the treatment effect when the score is unobservable to the econometrician while the treatment status is known for all units. In the first stage, we use a statistical model to predict a unit’s treatment status based on a continuous synthetic score. In the second stage, we apply a regression discontinuity design using the predicted synthetic score as the running variable to estimate the treatment effect on an outcome of interest. We establish conditions under which the method identifies the local treatment effect for a unit at the threshold of the unobservable score, the same parameter that a standard regression discontinuity design with known score would identify. We also examine the properties of the estimator using simulations, and propose the use machine learning algorithms to achieve high prediction accuracy. Finally, we apply the method to measure the effect of an investment grade rating on corporate bond prices by any of the three largest credit ratings agencies. We find an average 1% increase in the prices of corporate bonds that received an investment grade as opposed to a non-investment grade rating.
Abstract: I will first review the construction of the moduli space of tropical curves (or metric graphs), and its relation to graph complexes. The graph Laplacian may be interpreted as a tropical version of the classical Torelli map and its determinant is the Kirchhoff graph polynomial (also called 1st Symanzik), which is one of the two key components in Feynman integrals in high energy physics.The other component is the so-called 2nd Symanzik polynomial, which is defined for graphs with external half edges and involves particle masses (edge colourings). I will explain how this too may be interpreted as the determinant of a generalised graph Laplacian, and how it leads to a volumetric interpretation of a certain class of Feynman integrals.
Abstract: The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics, probability, computer science, and chemistry. The algorithm builds a maximal independent set by inspecting the graph’s vertices one at a time according to a random order, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge.
In this talk, I will present a simple yet general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a valuable notion of local convergence. I will demonstrate the applicability of this framework by giving short and straightforward proofs for results on previously studied families of graphs, such as paths and various random graphs, and by providing new results for other models such as random trees.
If time allows, I will discuss a more delicate (and combinatorial) result, according to which, in expectation, the cardinality of a random greedy independent set in the path is no larger than that in any other tree of the same order.
The talk is based on joint work with Michael Krivelevich, Tamás Mészáros and Clara Shikhelman.
Abstract: For many classes of graphs that arise naturally in discrete geometry (for example intersection graphs of segments or disks in the plane), the edges of these graphs can be defined algebraically using the signs of a finite list of fixed polynomials. We investigate the number of n-vertex graphs in such an algebraically defined class of graphs. Warren’s theorem (a variant of a theorem of Milnor and Thom) implies upper bounds for the number of n-vertex graphs in such graph classes, but all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving a lower bound for this number (under some reasonable assumptions on the fixed list of polynomials), and this lower bound essentially matches the upper bound from Warren’s theorem.
On March 4-6, 2020 the CMSA will be hosting a three-day workshop on Mirror symmetry, Gauged linear sigma models, Matrix factorizations, and related topics as part of the Simons Collaboration on Homological Mirror Symmetry. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Title: Hodge structures and the topology of algebraic varieties
Abstract: We review the major progress made since the 50’s in our understanding of the topology of complex algebraic varieties. Most of the results we will discuss rely on Hodge theory, which has some analytic aspects giving the Hodge and Lefschetz decompositions, and the Hodge-Riemann relations. We will see that a crucial ingredient, the existence of a polarization, is missing in the general Kaehler context. We will also discuss some results and problems related to algebraic cycles and motives.
Abstract: Moduli spaces of various gauge theory equations and of various versions of (pseudo) holomorphic curve equations have played important role in geometry in these 40 years. Started with Floer’s work people start to obtain more sophisticated object such as groups, rings, or categories from (system of) moduli spaces. I would like to survey some of those works and the methods to study family of moduli spaces systematically.
Abstract: In 1973, Lemmens and Seidel asked to determine N_alpha(r), the maximum number of equiangular lines in R^r with common angle arccos(alpha). Recently, this problem has been almost completely settled when r is exponentially large relative to 1/alpha, with the approach both relying on Ramsey’s theorem, as well as being limited by it. In this talk, we will show how orthogonal projections of matrices with respect to the Frobenius inner product can be used to overcome this limitation, thereby obtaining significantly improved upper bounds on N_alpha(r) when r is polynomial in 1/alpha. In particular, our results imply that N_alpha(r) = Theta(r) for alpha >= Omega(1 / r^1/5).
Our projection method generalizes to complex equiangular lines in C^r, which may be of independent interest in quantum theory. Applying this method also allows us to obtain the first universal bound on the maximum number of complex equiangular lines in C^r with common Hermitian angle arccos(alpha), an extension of the Alon-Boppana theorem to dense regular graphs, which is tight for strongly regular graphs corresponding to r(r+1)/2 equiangular lines in R^r, an improvement to Welch’s bound in coding theory.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Abstract: In the early 1980s Michael Atiyah and Raoul Bott wrote two influential papers, ‘The Yang-Mills equations over Riemann surfaces’ and ‘The moment map and equivariant cohomology’, bringing together ideas ranging from algebraic and symplectic geometry through algebraic topology to mathematical physics and number theory. The aim of this talk is to explain their key insights and some of the new directions towards which these papers led.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, aMemorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
Abstract: I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.
Title: Discrepancy Theory and Randomized Controlled Trials
Abstract: Discrepancy theory tells us that it is possible to partition vectors into sets so that each set looks surprisingly similar to every other. By “surprisingly similar” we mean much more similar than a random partition. I will begin by surveying fundamental results in discrepancy theory, including Spencer’s famous existence proofs and Bansal’s recent algorithmic realizations of them. Randomized Controlled Trials are used to test the effectiveness of interventions, like medical treatments. Randomization is used to ensure that the test and control groups are probably similar. When we know nothing about the experimental subjects, uniform random assignment is the best we can do. When we know information about the experimental subjects, called covariates, we can combine the strengths of randomization with the promises of discrepancy theory. This should allow us to obtain more accurate estimates of the effectiveness of treatments, or to conduct trials with fewer experimental subjects. I will introduce the Gram-Schmidt Walk algorithm of Bansal, Dadush, Garg, and Lovett, which produces random solutions to discrepancy problems. I will then explain how Chris Harshaw, Fredrik Sävje, Peng Zhang, and I use this algorithm to improve the design of randomized controlled trials. Our Gram-Schmidt Walk Designs have increased accuracy when the experimental outcomes are correlated with linear functions of the covariates, and are comparable to uniform random assignments in the worst case.
Abstract: In a graph G = (V, E) we consider a system of paths S so that for every two vertices u,v in V there is a unique uv path in S connecting them. The path system is said to be consistent if it is closed under taking subpaths, i.e. if P is a path in S then any subpath of P is also in S. Every positive weight function w: E–>R^+ gives rise to a consistent path system in G by taking the paths in S to be geodesics w.r.t. w. In this case, we say w induces S. We say a graph G is metrizable if every consistent path system in G is induced by some such w.
We’ll discuss the concept of graph metrizability, and, in particular, we’ll see that while metrizability is a rare property, there exists infinitely many 2-connected metrizable graphs.
Abstract: The story of the index theorem ties together the Gang of Four—Atiyah, Bott, Hirzebruch, and Singer—and lies at the intersection of analysis, geometry, and topology. In the first part of the talk I will recount high points in the early developments. Then I turn to subsequent variations and applications. Throughout I emphasize the role of the Dirac operator.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, aMemorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
Abstract: There are several properties of closed geodesics which are proven using its Hamiltonian formulation, which has no analogue for minimal surfaces. I will talk about some recent progress in proving some of these properties for minimal surfaces.
Abstract:In this talk I will discuss a couple of research directions for robust AI beyond deep neural networks. The first is the need to understand what we are learning, by shifting the focus from targeting effects to understanding causes. The second is the need for a hybrid neural/symbolic approach that leverages both commonsense knowledge and massive amount of data. Specifically, as an example, I will present some latest work at Microsoft Research on building a pre-trained grounded text generator for task-oriented dialog. It is a hybrid architecture that employs a large-scale Transformer-based deep learning model, and symbol manipulation modules such as business databases, knowledge graphs and commonsense rules. Unlike GPT or similar language models learnt from data, it is a multi-turn decision making system which takes user input, updates the belief state, retrieved from the database via symbolic reasoning, and decides how to complete the task with grounded response.
Title:Area-minimizing integral currents and their regularity
Abstract: Caccioppoli sets and integral currents (their generalization in higher codimension) were introduced in the late fifties and early sixties to give a general geometric approach to the existence of area-minimizing oriented surfaces spanning a given contour. These concepts started a whole new subject which has had tremendous impacts in several areas of mathematics: superficially through direct applications of the main theorems, but more deeply because of the techniques which have been invented to deal with related analytical and geometrical challenges. In this lecture I will review the basic concepts, the related existence theory of solutions of the Plateau problem, and what is known about their regularity. I will also touch upon several fundamental open problems which still defy our understanding.
Abstract: Eugene Wachspress introduced polypols as real bounded semialgebraic sets in the plane that generalize polygons. He aimed to generalize barycentric coordinates from triangles to arbitrary polygons and further to polypols. For this, he defined the adjoint curve of a rational polypol. In the study of scattering amplitudes in physics, positive geometries are real semialgebraic sets together with a rational canonical form. We combine these two worlds by providing an explicit formula for the canonical form of a rational polypol in terms of defining equations of the adjoint curve and the facets of the polypol. For the special case of polygons, we show that the adjoint curve is hyperbolic and provide an explicit description of its nested ovals. Finally, we discuss the map that associates the adjoint curve to a given rational polypol, in particular the cases where this map is finite. For instance, using monodromy we find that a general quartic curve is the adjoint of 864 heptagons.
