Superconductivity in infinite-layer nickelates
2022-04-20 10:30 - 13:00 (America/New_York)
Abstract: Since its discovery, unconventional superconductivity in cuprates has motivated the search for materials with analogous electronic or atomic structure. We have used soft chemistry approaches to synthesize superconducting infinite layer nickelates from their perovskite precursor phase. We will present the synthesis and transport properties of the nickelates, observation of a doping-dependent superconducting dome, and our current understanding of their electronic and magnetic structure.

Secure Multi-Party Computation: from Theory to Practice
2022-04-21 09:00 - 10:00
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.

Future stability of the $1+3$ Milne model for the Einstein-Klein-Gordon system
2022-04-21 10:00 - 11:00
Abstract: We study the small perturbations of the $1+3$-dimensional Milne model for the Einstein-Klein-Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on the $1+3$ splitting of the Bianchi-Klein-Gordon equations. Moreover, we treat the Bianchi-Klein-Gordon equations as evolution equations and establish the energy scheme in the sense that we only commute the Bianchi-Klein-Gordon equations with spatially covariant derivatives while normal derivative is not allowed. We propose some refined estimates for lapse and the hierarchies of energy estimates to close the energy argument.

CMSA/Tsinghua Math-Science Literature Lecture: Three Introductory Lectures on Game Theory for Mathematicians: Auction Theory
2022-04-22 09:30 - 23:00 | Virtual
Eric Maskin (Harvard University) Three Introductory Lectures on Game Theory for Mathematicians
April 22, 2022 | 9:30 – 11:00 am ET
Title: Auction Theory
Abstract: Equivalences among four standard auctions: the high-bid auction (the high bidder wins and pays her bid); the second-bid auction (the high bidder wins and pays the second-highest bid); the Dutch auction (the auctioneer lowers the price successively until some bidder is willing to pay); and the English auction (bidders raise their bids successively until no one wants to bid higher).
Talk chairs: Scott Kominers, Sergiy Verstyuk
SLIDES | VIDEO Answers to Questions from Talks 2 and 3

Higgs = SPT
2022-04-22 15:30 - 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Ruben Verresen
Title: Higgs = SPT
Abstract: The Higgs phase of a gauge theory is important to both fundamental physics (e.g., electroweak theory) as well as condensed matter systems (superconductors and other emergent phenomena). However, such a charge condensate seems subtle and is sometimes described as the spontaneous breaking of gauge symmetry (or a global subgroup). In this talk, I will argue that the Higgs phase is best understood as a symmetry-protected topological (SPT) phase. The concept of SPT phases arose out of the condensed matter community, to describe systems with short-range entanglement and edge modes which cannot be removed in the presence of certain symmetries. The perspective that the Higgs phase is an SPT phase recovers known properties of the Higgs phase and provides new insights. In particular, we revisit the Fradkin-Shenker model and the distinction between the Higgs and confined phases of a gauge theory.

Algebraic Statistics with a View towards Physics
2022-04-26 09:00 - 10:00
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.
LINK TO FORM

Modularity of mirror families of log Calabi–Yau surfaces
2022-04-26 09:30 - 10:30 | Virtual
Abstract: In “Mirror symmetry for log Calabi–Yau surfaces I,” given a smooth log Calabi–Yau surface pair (Y,D), Gross–Hacking–Keel constructed its mirror family as the spectrum of an explicit algebra whose structure coefficients are determined by the enumerative geometry of (Y,D). As a follow-up of the work of Gross–Hacking–Keel, when (Y,D) is positive, we prove the modularity of the mirror family as the universal family of log Calabi-Yau surface pairs deformation equivalent to (Y,D) with at worst du Val singularities. As a corollary, we show that the ring of regular functions of a smooth affine log Calabi–Yau surface has a canonical basis of theta functions. The key step towards the proof of the main theorem is the application of the tropical construction of singular cycles and explicit formulas of period integrals given in the work of Helge–Siebert. This is joint work with Jonathan Lai.

Workshop on Nonlinear Algebra and Combinatorics from Physics
2022-04-27 09:00 - 2022-04-29 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
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.
Speakers:

Federico Ardila (San Francisco State)
Nima Arkani-Hamed (IAS)
Madeline Brandt (Brown)
Nick Early (Max Planck Institute)
Chris Eur (Harvard)
Claudia Fevola (Max Planck Institute)
Christian Gaetz (Harvard)
Yuji Kodama (Ohio State University)
Yelena Mandelshtam (Berkeley)
Sebastian Mizera (IAS)
Matteo Parisi (Harvard CMSA)
Emma Previato (Boston University)
Anna Seigal (Harvard)
Melissa Sherman-Bennett (University of Michigan)
Simon Telen (Max Planck Institute)
Charles Wang (Harvard)

Schedule
Wednesday, April 27, 2022

9:30 am–10:30 am
Federico Ardila
Title: Nonlinear spaces from linear spaces
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.