This talk is based on joint work with R. Piene, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn, M.-S. Sorea, and S. Telen.
Abstract: The (tree) amplituhedron was introduced in 2013 by Arkani-Hamed and Trnka in their study of N=4 SYM scattering amplitudes. A central conjecture in the field was to prove that the m=4 amplituhedron is triangulated by the images of certain positroid cells, called the BCFW cells. In this talk I will describe a resolution of this conjecture. The seminar is based on a recent joint work with Chaim Even-Zohar and Tsviqa Lakrec.
Abstract: We will go over the fundamentals of quantum error correction and fault tolerance and survey some of the recent developments in the field. Talk chair: Zhengwei Liu
Abstract: The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. The problem has a storied history and was studied by such eminent mathematicians as Gauss and Polya. The problem has also found applications in fields such as algorithm design and circuit development.
Despite much study, until recently very little was known regarding the asymptotics of Q(n). We apply modern methods from probabilistic combinatorics to reduce understanding Q(n) to the study of a particular infinite-dimensional convex optimization problem. The chief implication is that (in an appropriate sense) for a~1.94, Q(n) is approximately (ne^(-a))^n. Furthermore, our methods allow us to study the typical “shape” of n-queens configurations.
On March 24-26, The Center of Mathematical Sciences and Applications will be hosting a workshop on Geometry, Imaging, and Computing, based off the journal of the same name. The workshop will take place in CMSA building, G10.
Abstract: Recent years have seen tremendous advances in our understanding of perturbative quantum field theory—fueled largely by discoveries (and eventual explanations and exploitation) of shocking simplicity in the mathematical form of the predictions made for experiment. Among the most important frontiers in this progress is the understanding of loop amplitudes—their mathematical form, underlying geometric structure, and how best to manifest the physical properties of finite observables in general quantum field theories. This work is motivated in part by the desire to simplify the difficult work of doing Feynman integrals. I review some of the examples of this progress, and describe some ongoing efforts to recast perturbation theory in terms that expose as much simplicity (and as much physics) as possible.
The workshop on Probabilistic and Extremal Combinatorics will take place February 5-9, 2018 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
Extremal and Probabilistic Combinatorics are two of the most central branches of modern combinatorial theory. Extremal Combinatorics deals with problems of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects satisfying certain requirements. Such problems are often related to other areas including Computer Science, Information Theory, Number Theory and Geometry. This branch of Combinatorics has developed spectacularly over the last few decades. Probabilistic Combinatorics can be described informally as a (very successful) hybrid between Combinatorics and Probability, whose main object of study is probability distributions on discrete structures.
There are many points of interaction between these fields. There are deep similarities in methodology. Both subjects are mostly asymptotic in nature. Quite a few important results from Extremal Combinatorics have been proven applying probabilistic methods, and vice versa. Such emerging subjects as Extremal Problems in Random Graphs or the theory of graph limits stand explicitly at the intersection of the two fields and indicate their natural symbiosis.
The symposia will focus on the interactions between the above areas. These topics include Extremal Problems for Graphs and Set Systems, Ramsey Theory, Combinatorial Number Theory, Combinatorial Geometry, Random Graphs, Probabilistic Methods and Graph Limits.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Don Zagier (Max Planck Institute for Mathematics and International Centre for Theoretical Physics)
Title: Quantum topology and new types of modularity
Abstract: The talk concerns two fundamental themes of modern 3-dimensional topology and their unexpected connection with a theme coming from number theory. A deep insight of William Thurston in the mid-1970s is that the vast majority of complements of knots in the 3-sphere, or more generally of 3-manifolds, have a unique metric structure as hyperbolic manifolds of constant curvature -1, so that 3-dimensional topology is in some sense not really a branch of topology at all, but of differential geometry. In a different direction, the work of Vaughan Jones and Ed Witten in the late 1980s gave rise to the field of Quantum Topology, in which new types of invariants of knot complements and 3-manifolds are introduced that have their origins in ideas coming from quantum field theory. These two themes then became linked by Kashaev’s famous Volume Conjecture, now some 25 years old, which says that the Kashaev invariant _N of a hyperbolic knot K (this is a quantum invariant defined for each positive integer N and whose values are algebraic numbers) grows exponentially as N tends to infinity with an exponent proportional to the hyperbolic volume of the knot complement. About 10 years ago, I was led by numerical experiments to the discovery that Kashaev’s invariant could be upgraded to an invariant having rational numbers as its argument (with the original invariant being the value at 1/N) and that the Volume Conjecture then became part of a bigger story saying that the new invariant has some sort of strange transformation property under the action x -> (ax+b)/(cx+d) of the modular group SL(2,Z) on the argument. This turned out to be only the beginning of a fascinating and multi-faceted story relating quantum invariants, q-series, modularity, and many other topics. In the talk, which is intended for a general mathematical audience, I would like to recount some parts of this story, which is joint work with Stavros Garoufalidis (and of course involving contributions from many other authors). The “new types of modularity” in the title refer to a specific byproduct of these investigations, namely that there is a generalization of the classical notion of holomorphic modular form – which plays an absolutely central role in modern number theory – to a new class of holomorphic functions in the upper half-plane that no longer satisfy a transformation law under the action of the modular group, but a weaker extendability property instead. This new class, called “holomorphic quantum modular forms”, turns out to contain many other functions of a more number-theoretical nature as well as the original examples coming from quantum invariants.
Abstract: I will review two famous papers of Ray and Singer on analytic torsion written approximately half a century ago. Then I will sketch the influence of analytic torsion in a variety of areas of physics including anomalies, topological field theory, and string theory.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, aMemorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer.
The CMSA will be hosting a four-day Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory on January 10-13, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
We may be able to provide some financial support for grad students and postdocs interested in this event. If you are interested in funding, please send a letter of support from your mentor to Hansol Hong at hansol84@gmail.com.
In 2021, the CMSA hosted a lecture series on the literature of the mathematical sciences. This series highlights significant accomplishments in the intersection between mathematics and the sciences. Speakers include Edward Witten, Lydia Bieri, Simon Donaldson, Michael Freedman, Dan Freed, and many more.
Videos of these talks can be found in this Youtube playlist.
https://youtu.be/vb_JEhUW9t4
In the Spring 2021 semester, the CMSA hosted a sub-program on this series titled A Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer. Below is the schedule for talks in that subprogram
Eduard Jacob Neven Looijenga(Tsinghua University & Utrecht University)
Title: Theorems of Torelli type
Abstract: Given a closed manifold of even dimension 2n, then Hodge showed around 1950 that a kählerian complex structure on that manifold determines a decomposition of its complex cohomology. This decomposition, which can potentially vary continuously with the complex structure, extracts from a non-linear given, linear data. It can contain a lot of information. When there is essentially no loss of data in this process, we say that the Torelli theorem holds. We review the underlying theory and then survey some cases where this is the case. This will include the classical case n=1, but the emphasis will be on K3 manifolds (n=2) and more generally, on hyperkählerian manifolds. These cases stand out, since one can then also tell which decompositions occur.
The Center of Mathematical Sciences and Applications will be hosting a Mini-school on Nonlinear Equations on December 3-4, 2016. The conference will have speakers and will be hosted at Harvard CMSA Building: Room G1020 Garden Street, Cambridge, MA 02138.
The mini-school will consist of lectures by experts in geometry and analysis detailing important developments in the theory of nonlinear equations and their applications from the last 20-30 years. The mini-school is aimed at graduate students and young researchers working in geometry, analysis, physics and related fields.
Please click Mini-School Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Abstract: A main challenge in analyzing single-cell RNA sequencing (scRNA-seq) data is to reduce technical variations yet retain cell heterogeneity. Due to low mRNAs content per cell and molecule losses during the experiment (called ‘dropout’), the gene expression matrix has a substantial amount of zero read counts. Existing imputation methods treat either each cell or each gene as independently and identically distributed, which oversimplifies the gene correlation and cell type structure. We propose a statistical model-based approach, called SIMPLEs (SIngle-cell RNA-seq iMPutation and celL clustErings), which iteratively identifies correlated gene modules and cell clusters and imputes dropouts customized for individual gene module and cell type. Simultaneously, it quantifies the uncertainty of imputation and cell clustering via multiple imputations. In simulations, SIMPLEs performed significantly better than prevailing scRNA-seq imputation methods according to various metrics. By applying SIMPLEs to several real datasets, we discovered gene modules that can further classify subtypes of cells. Our imputations successfully recovered the expression trends of marker genes in stem cell differentiation and can discover putative pathways regulating biological processes.
Abstract: Encryption is the backbone of cybersecurity. While encryption can secure data both in transit and at rest, in the new era of ubiquitous computing, modern cryptography also aims to protect data during computation. Secure multi-party computation (MPC) is a powerful technology to tackle this problem, which enables distrustful parties to jointly perform computation over their private data without revealing their data to each other. Although it is theoretically feasible and provably secure, the adoption of MPC in real industry is still very much limited as of today, the biggest obstacle of which boils down to its efficiency.
My research goal is to bridge the gap between the theoretical feasibility and practical efficiency of MPC. Towards this goal, my research spans both theoretical and applied cryptography. In theory, I develop new techniques for achieving general MPC with the optimal complexity, bringing theory closer to practice. In practice, I design tailored MPC to achieve the best concrete efficiency for specific real-world applications. In this talk, I will discuss the challenges in both directions and how to overcome these challenges using cryptographic approaches. I will also show strong connections between theory and practice.
Biography: Peihan Miao is an assistant professor of computer science at the University of Illinois Chicago (UIC). Before coming to UIC, she received her Ph.D. from the University of California, Berkeley in 2019 and had brief stints at Google, Facebook, Microsoft Research, and Visa Research. Her research interests lie broadly in cryptography, theory, and security, with a focus on secure multi-party computation — especially in incorporating her industry experiences into academic research.
Location: Room G10, 20 Garden Street, Cambridge, MA 02138.
Organizers: Michael R. Douglas (CMSA/Stony Brook/IAIFI) and Peter Chin (CMSA/BU).
Machine learning has driven many exciting recent scientific advances. It has enabled progress on long-standing challenges such as protein folding, and it has helped mathematicians and mathematical physicists create new conjectures and theorems in knot theory, algebraic geometry, and representation theory.
At this workshop, we will bring together mathematicians, theoretical physicists, and machine learning researchers to review the state of the art in machine learning, discuss how ML results can be used to inspire, test and refine precise conjectures, and identify mathematical questions which may be suitable for this approach.
Speakers:
James Halverson, Northeastern University Dept. of Physics and IAIFI
Fabian Ruehle, Northeastern University Dept. of Physics and Mathematics and IAIFI
The Center of Mathematical Sciences and Applications will be hosting a workshop on Optimization in Image Processing on June 27 – 30, 2016. This 4-day workshop aims to bring together researchers to exchange and stimulate ideas in imaging sciences, with a special focus on new approaches based on optimization methods. This is a cutting-edge topic with crucial impact in various areas of imaging science including inverse problems, image processing and computer vision. 16 speakers will participate in this event, which we think will be a very stimulating and exciting workshop. The workshop will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
Titles, abstracts and schedule will be provided nearer to the event.
Speakers:
Antonin Chambolle, CMAP, Ecole Polytechnique
Raymond Chan, The Chinese University of Hong Kong
Ke Chen, University of Liverpool
Patrick Louis Combettes, Université Pierre et Marie Curie
Mario Figueiredo, Instituto Superior Técnico
Alfred Hero, University of Michigan
Ronald Lok Ming Lui, The Chinese University of Hong Kong
Mila Nikolova, Ecole Normale Superieure Cachan
Shoham Sabach, Israel Institute of Technology
Martin Benning, University of Cambridge
Jin Keun Seo, Yonsei University
Fiorella Sgallari, University of Bologna
Gabriele Steidl, Kaiserslautern University of Technology
Joachim Weickert, Saarland University
Isao Yamada, Tokyo Institute of Technology
Wotao Yin, UCLA
Please click Workshop Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
The Center of Mathematical Sciences and Applications will be hosting a 3-day workshop on Homological Mirror Symmetry and related areas on May 6 – May 8, 2016 at Harvard CMSA Building: Room G1020 Garden Street, Cambridge, MA 02138
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Schedule:
May 6 – Day 1
9:00am
Breakfast
9:35am
Opening remarks
9:45am – 10:45am
Si Li, “Quantum master equation, chiral algebra, and integrability”
Title: Ising model, total positivity, and criticality
Abstract: The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality. The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig’s theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes. I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with P. Pylyavskyy. Building on this connection, I will give a new boundary correlation formula for the critical Ising model
On April 27–29, 2022, the CMSA hosted a workshop on Nonlinear Algebra and Combinatorics.
Organizers: Bernd Sturmfels (MPI Leipzig) and Lauren Williams (Harvard).
In recent years, ideas from integrable systems and scattering amplitudes have led to advances in nonlinear algebra and combinatorics. In this short workshop, aimed at younger participants in the field, we will explore some of the interactions between the above topics.
Abstract: Matroid theory provides a combinatorial model for linearity, but it plays useful roles beyond linearity. In the classical setup, a linear subspace V of an n-dimensional vector space gives rise to a matroid M(V) on {1,…,n}. However, the matroid M(V) also knows about some nonlinear geometric spaces related to V. Conversely, those nonlinear spaces teach us things we didn’t know about matroids. My talk will discuss some examples.
10:30 am–11:00 am
COFFEE BREAK
11:00 am–11:45 am
Chris Eur
Title: Tautological classes of matroids
Abstract: Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call “tautological bundles (classes)” of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.
11:45 am–2:00 pm
LUNCH BREAK
2:00 pm–2:45 pm
Nick Early
Title: Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes
Abstract: The associahedron is known to encapsulate physical properties such as the notion of tree-level factorization for one of the simplest Quantum Field Theories, the biadjoint scalar, which has only cubic interactions. I will discuss novel instances of the associahedron and the biadjoint scalar in a class of generalized amplitudes, discovered by Cachazo, Early, Guevara and Mizera, by taking certain limits thereof. This connection leads to a simple proof of a new realization of the associahedron involving a Minkowski sum of certain positroid polytopes in the second hypersimplex.
2:45 pm–3:30 pm
Anna Seigal
Title: Invariant theory for maximum likelihood estimation
Abstract: I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.
3:30 pm–4:00 pm
COFFEE BREAK
4:00 pm–4:45 pm
Matteo Parisi
Title: Amplituhedra, Scattering Amplitudes, and Triangulations
Abstract: In this talk I will discuss about Amplituhedra – generalizations of polytopes inside the Grassmannian – introduced by physicists to encode interactions of elementary particles in certain Quantum Field Theories. In particular, I will explain how the problem of finding triangulations of Amplituhedra is connected to computing scattering amplitudes of N=4 super Yang-Mills theory. Triangulations of polygons are encoded in the associahedron, studied by Stasheff in the sixties; in the case of polytopes, triangulations are captured by secondary polytopes, constructed by Gelfand et al. in the nineties. Whereas a “secondary” geometry describing triangulations of Amplituhedra is still not known, and we pave the way for such studies. I will discuss how the combinatorics of triangulations interplays with T-duality from String Theory, in connection with the Momentum Amplituhedron. A generalization of T-duality led us to discover a striking duality between Amplituhedra of “m=2” type and a seemingly unrelated object – the Hypersimplex. The latter is a polytope which appears in many contexts, from matroid theory to tropical geometry. Based on joint works with Lauren Williams, Melissa Sherman-Bennett, Tomasz Lukowski.
4:45 pm–5:30 pm
Melissa Sherman-Bennett
Title: The hypersimplex and the m=2 amplituhedron
Abstract: In this talk, I’ll continue where Matteo left off. I’ll give some more details about the curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). The hypersimplex decompositions are closely related to matroidal subdivisions. Along the way, we prove a nice description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers, inspired by an analogous triangulation of the hypersimplex into Eulerian-number-many simplices.
Thursday, April 28, 2022
9:30 am–10:30 am
Claudia Fevola
Title: Nonlinear Algebra meets Feynman integrals
Abstract: Feynman integrals play a central role in particle physics in the theory of scattering amplitudes. They form a finite-dimensional vector space and the elements of a basis are named “master integrals” in the physics literature. The number of master integrals has been interpreted in different ways: it equals the dimension of a twisted de Rham cohomology group, the Euler characteristic of a very affine variety, and the holonomic rank of a D-module. In this talk, we are interested in a more general family of integrals that contains Feynman integrals as a special case. We explore this setting using tools coming from nonlinear algebra. This is an ongoing project with Daniele Agostini, Anna-Laura Sattelberger, and Simon Telen.
10:30 am–11:00 am
COFFEE BREAK
11:00 am–11:45 am
Simon Telen
Title: Landau discriminants
Abstract: The Landau discriminant is a projective variety containing kinematic parameters for which a Feynman integral can have singularities. We present a definition and geometric properties. We discuss how to compute Landau discriminants using symbolic and numerical methods. Our methods can be used, for instance, to compute the Landau discriminant of the pentabox diagram, which is a degree 12 hypersurface in 6-space. This is joint work with Sebastian Mizera.
11:45 am–2:00 pm
LUNCH BREAK
2:00 pm–2:45 pm
Christian Gaetz
Title: 1-skeleton posets of Bruhat interval polytopes
Abstract: Bruhat interval polytopes are a well-studied class of generalized permutohedra which arise as moment map images of various toric varieties and totally positive spaces in the flag variety. I will describe work in progress in which I study the 1-skeleta of these polytopes, viewed as posets interpolating between weak order and Bruhat order. In many cases these posets are lattices and the polytopes, despite not being simple, have interesting h-vectors. In a special case, work of Williams shows that Bruhat interval polytopes are isomorphic to bridge polytopes, so that chains in the 1-skeleton poset correspond to BCFW-bridge decompositions of plabic graphs.
2:45 pm–3:30 pm
Madeleine Brandt
Title: Top Weight Cohomology of $A_g$
Abstract: I will discuss a recent project in computing the top weight cohomology of the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. This piece of the cohomology is controlled by the combinatorics of the boundary strata of a compactification of $A_g$. Thus, it can be computed combinatorially. This is joint work with Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.
3:30 pm–4:00 pm
COFFEE BREAK
4:00 pm–5:00 pm
Emma Previato
Title: Sigma function on curves with non-symmetric semigroup
Abstract: We start with an overview of the correspondence between spectral curves and commutative rings of differential operators, integrable hierarchies of non-linear PDEs and Jacobian vector fields. The coefficients of the operators can be written explicitly in terms of the Kleinian sigma function: Weierstrass’ sigma function was generalized to genus greater than one by Klein, and is a ubiquitous tool in integrability. The most accessible case is the sigma function of telescopic curves. In joint work with J. Komeda and S. Matsutani, we construct a curve with non-symmetric Weierstrass semigroup (equivalently, Young tableau), consequently non-telescopic, and its sigma function. We conclude with possible applications to commutative rings of differential operators.
Abstract: It is well-known that soliton solutions of the KdV hierarchy are obtained by singular limits of hyper-elliptic curves. However, there is no general results for soliton solutions of the KP hierarchy, KP solitons. In this talk, I will show that some of the KP solitons are related to the singular space curves associated with certain class of numerical semigroups.
10:00 am–10:30 am
COFFEE BREAK
10:30 am–11:15 am
Yelena Mandelshtam
Title: Curves, degenerations, and Hirota varieties
Abstract: The Kadomtsev-Petviashvili (KP) equation is a differential equation whose study yields interesting connections between integrable systems and algebraic geometry. In this talk I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes all KP solutions arising from such a sum. I will then discuss a special case, studying the Hirota variety of a rational nodal curve. Of particular interest is an irreducible subvariety that is the image of a parameterization map. Proving that this is a component of the Hirota variety entails solving a weak Schottky problem for rational nodal curves. This talk is based on joint work with Daniele Agostini, Claudia Fevola, and Bernd Sturmfels.
11:15 am–12:00 pm
Charles Wang
Title: Differential Algebra of Commuting Operators
Abstract: In this talk, we will give an overview of the problem of finding the centralizer of a fixed differential operator in a ring of differential operators, along with connections to integrable hierarchies and soliton solutions to e.g. the KdV or KP equations. Given these interesting connections, it is important to be able to compute centralizers of differential operators, and we discuss how to use techniques from differential algebra to approach this question, as well as how having these computational tools can help in understanding the structure of soliton solutions to these equations.
12:00 pm–2:00 pm
LUNCH BREAK
2:00 pm–3:00 pm
Sebastian Mizera
Title: Feynman Polytopes
Abstract: I will give an introduction to a class of polytopes that recently emerged in the study of scattering amplitudes in quantum field theory.
3:00 pm–3:30 pm
COFFEE BREAK
3:30 pm–4:30 pm
Nima Arkani-Hamed
Title: Spacetime, Quantum Mechanics and Combinatorial Geometries at Infinity
Title: Diffusive growth sourced by topological defects
Abstract: In this talk, we develop a minimal model of morphogenesis of a surface where the dynamics of the intrinsic geometry is diffusive growth sourced by topological defects. We show that a positive (negative) defect can dynamically generate a cone (hyperbolic cone). We analytically explain features of the growth profile as a function of position and time, and predict that in the presence of a positive defect, a bump forms with height profile h(t) ~ t^(1/2) for early times t. To incorporate the effect of the mean curvature, we exploit the fact that for axisymmetric surfaces, the extrinsic geometry can be deduced entirely by the intrinsic geometry. We find that the resulting stationary geometry, for polar order and small bending modulus, is a deformed football. We apply our framework to various biological systems. In an ex-vivo setting of cultured murine neural progenitor cells, we show that our framework is consistent with the observed cell accumulation at positive defects and depletion at negative defects. In an in-vivo setting, we show that the defect configuration consisting of a bound +1 defect state, which is stabilized by activity, surrounded by two -1/2 defects can create a stationary ring configuration of tentacles, consistent with observations of a basal marine invertebrate Hydra
The workshop on coding and information theory will take place April 9-13, 2018 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
This workshop will focus on new developments in coding and information theory that sit at the intersection of combinatorics and complexity, and will bring together researchers from several communities — coding theory, information theory, combinatorics, and complexity theory — to exchange ideas and form collaborations to attack these problems.
Squarely in this intersection of combinatorics and complexity, locally testable/correctable codes and list-decodable codes both have deep connections to (and in some cases, direct motivation from) complexity theory and pseudorandomness, and recent progress in these areas has directly exploited and explored connections to combinatorics and graph theory. One goal of this workshop is to push ahead on these and other topics that are in the purview of the year-long program. Another goal is to highlight (a subset of) topics in coding and information theory which are especially ripe for collaboration between these communities. Examples of such topics include polar codes; new results on Reed-Muller codes and their thresholds; coding for distributed storage and for DNA memories; coding for deletions and synchronization errors; storage capacity of graphs; zero-error information theory; bounds on codes using semidefinite programming; tensorization in distributed source and channel coding; and applications of information-theoretic methods in probability and combinatorics. All these topics have attracted a great deal of recent interest in the coding and information theory communities, and have rich connections to combinatorics and complexity which could benefit from further exploration and collaboration.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
On December 2-4, 2019 the CMSA will be hosting a workshop on Quantum Matter as part of our program on Quantum Matter in Mathematics and Physics. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
During the 2021–22 academic year, the CMSA will be hosting a seminar on Combinatorics, Physics and Probability, organized by Matteo Parisi and Michael Simkin. This seminar will take place on Tuesdays at 9:00 am – 10:00 am (Boston time). The meetings will take place virtually on Zoom. To learn how to attend, please fill out this form, or contact the organizers Matteo (mparisi@cmsa.fas.harvard.edu) and Michael (msimkin@cmsa.fas.harvard.edu).
The schedule below will be updated as talks are confirmed.
Spring 2022
Date
Speaker
Title/Abstract
1/25/2022 *note special time 9:00–10:00 AM ET
Jacob Bourjaily (Penn State University, Eberly College of Science
Abstract: Recent years have seen tremendous advances in our understanding of perturbative quantum field theory—fueled largely by discoveries (and eventual explanations and exploitation) of shocking simplicity in the mathematical form of the predictions made for experiment. Among the most important frontiers in this progress is the understanding of loop amplitudes—their mathematical form, underlying geometric structure, and how best to manifest the physical properties of finite observables in general quantum field theories. This work is motivated in part by the desire to simplify the difficult work of doing Feynman integrals. I review some of the examples of this progress, and describe some ongoing efforts to recast perturbation theory in terms that expose as much simplicity (and as much physics) as possible.
Abstract: The (tree) amplituhedron was introduced in 2013 by Arkani-Hamed and Trnka in their study of N=4 SYM scattering amplitudes. A central conjecture in the field was to prove that the m=4 amplituhedron is triangulated by the images of certain positroid cells, called the BCFW cells. In this talk I will describe a resolution of this conjecture. The seminar is based on a recent joint work with Chaim Even-Zohar and Tsviqa Lakrec.
Abstract: I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.
2/15/2022
Igor Balla, Hebrew University of Jerusalem
Title: Equiangular lines and regular graphs
Abstract: In 1973, Lemmens and Seidel asked to determine N_alpha(r), the maximum number of equiangular lines in R^r with common angle arccos(alpha). Recently, this problem has been almost completely settled when r is exponentially large relative to 1/alpha, with the approach both relying on Ramsey’s theorem, as well as being limited by it. In this talk, we will show how orthogonal projections of matrices with respect to the Frobenius inner product can be used to overcome this limitation, thereby obtaining significantly improved upper bounds on N_alpha(r) when r is polynomial in 1/alpha. In particular, our results imply that N_alpha(r) = Theta(r) for alpha >= Omega(1 / r^1/5).
Our projection method generalizes to complex equiangular lines in C^r, which may be of independent interest in quantum theory. Applying this method also allows us to obtain the first universal bound on the maximum number of complex equiangular lines in C^r with common Hermitian angle arccos(alpha), an extension of the Alon-Boppana theorem to dense regular graphs, which is tight for strongly regular graphs corresponding to r(r+1)/2 equiangular lines in R^r, an improvement to Welch’s bound in coding theory.
Fall 2021
Date
Speaker
Title/Abstract
9/21/2021
Nima Arkani-Hamed IAS (Institute for Advanced Study), School of Natural Sciences
Title: Surfacehedra and the Binary Positive Geometry of Particle and “String” Amplitudes
9/28/2021
Melissa Sherman-Bennett University of Michigan, Department of Mathematics
Title: The hypersimplex and the m=2 amplituhedron
Abstract: I’ll discuss a curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). Along the way, we prove the sign-flip description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers (inspired by an analogous hypersimplex decomposition).
10/5/2021
Daniel Cizma, Hebrew University
Title: Geodesic Geometry on Graphs
Abstract: In a graph G = (V, E) we consider a system of paths S so that for every two vertices u,v in V there is a unique uv path in S connecting them. The path system is said to be consistent if it is closed under taking subpaths, i.e. if P is a path in S then any subpath of P is also in S. Every positive weight function w: E–>R^+ gives rise to a consistent path system in G by taking the paths in S to be geodesics w.r.t. w. In this case, we say w induces S. We say a graph G is metrizable if every consistent path system in G is induced by some such w.
We’ll discuss the concept of graph metrizability, and, in particular, we’ll see that while metrizability is a rare property, there exists infinitely many 2-connected metrizable graphs.
Joint work with Nati Linial.
10/12/2021
Lisa Sauermann, MIT
Title: On counting algebraically defined graphs
Abstract: For many classes of graphs that arise naturally in discrete geometry (for example intersection graphs of segments or disks in the plane), the edges of these graphs can be defined algebraically using the signs of a finite list of fixed polynomials. We investigate the number of n-vertex graphs in such an algebraically defined class of graphs. Warren’s theorem (a variant of a theorem of Milnor and Thom) implies upper bounds for the number of n-vertex graphs in such graph classes, but all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving a lower bound for this number (under some reasonable assumptions on the fixed list of polynomials), and this lower bound essentially matches the upper bound from Warren’s theorem.
10/19/2021
Pavel Galashin UCLA, Department of Mathematics
Title: Ising model, total positivity, and criticality
Abstract: The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality. The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig’s theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes. I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with P. Pylyavskyy. Building on this connection, I will give a new boundary correlation formula for the critical Ising model.
10/26/2021
Candida Bowtell, University of Oxford
Title: The n-queens problem
Abstract: The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n).
In this talk we’ll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost’ configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.
This is joint work with Peter Keevash.
11/9/2021
Steven Karp Universite du Quebec a Montreal, LaCIM (Laboratoire de combinatoire et d’informatique mathématique)
Abstract: One can view a partial flag variety in C^n as an adjoint orbit inside the Lie algebra of n x n skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. We classify gradient flows on adjoint orbits in various metrics which are compatible with total positivity. As applications, we show how the classical Toda flow fits into this framework, and prove that a new family of amplituhedra are homeomorphic to closed balls. This is joint work with Anthony Bloch.
Abstract: From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees. Joint work with Omer Angel and Gourab Ray.
Abstract: The Prague dimension of graphs was introduced by Nesetril, Pultr and Rodl in the 1970s: as a combinatorial measure of complexity, it is closely related to clique edges coverings and partitions. Proving a conjecture of Furedi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order n/(log n) for constant edge-probabilities. The main new proof ingredient is a Pippenger-Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size O(log n).
Abstract: The last few decades have seen a surge of interest in building towards a theory of discrete curvature that attempts to translate the key properties of curvature in differential geometry to the setting of discrete objects and spaces. In the case of graphs there have been several successful proposals, for instance by Lin-Lu-Yau, Forman and Ollivier, that replicate important curvature theorems and have inspired applications in a variety of practical settings. In this talk, I will introduce a new notion of discrete curvature on graphs, which we call the resistance curvature, and discuss some of its basic properties. The resistance curvature is defined based on the concept of effective resistance which is a metric between the vertices of a graph and has many other properties such as a close relation to random spanning trees. The rich theory of these effective resistances allows to study the resistance curvature in great detail; I will for instance show that “Lin-Lu-Yau >= resistance >= Forman curvature” in a specific sense, show strong evidence that the resistance curvature converges to zero in expectation for Euclidean random graphs, and give a connectivity theorem for positively curved graphs. The resistance curvature also has a naturally associated discrete Ricci flow which is a gradient flow and has a closed-form solution in the case of vertex-transitive and path graphs. Finally, if time permits I will draw a connection with the geometry of hyperacute simplices, following the work of Miroslav Fiedler. This work was done in collaboration with Renaud Lambiotte.
Abstract: Let M_n be drawn uniformly from all n by n symmetric matrices with entries in {-1,1}. In this talk I’ll consider the following basic question: what is the probability that M_n is singular? I’ll discuss recent joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe where we show that this probability is exponentially small. I hope to make the talk accessible to a fairly general audience.
Abstract: A long-standing problem in random graph theory has been to determine asymptotically the length of a longest induced path in sparse random graphs. Independent work of Luczak and Suen from the 90s showed the existence of an induced path of roughly half the optimal size, which seems to be a barrier for certain natural approaches. Recently, in joint work with Draganic and Krivelevich, we solved this problem. In the talk, I will discuss the history of the problem and give an overview of the proof.
12/21/2021
01/25/2022
Jacob Bourjaily Penn State University, Department of Physics
Abstract: We discuss the algebraic geometry of maximum likelihood estimation from the perspective of scattering amplitudes in particle physics. A guiding examples the moduli space of n-pointed rational curves. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions to rational function equations. Their number is an Euler characteristic. Soft limit degenerations are combined with certified numerical methods for concrete computations.
**This talk will be hybrid. Talk will be held at CMSA (20 Garden St) Room G10.
All non-Harvard affiliated visitors to the CMSA building will need to complete this covid form prior to arrival.
Abstract: About 30 years ago, string theorists made remarkable discoveries of hidden structures in algebraic geometry. First, the usual cup-product on the cohomology of a complex projective variety admits a canonical multi-parameter deformation to so-called quantum product, satisfying a nice system of differential equations (WDVV equations). The second discovery, even more striking, is Mirror Symmetry, a duality between families of Calabi-Yau varieties acting as a mirror reflection on the Hodge diamond.
Later it was realized that the quantum product belongs to the realm of symplectic geometry, and a half of mirror symmetry (called Homological Mirror Symmetry) is a duality between complex algebraic and symplectic varieties. The search of correct definitions and possible generalizations lead to great advances in many domains, giving mathematicians new glasses, through which they can see familiar objects in a completely new way.
I will review the history of major mathematical advances in the subject of HMS, and the swirl of ideas around it.
Abstract: The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n).
In this talk we’ll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost’ configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.
As part of the program on Mathematical Biology a workshop on Invariance and Geometry in Sensation, Action and Cognition will take place on April 15-17, 2019.
Legend has it that above the door to Plato’s Academy was inscribed “Μηδείς άγεωµέτρητος είσίτω µον τήν στέγην”, translated as “Let no one ignorant of geometry enter my doors”. While geometry and invariance has always been a cornerstone of mathematics, it has traditionally not been an important part of biology, except in the context of aspects of structural biology. The premise of this meeting is a tantalizing sense that geometry and invariance are also likely to be important in (neuro)biology and cognition. Since all organisms interact with the physical world, this implies that as neural systems extract information using the senses to guide action in the world, they need appropriately invariant representations that are stable, reproducible and capable of being learned. These invariances are a function of the nature and type of signal, its corruption via noise, and the method of storage and use.
This hypothesis suggests many puzzles and questions: What representational geometries are reflected in the brain? Are they learned or innate? What happens to the invariances under realistic assumptions about noise, nonlinearity and finite computational resources? Can cases of mental disorders and consequences of brain damage be characterized as break downs in representational invariances? Can we harness these invariances and sensory contingencies to build more intelligent machines? The aim is to revisit these old neuro-cognitive problems using a series of modern lenses experimentally, theoretically and computationally, with some tutorials on how the mathematics and engineering of invariant representations in machines and algorithms might serve as useful null models.
In addition to talks, there will be a set of tutorial talks on the mathematical description of invariance (P.J. Olver), the computer vision aspects of invariant algorithms (S. Soatto), and the neuroscientific and cognitive aspects of invariance (TBA). The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by L. Mahadevan (Harvard), Talia Konkle (Harvard), Samuel Gershman (Harvard), and Vivek Jayaraman (HHMI).
Title: Insect cognition: Small tales of geometry & invariance
Abstract: Decades of field and laboratory experiments have allowed ethologists to discover the remarkable sophistication of insect behavior. Over the past couple of decades, physiologists have been able to peek under the hood to uncover sophistication in insect brain dynamics as well. In my talk, I will describe phenomena that relate to the workshop’s theme of geometry and invariance. I will outline how studying insects —and flies in particular— may enable an understanding of the neural mechanisms underlying these intriguing phenomena.
10:00 – 10:45am
Elizabeth Torres
Title: Connecting Cognition and Biophysical Motions Through Geometric Invariants and Motion Variability
Abstract: In the 1930s Nikolai Bernstein defined the degrees of freedom (DoF) problem. He asked how the brain could control abundant DoF and produce consistent solutions, when the internal space of bodily configurations had much higher dimensions than the space defining the purpose(s) of our actions. His question opened two fundamental problems in the field of motor control. One relates to the uniqueness or consistency of a solution to the DoF problem, while the other refers to the characterization of the diverse patterns of variability that such solution produces.
In this talk I present a general geometric solution to Bernstein’s DoF problem and provide empirical evidence for symmetries and invariances that this solution provides during the coordination of complex naturalistic actions. I further introduce fundamentally different patterns of variability that emerge in deliberate vs. spontaneous movements discovered in my lab while studying athletes and dancers performing interactive actions. I here reformulate the DoF problem from the standpoint of the social brain and recast it considering graph theory and network connectivity analyses amenable to study one of the most poignant developmental disorders of our times: Autism Spectrum Disorders.
I offer a new unifying framework to recast dynamic and complex cognitive and social behaviors of the full organism and to characterize biophysical motion patterns during migration of induced pluripotent stem cell colonies on their way to become neurons.
10:45 – 11:15am
Coffee Break
11:15 – 12:00pm
Peter Olver
Title: Symmetry and invariance in cognition — a mathematical perspective”
Abstract: Symmetry recognition and appreciation is fundamental in human cognition. (It is worth speculating as to why this may be so, but that is not my intent.) The goal of these two talks is to survey old and new mathematical perspectives on symmetry and invariance. Applications will arise from art, computer vision, geometry, and beyond, and will include recent work on 2D and 3D jigsaw puzzle assembly and an ongoing collaboration with anthropologists on the analysis and refitting of broken bones. Mathematical prerequisites will be kept to a bare minimum.
12:00 – 12:45pm
Stefano Soatto/Alessandro Achille
Title: Information in the Weights and Emergent Properties of Deep Neural Networks
Abstract: We introduce the notion of information contained in the weights of a Deep Neural Network and show that it can be used to control and describe the training process of DNNs, and can explain how properties, such as invariance to nuisance variability and disentanglement, emerge naturally in the learned representation. Through its dynamics, stochastic gradient descent (SGD) implicitly regularizes the information in the weights, which can then be used to bound the generalization error through the PAC-Bayes bound. Moreover, the information in the weights can be used to defined both a topology and an asymmetric distance in the space of tasks, which can then be used to predict the training time and the performance on a new task given a solution to a pre-training task.
While this information distance models difficulty of transfer in first approximation, we show the existence of non-trivial irreversible dynamics during the initial transient phase of convergence when the network is acquiring information, which makes the approximation fail. This is closely related to critical learning periods in biology, and suggests that studying the initial convergence transient can yield important insight beyond those that can be gleaned from the well-studied asymptotics.
12:45 – 2:00pm
Lunch
2:00 – 2:45pm
Anitha Pasupathy
Title: Invariant and non-invariant representations in mid-level ventral visual cortex
My laboratory investigates how visual form is encoded in area V4, a critical mid-level stage of form processing in the macaque monkey. Our goal is to reveal how V4 representations underlie our ability to segment visual scenes and recognize objects. In my talk I will present results from two experiments that highlight the different strategies used by the visual to achieve these goals. First, most V4 neurons exhibit form tuning that is exquisitely invariant to size and position, properties likely important to support invariant object recognition. On the other hand, form tuning in a majority of neurons is also highly dependent on the interior fill. Interestingly, unlike primate V4 neurons, units in a convolutional neural network trained to recognize objects (AlexNet) overwhelmingly exhibit fill-outline invariance. I will argue that this divergence between real and artificial circuits reflects the importance of local contrast in parsing visual scenes and overall scene understanding.
2:45 – 3:30pm
Jacob Feldman
Title: Bayesian skeleton estimation for shape representation and perceptual organization
Abstract: In this talk I will briefly summarize a framework in which shape representation and perceptual organization are reframed as probabilistic estimation problems. The approach centers around the goal of identifying the skeletal model that best “explains” a given shape. A Bayesian solution to this problem requires identifying a prior over shape skeletons, which penalizes complexity, and a likelihood model, which quantifies how well any particular skeleton model fits the data observed in the image. The maximum-posterior skeletal model thus constitutes the most “rational” interpretation of the image data consistent with the given assumptions. This approach can easily be extended and generalized in a number of ways, allowing a number of traditional problems in perceptual organization to be “probabilized.” I will briefly illustrate several such extensions, including (1) figure/ground and grouping (3) 3D shape and (2) shape similarity.
3:30 – 4:00pm
Tea Break
4:00 – 4:45pm
Moira Dillon
Title: Euclid’s Random Walk: Simulation as a tool for geometric reasoning through development
Abstract: Formal geometry lies at the foundation of millennia of human achievement in domains such as mathematics, science, and art. While formal geometry’s propositions rely on abstract entities like dimensionless points and infinitely long lines, the points and lines of our everyday world all have dimension and are finite. How, then, do we get to abstract geometric thought? In this talk, I will provide evidence that evolutionarily ancient and developmentally precocious sensitivities to the geometry of our everyday world form the foundation of, but also limit, our mathematical reasoning. I will also suggest that successful geometric reasoning may emerge through development when children abandon incorrect, axiomatic-based strategies and come to rely on dynamic simulations of physical entities. While problems in geometry may seem answerable by immediate inference or by deductive proof, human geometric reasoning may instead rely on noisy, dynamic simulations.
4:45 – 5:30pm
Michael McCloskey
Title: Axes and Coordinate Systems in Representing Object Shape and Orientation
Abstract: I describe a theoretical perspective in which a) object shape is represented in an object-centered reference frame constructed around orthogonal axes; and b) object orientation is represented by mapping the object-centered frame onto an extrinsic (egocentric or environment-centered) frame. I first show that this perspective is motivated by, and sheds light on, object orientation errors observed in neurotypical children and adults, and in a remarkable case of impaired orientation perception. I then suggest that orientation errors can be used to address questions concerning how object axes are defined on the basis of object geometry—for example, what aspects of object geometry (e.g., elongation, symmetry, structural centrality of parts) play a role in defining an object principal axis?
5:30 – 6:30pm
Reception
Tuesday, April 16
Time
Speaker
Title/Abstract
8:30 – 9:00am
Breakfast
9:00 – 9:45am
Peter Olver
Title: Symmetry and invariance in cognition — a mathematical perspective”
Abstract: Symmetry recognition and appreciation is fundamental in human cognition. (It is worth speculating as to why this may be so, but that is not my intent.) The goal of these two talks is to survey old and new mathematical perspectives on symmetry and invariance. Applications will arise from art, computer vision, geometry, and beyond, and will include recent work on 2D and 3D jigsaw puzzle assembly and an ongoing collaboration with anthropologists on the analysis and refitting of broken bones. Mathematical pre
9:45 – 10:30am
Stefano Soatto/Alessandro Achille
Title: Information in the Weights and Emergent Properties of Deep Neural Networks
Abstract: We introduce the notion of information contained in the weights of a Deep Neural Network and show that it can be used to control and describe the training process of DNNs, and can explain how properties, such as invariance to nuisance variability and disentanglement, emerge naturally in the learned representation. Through its dynamics, stochastic gradient descent (SGD) implicitly regularizes the information in the weights, which can then be used to bound the generalization error through the PAC-Bayes bound. Moreover, the information in the weights can be used to defined both a topology and an asymmetric distance in the space of tasks, which can then be used to predict the training time and the performance on a new task given a solution to a pre-training task.
While this information distance models difficulty of transfer in first approximation, we show the existence of non-trivial irreversible dynamics during the initial transient phase of convergence when the network is acquiring information, which makes the approximation fail. This is closely related to critical learning periods in biology, and suggests that studying the initial convergence transient can yield important insight beyond those that can be gleaned from the well-studied asymptotics.
10:30 – 11:00am
Coffee Break
11:00 – 11:45am
Jeannette Bohg
Title: On perceptual representations and how they interact with actions and physical representations
Abstract: I will discuss the hypothesis that perception is active and shaped by our task and our expectations on how the world behaves upon physical interaction. Recent approaches in robotics follow this insight that perception is facilitated by physical interaction with the environment. First, interaction creates a rich sensory signal that would otherwise not be present. And second, knowledge of the regularity in the combined space of sensory data and action parameters facilitate the prediction and interpretation of the signal. In this talk, I will present two examples from our previous work where a predictive task facilitates autonomous robot manipulation by biasing the representation of the raw sensory data. I will present results on visual but also haptic data.
11:45 – 12:30pm
Dagmar Sternad
Title: Exploiting the Geometry of the Solution Space to Reduce Sensitivity to Neuromotor Noise
Abstract: Control and coordination of skilled action is frequently examined in isolation as a neuromuscular problem. However, goal-directed actions are guided by information that creates solutions that are defined as a relation between the actor and the environment. We have developed a task-dynamic approach that starts with a physical model of the task and mathematical analysis of the solution spaces for the task. Based on this analysis we can trace how humans develop strategies that meet complex demands by exploiting the geometry of the solution space. Using three interactive tasks – throwing or bouncing a ball and transporting a “cup of coffee” – we show that humans develop skill by: 1) finding noise-tolerant strategies and channeling noise into task-irrelevant dimensions, 2) exploiting solutions with dynamic stability, and 3) optimizing predictability of the object dynamics. These findings are the basis for developing propositions about the controller: complex actions are generated with dynamic primitives, attractors with few invariant types that overcome substantial delays and noise in the neuro-mechanical system.
12:30 – 2:00pm
Lunch
2:00 – 2:45pm
Sam Ocko
Title: Emergent Elasticity in the Neural Code for Space
Abstract: To navigate a novel environment, animals must construct an internal map of space by combining information from two distinct sources: self-motion cues and sensory perception of landmarks. How do known aspects of neural circuit dynamics and synaptic plasticity conspire to construct such internal maps, and how are these maps used to maintain representations of an animal’s position within an environment. We demonstrate analytically how a neural attractor model that combines path integration of self-motion with Hebbian plasticity in synaptic weights from landmark cells can self-organize a consistent internal map of space as the animal explores an environment. Intriguingly, the emergence of this map can be understood as an elastic relaxation process between landmark cells mediated by the attractor network during exploration. Moreover, we verify several experimentally testable predictions of our model, including: (1) systematic deformations of grid cells in irregular environments, (2) path-dependent shifts in grid cells towards the most recently encountered landmark, (3) a dynamical phase transition in which grid cells can break free of landmarks in altered virtual reality environments and (4) the creation of topological defects in grid cells. Taken together, our results conceptually link known biophysical aspects of neurons and synapses to an emergent solution of a fundamental computational problem in navigation, while providing a unified account of disparate experimental observations.
2:45 – 3:30pm
Tatyana Sharpee
Title: Hyperbolic geometry of the olfactory space
Abstract: The sense of smell can be used to avoid poisons or estimate a food’s nutrition content because biochemical reactions create many by-products. Thus, the production of a specific poison by a plant or bacteria will be accompanied by the emission of certain sets of volatile compounds. An animal can therefore judge the presence of poisons in the food by how the food smells. This perspective suggests that the nervous system can classify odors based on statistics of their co-occurrence within natural mixtures rather than from the chemical structures of the ligands themselves. We show that this statistical perspective makes it possible to map odors to points in a hyperbolic space. Hyperbolic coordinates have a long but often underappreciated history of relevance to biology. For example, these coordinates approximate distance between species computed along dendrograms, and more generally between points within hierarchical tree-like networks. We find that both natural odors and human perceptual descriptions of smells can be described using a three-dimensional hyperbolic space. This match in geometries can avoid distortions that would otherwise arise when mapping odors to perception. We identify three axes in the perceptual space that are aligned with odor pleasantness, its molecular boiling point and acidity. Because the perceptual space is curved, one can predict odor pleasantness by knowing the coordinates along the molecular boiling point and acidity axes.
3:30 – 4:00pm
Tea Break
4:00 – 4:45pm
Ed Connor
Title: Representation of solid geometry in object vision cortex
Abstract: There is a fundamental tension in object vision between the 2D nature of retinal images and the 3D nature of physical reality. Studies of object processing in the ventral pathway of primate visual cortex have focused mainly on 2D image information. Our latest results, however, show that representations of 3D geometry predominate even in V4, the first object-specific stage in the ventral pathway. The majority of V4 neurons exhibit strong responses and clear selectivity for solid, 3D shape fragments. These responses are remarkably invariant across radically different image cues for 3D shape: shading, specularity, reflection, refraction, and binocular disparity (stereopsis). In V4 and in subsequent stages of the ventral pathway, solid shape geometry is represented in terms of surface fragments and medial axis fragments. Whole objects are represented by ensembles of neurons signaling the shapes and relative positions of their constituent parts. The neural tuning dimensionality of these representations includes principal surface curvatures and their orientations, surface normal orientation, medial axis orientation, axial curvature, axial topology, and position relative to object center of mass. Thus, the ventral pathway implements a rapid transformation of 2D image data into explicit representations 3D geometry, providing cognitive access to the detailed structure of physical reality.
4:45 – 5:30pm
L. Mahadevan
Title: Simple aspects of geometry and probability in perception
Abstract: Inspired by problems associated with noisy perception, I will discuss two questions: (i) how might we test people’s perception of probability in a geometric context ? (ii) can one construct invariant descriptions of 2D images using simple notions of probabilistic geometry? Along the way, I will highlight other questions that the intertwining of geometry and probability raises in a broader perceptual context.
Wednesday, April 17
Time
Speaker
Title/Abstract
8:30 – 9:00am
Breakfast
9:00 – 9:45am
Gily Ginosar
Title: The 3D geometry of grid cells in flying bats
Abstract: The medial entorhinal cortex (MEC) contains a variety of spatial cells, including grid cells and border cells. In 2D, grid cells fire when the animal passes near the vertices of a 2D spatial lattice (or grid), which is characterized by circular firing-fields separated by fixed distances, and 60 local angles – resulting in a hexagonal structure. Although many animals navigate in 3D space, no studies have examined the 3D volumetric firing of MEC neurons. Here we addressed this by training Egyptian fruit bats to fly in a large room (5.84.62.7m), while we wirelessly recorded single neurons in MEC. We found 3D border cells and 3D head-direction cells, as well as many neurons with multiple spherical firing-fields. 20% of the multi-field neurons were 3D grid cells, exhibiting a narrow distribution of characteristic distances between neighboring fields – but not a perfect 3D global lattice. The 3D grid cells formed a functional continuum with less structured multi-field neurons. Both 3D grid cells and multi-field cells exhibited an anatomical gradient of spatial scale along the dorso-ventral axis of MEC, with inter-field spacing increasing ventrally – similar to 2D grid cells in rodents. We modeled 3D grid cells and multi-field cells as emerging from pairwise-interactions between fields, using an energy potential that induces repulsion at short distances and attraction at long distances. Our analysis shows that the model explains the data significantly better than a random arrangement of fields. Interestingly, simulating the exact same model in 2D yielded a hexagonal-like structure, akin to grid cells in rodents. Together, the experimental data and preliminary modeling suggest that the global property of grid cells is multiple fields that repel each other with a characteristic distance-scale between adjacent fields – which in 2D yields a global hexagonal lattice while in 3D yields only local structure but no global lattice.
(1) Department of Neurobiology, Weizmann Institute of Science, Rehovot 76100, Israel
(2) Department of Bioengineering, Imperial College London, London, SW7 2AZ, UK
(3) The Edmond and Lily Safra Center for Brain Sciences, and Racah Institute of Physics, The Hebrew
University of Jerusalem, Jerusalem, 91904, Israel
9:45 – 10:30am
Sandro Romani
Title: Neural networks for 3D rotations
Abstract: Studies in rodents, bats, and humans have uncovered the existence of neurons that encode the orientation of the head in 3D. Classical theories of the head-direction (HD) system in 2D rely on continuous attractor neural networks, where neurons with similar heading preference excite each other, while inhibiting other HD neurons. Local excitation and long-range inhibition promote the formation of a stable “bump” of activity that maintains a representation of heading. The extension of HD models to 3D is hindered by complications (i) 3D rotations are non-commutative (ii) the space described by all possible rotations of an object has a non-trivial topology. This topology is not captured by standard parametrizations such as Euler angles (e.g. yaw, pitch, roll). For instance, with these parametrizations, a small change of the orientation of the head could result in a dramatic change of neural representation. We used methods from the representation theory of groups to develop neural network models that exhibit patterns of persistent activity of neurons mapped continuously to the group of 3D rotations. I will further discuss how these networks can (i) integrate vestibular inputs to update the representation of heading, and (ii) be used to interpret “mental rotation” experiments in humans.
This is joint work with Hervé Rouault (CENTURI) and Alon Rubin (Weizmann Institute of Science).
10:30 – 11:00am
Coffee Break
11:00 – 11:45am
Sam Gershman
Title: The hippocampus as a predictive map
Abstract: A cognitive map has long been the dominant metaphor for hippocampal function, embracing the idea that place cells encode a geometric representation of space. However, evidence for predictive coding, reward sensitivity and policy dependence in place cells suggests that the representation is not purely spatial. I approach this puzzle from a reinforcement learning perspective: what kind of spatial representation is most useful for maximizing future reward? I show that the answer takes the form of a predictive representation. This representation captures many aspects of place cell responses that fall outside the traditional view of a cognitive map. Furthermore, I argue that entorhinal grid cells encode a low-dimensionality basis set for the predictive representation, useful for suppressing noise in predictions and extracting multiscale structure for hierarchical planning.
11:45 – 12:30pm
Lucia Jacobs
Title: The adaptive geometry of a chemosensor: the origin and function of the vertebrate nose
Abstract: A defining feature of a living organism, from prokaryotes to plants and animals, is the ability to orient to chemicals. The distribution of chemicals, whether in water, air or on land, is used by organisms to locate and exploit spatially distributed resources, such as nutrients and reproductive partners. In animals, the evolution of a nervous system coincided with the evolution of paired chemosensors. In contemporary insects, crustaceans, mollusks and vertebrates, including humans, paired chemosensors confer a stereo olfaction advantage on the animal’s ability to orient in space. Among vertebrates, however, this function faced a new challenge with the invasion of land. Locomotion on land created a new conflict between respiration and spatial olfaction in vertebrates. The need to resolve this conflict could explain the current diversity of vertebrate nose geometries, which could have arisen due to species differences in the demand for stereo olfaction. I will examine this idea in more detail in the order Primates, focusing on Old World primates, in particular, the evolution of an external nose in the genus Homo.
12:30 – 1:30pm
Lunch
1:30 – 2:15pm
Talia Konkle
Title: The shape of things and the organization of object-selective cortex
Abstract: When we look at the world, we effortlessly recognize the objects around us and can bring to mind a wealth of knowledge about their properties. In part 1, I’ll present evidence that neural responses to objects are organized by high-level dimensions of animacy and size, but with underlying neural tuning to mid-level shape features. In part 2, I’ll present evidence that representational structure across much of the visual system has the requisite structure to predict visual behavior. Together, these projects suggest that there is a ubiquitous “shape space” mapped across all of occipitotemporal cortex that underlies our visual object processing capacities. Based on these findings, I’ll speculate that the large-scale spatial topography of these neural responses is critical for pulling explicit content out of a representational geometry.
2:15 – 3:00pm
Vijay Balasubramanian
Title: Becoming what you smell: adaptive sensing in the olfactory system
Abstract: I will argue that the circuit architecture of the early olfactory system provides an adaptive, efficient mechanism for compressing the vast space of odor mixtures into the responses of a small number of sensors. In this view, the olfactory sensory repertoire employs a disordered code to compress a high dimensional olfactory space into a low dimensional receptor response space while preserving distance relations between odors. The resulting representation is dynamically adapted to efficiently encode the changing environment of volatile molecules. I will show that this adaptive combinatorial code can be efficiently decoded by systematically eliminating candidate odorants that bind to silent receptors. The resulting algorithm for “estimation by elimination” can be implemented by a neural network that is remarkably similar to the early olfactory pathway in the brain. The theory predicts a relation between the diversity of olfactory receptors and the sparsity of their responses that matches animals from flies to humans. It also predicts specific deficits in olfactory behavior that should result from optogenetic manipulation of the olfactory bulb.
3:00 – 3:45pm
Ila Feite
Title: Invariance, stability, geometry, and flexibility in spatial navigation circuits
Abstract: I will describe how the geometric invariances or symmetries of the external world are reflected in the symmetries of neural circuits that represent it, using the example of the brain’s networks for spatial navigation. I will discuss how these symmetries enable spatial memory, evidence integration, and robust representation. At the same time, I will discuss how these seemingly rigid circuits with their inscribed symmetries can be harnessed to represent a range of spatial and non-spatial cognitive variables with high flexibility.
Jointly organized by Harvard University, Massachusetts Institute of Technology, and Microsoft Research New England, the Charles River Lectures on Probability and Related Topics is a one-day event for the benefit of the greater Boston area mathematics community.
The 2017 lectures will take place 9:15am – 5:30pm on Monday, October 2 at Harvard University in the Harvard Science Center.
Title: Noise stability of the spectrum of large matrices
Abstract: The spectrum of large non-normal matrices is notoriously sensitive to perturbations, as the example of nilpotent matrices shows. Remarkably, the spectrum of these matrices perturbed by polynomially(in the dimension) vanishing additive noise is remarkably stable. I will describe some results and the beginning of a theory.
The talk is based on joint work with Anirban Basak and Elliot Paquette, and earlier works with Feldheim, Guionnet, Paquette and Wood.
10:20 am – 11:20 am:Andrea Montanari
Title: Algorithms for estimating low-rank matrices
Abstract: Many interesting problems in statistics can be formulated as follows. The signal of interest is a large low-rank matrix with additional structure, and we are given a single noisy view of this matrix. We would like to estimate the low rank signal by taking into account optimally the signal structure. I will discuss two types of efficient estimation procedures based on message-passing algorithms and semidefinite programming relaxations, with an emphasis on asymptotically exact results.
11:20 am – 11:45 am: Break
11:45 am – 12:45 pm:Paul Bourgade
Title: Random matrices, the Riemann zeta function and trees
Abstract: Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on joint works with Arguin, Belius, Radziwill and Soundararajan.
1:00 pm – 2:30 pm: Lunch
In Harvard Science Center Hall E:
2:45 pm – 3:45 pm: Roman Vershynin
Title: Deviations of random matrices and applications
Abstract: Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This lecture will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in high dimensional geometry and data science.
3:45 pm – 4:15 pm: Break
4:15 pm – 5:15 pm:Massimiliano Gubinelli
Title: Weak universality and Singular SPDEs
Abstract: Mesoscopic fluctuations of microscopic (discrete or continuous) dynamics can be described in terms of nonlinear stochastic partial differential equations which are universal: they depend on very few details of the microscopic model. This universality comes at a price: due to the extreme irregular nature of the random field sample paths, these equations turn out to not be well-posed in any classical analytic sense. I will review recent progress in the mathematical understanding of such singular equations and of their (weak) universality and their relation with the Wilsonian renormalisation group framework of theoretical physics.
On September 10-11, 2019, the CMSA will be hosting a second workshop on Topological Aspects of Condensed Matter.
New ideas rooted in topology have recently had a major impact on condensed matter physics, and have led to new connections with high energy physics, mathematics and quantum information theory. The aim of this program will be to deepen these connections and spark new progress by fostering discussion and new collaborations within and across disciplines.
Topics include i) the classification of topological states ii) topological orders in two and three dimensions including quantum spin liquids, quantum Hall states and fracton phases and iii) interplay of symmetry and topology in quantum many body systems, including symmetry protected topological phases, symmetry fractionalization and anomalies iv) topological phenomena in quantum systems driven far from equlibrium v) quantum field theory approaches to topological matter.
On August 27-28, 2018, the CMSA will be hosting a Kickoff workshop on Topology and Quantum Phases of Matter. New ideas rooted in topology have recently had a big impact on condensed matter physics, and have highlighted new connections with high energy physics, mathematics and quantum information theory. Additionally, these ideas have found applications in the design of photonic systems and of materials with novel mechanical properties. The aim of this program will be to deepen these connections by fostering discussion and seeding new collaborations within and across disciplines.
Abstract: A long-standing problem in random graph theory has been to determine asymptotically the length of a longest induced path in sparse random graphs. Independent work of Luczak and Suen from the 90s showed the existence of an induced path of roughly half the optimal size, which seems to be a barrier for certain natural approaches. Recently, in joint work with Draganic and Krivelevich, we solved this problem. In the talk, I will discuss the history of the problem and give an overview of the proof.
Title: The Noether Theorems in Geometry: Then and Now
Abstract: The 1918 Noether theorems were a product of the general search for energy and momentum conservation in Einstein’s newly formulated theory of general relativity. Although widely referred to as the connection between symmetry and conservation laws, the theorems themselves are often not understood properly and hence have not been as widely used as they might be. In the first part of the talk, I outline a brief history of the theorems, explain a bit of the language, translate the first theorem into coordinate invariant language and give a few examples. I will mention only briefly their importance in physics and integrable systems. In the second part of the talk, I describe why they are still relevant in geometric analysis: how they underlie standard techniques and why George Daskalopoulos and I came to be interested in them for our investigation into the best Lipschitz maps of Bill Thurston. Some applications to integrals on a domain a hyperbolic surface leave open possibilities for applications to integrals on domains which are locally symmetric spaces of higher dimension. The talk finishes with an example or two from the literature.
Title: On the solution space of the Ising perceptron model
Abstract: Consider the discrete cube $\{-1,1\}^N$ and a random collection of half spaces which includes each half space $H(x) := \{y \in \{-1,1\}^N: x \cdot y \geq \kappa \sqrt{N}\}$ for $x \in \{-1,1\}^N$ independently with probability $p$. The solution space is the intersection of these half spaces. In this talk, we will talk about its sharp threshold phenomenon, the frozen structure of the solution space, and the Gardner formula.
Title: Black Holes, 2D Gravity, and Random Matrices
Abstract: I will discuss old and new connections between black hole physics, 2D quantum gravity, and random matrix theory. Black holes are believed to be very complicated, strongly interacting quantum mechanical systems, and certain aspects of their Hamiltonians should be well approximated by random matrix theory. The near-horizon effective dynamics of near-extremal black holes is two-dimensional, and many theories of 2D quantum gravity are known to have random matrix descriptions. All of these expectations were recently borne out in surprising detail with the solution of the Jackiw-Teitelboim (JT) model, but this result raises more questions than it answers. If time permits, I will discuss some extensions of these results and possible future directions.
Beginning in Spring 2020, the CMSA will be hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars. Talks will take place throughout the semester. All talks will take place virtually. You must register to attend. Recordings will be posted to this page.
Abstract: Let M_n be drawn uniformly from all n by n symmetric matrices with entries in {-1,1}. In this talk I’ll consider the following basic question: what is the probability that M_n is singular? I’ll discuss recent joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe where we show that this probability is exponentially small. I hope to make the talk accessible to a fairly general audience.
Title: Asymptotic localization, massive fields, and gravitational singularities
Abstract: I will review three recent developments on Einstein’s field equations under low decay or low regularity conditions. First, the Seed-to-Solution Method for Einstein’s constraint equations, introduced in collaboration with T.-C. Nguyen generates asymptotically Euclidean manifolds with the weakest or strongest possible decay (infinite ADM mass, Schwarzschild decay, etc.). The ‘asymptotic localization problem’ is also proposed an alternative to the ‘optimal localization problem’ by Carlotto and Schoen. We solve this new problem at the harmonic level of decay. Second, the Euclidian-Hyperboloidal Foliation Method, introduced in collaboration with Yue Ma, applies to nonlinear wave systems which need not be asymptotically invariant under Minkowski’s scaling field and to solutions with low decay in space. We established the global nonlinear stability of self-gravitating massive matter field in the regime near Minkowski spacetime. Third, in collaboration with Bruno Le Floch and Gabriele Veneziano, I studied spacetimes in the vicinity of singularity hypersurfaces and constructed bouncing cosmological spacetimes of big bang-big crunch type. The notion of singularity scattering map provides a flexible tool for formulating junction conditions and, by analyzing Einstein’s constraint equations, we established a surprising classification of all gravitational bouncing laws. Blog: philippelefloch.org
Title: Threshold phenomena in random graphs and hypergraphs
Abstract: In 1959 Paul Erdos and Alfred Renyi introduced a model of random graphs that is the cornerstone of modern probabilistic combinatorics. Now known as the “Erdos-Renyi” model of random graphs it has far-reaching applications in combinatorics, computer science, and other fields.
The model is defined as follows: Given a natural number $n$ and a parameter $p \in [0,1]$, let $G(n;p)$ be the distribution on graphs with $n$ vertices in which each of the $\binom{n}{2}$ possible edges is present with probability $p$, independent of all others. Despite their apparent simplicity, the study of Erdos-Renyi random graphs has revealed many deep and non-trivial phenomena.
A central feature is the appearance of threshold phenomena: For all monotone properties (e.g., connectivity and Hamiltonicity) there is a critical probability $p_c$ such that if $p >> p_c$ then $G(n;p)$ possesses the property with high probability (i.e., with probability tending to 1 as $n \to \infty$) whereas if $p << p_c$ then with high probability $G(n;p)$ does not possess the property. In this talk we will focus on basic properties such as connectivity and containing a perfect matching. We will see an intriguing connection between these global properties and the local property of having no isolated vertices. We will then generalize the Erdos-Renyi model to higher dimensions where many open problems remain.
Title: Machine Learning the Gravity Equation for International Trade
Abstract: We will go through modern deep learning methods and existing approaches to their interpretation. Next, I will describe a graph neural network framework. You will also be introduced to an economic analog of gravity. Finally, we will see how these tools can help understand observed trade flows between 181 countries over 68 years. [Joint work with Michael R. Douglas.]
Title: Stable Big Bang formation for the Einstein equations
Abstract: I will discuss recent work concerning stability of cosmological singularities described by the generalized Kasner solutions. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be stable under perturbations of the Kasner initial data, as long as the Kasner exponents are “sub-critical”. We prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the full regime where stable monotonic-type curvature blowup is expected. We treat the 3+1-dimensional Einstein-scalar field system and the D+1-dimensional Einstein-vacuum equations for D≥10. This is joint work with Speck and Fournodavlos.
Abstract: Penrose et al. investigated the physical incoherence of the space-time with negative mass via the bending of light. Precise estimates of the time-delay of null geodesics were needed and played a pivotal role in their proof. In this paper, we construct an intermediate diagonal metric and reduce this problem to a causality comparison in the compactified space-time regarding time-like connectedness near conformal infinities. This different approach allows us to avoid encountering the difficulties and subtle issues that Penrose et al. met. It provides a new, substantially simple, and physically natural non-partial differential equation viewpoint to understand the positive mass theorem. This elementary argument modestly applies to asymptotically flat solutions that are vacuum and stationary near infinity
Abstract: The first known construction of mirror pairs of Calabi-Yau manifolds was the Greene-Plesser “quotient and resolve” procedure which applies to pencils of hypersurfaces in projective space. We’ll review this approach, uncover the hints it gives for some more general mirror constructions, and describe a brand-new variant that applies to pencils of hypersurfaces in Grassmannians. This last is joint work with Tom Coates and Elana Kalashnikov (arXiv:2110.0727).