6:00 pm

Dinner Banquet, Gran Gusto Trattoria

Friday, April 29, 2022

9:00 am–10:00 am
Yuji Kodama
Title: KP solitons and algebraic curves
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

Long common subsequences between bit-strings and the zero-rate threshold of deletion-correcting codes
2022-04-27 09:30 - 10:30
Speaker: Venkatesan Guruswami, UC Berkeley
Title: Long common subsequences between bit-strings and the zero-rate threshold of deletion-correcting codes
Abstract: Suppose we transmit n bits on a noisy channel that deletes some fraction of the bits arbitrarily. What’s the supremum p* of deletion fractions that can be corrected with a binary code of non-vanishing rate? Evidently p* is at most 1/2 as the adversary can delete all occurrences of the minority bit. It was unknown whether this simple upper bound could be improved, or one could in fact correct deletion fractions approaching 1/2.
We show that there exist absolute constants A and delta > 0 such that any subset of n-bit strings of size exp((log n)^A) must contain two strings with a common subsequence of length (1/2+delta)n. This immediately implies that the zero-rate threshold p* of worst-case bit deletions is bounded away from 1/2.
Our techniques include string regularity arguments and a structural lemma that classifies bit-strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence.
This is joint work with Xiaoyu He and Ray Li.

Aspects of 4d supersymmetric dynamics and geometry
2022-04-28 10:30 - 12:00 | Virtual
Abstract: We will overview the program of geometrically engineering four dimensional supersymmetric QFTs as compactifications of six dimensional SCFTs. In particular we will discuss how strong coupling phenomena in four dimensions, such as duality and emergence of symmetry, can be better understood in such geometric constructions.

Building active nematic and active polar liquids out of biological machines
2022-04-28 13:00 - 14:30 | Virtual
Speaker: Guillaume Duclos (Brandeis)

Title: Building active nematic and active polar liquids out of biological machines
Abstract: Active matter describes out-of-equilibrium materials composed of motile building blocks that convert free energy into mechanical work. The continuous input of energy at the particle scale liberates these systems from the constraints of thermodynamic equilibrium, leading to emergent collective behaviors not found in passive materials. In this talk, I will describe our recent efforts to build simple active systems composed of purified proteins and identify generic emergent behaviors in active systems. I will first discuss two distinct activity-driven instabilities in suspensions of microtubules and molecular motors. Second, I will describe a new model system for polar fluid whose collective dynamics are driven by the non-equilibrium turnover of actin filaments. Our results illustrate how biomimetic materials can serve as a platform for studying non-equilibrium statistical mechanics, as well as shine light on the physical mechanisms that regulate self-organization in living matter.

Video (Youtube)

Intersection number and systole on hyperbolic surfaces
2022-04-28 15:33 - 17:33
Abstract: Let X be a compact hyperbolic surface. We can see that there is a constant C(X) such that the intersection number of the closed geodesics is leq C(X) times the product of their lengths. Consider the optimum constant C(X). In this talk, we describe its asymptotic behavior in terms of systole, length of the shortest closed geodesic on X.

A new proof for the nonlinear stability of slowly-rotating Kerr-de Sitter
2022-04-28 15:35 - 16:35
Abstract: The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vectorfield techniques to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.

Machine Learning the Gravity Equation for International Trade
2022-04-29 09:30 - 11:00 | Virtual
Member Seminar
Speaker: Sergiy Verstyuk
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.]

General Relativity Workshop
2022-05-02 09:00 - 2022-05-05 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Workshop on scalar curvature, minimal surfaces, and initial data sets
Dates: May 2–5, 2022
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA 02138 and via Zoom webinar.
Advanced registration for in-person components is required.
Organizers: Dan Lee (CMSA/CUNY), Martin Lesourd (CMSA/BHI), and Lan-Hsuan Huang (University of Connecticut).
Speakers:

Zhongshan An, University of Connecticut
Paula Burkhardt-Guim, NYU
Hyun Chul Jang, University of Miami
Chao Li, NYU
Christos Mantoulidis, Rice University
Robin Neumayer, Carnegie Mellon University
Andre Neves, University of Chicago
Tristan Ozuch, MIT
Annachiara Piubello, University of Miami
Antoine Song, UC Berkeley
Tin-Yau Tsang, UC Irvine
Ryan Unger, Princeton
Zhizhang Xie, Texas A & M
Xin Zhou, Cornell University
Jonathan Zhu, Princeton University

Schedule
Monday, May 2, 2022

9:30–10:30 am
Hyun Chul Jang
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 threshold for stacked triangulations
2022-05-03 09:30 - 10:30 | Hybrid
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.
Joint work with Eyal Lubetzky.

Qianfang: a type-safe and data-driven healthcare system starting from Traditional Chinese Medicine
2022-05-05 15:36 - 17:36
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.

2022 NSF FRG Workshop on Discrete Shapes
2022-05-06 10:00 - 2022-05-08 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
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.

Workshop on Discrete Shapes
May 6–8, 2022
Organizers:

David Glickenstein (University of Arizona)
Joel Hass (University of California, Davis)
Patrice Koehl (University of California, Davis)
Feng Luo (Rutgers University, New Brunswick)
Maria Trnkova (University of California, Davis)
Shing-Tung Yau (Harvard)

Speakers:

Miri Ben-Chen (Technion)
Alexander Bobenko (TU Berlin)
John Bowers (James Madison)
Steven Gortler (Harvard)
David Gu (Stony Brook)
Anil Hirani (UIUC)
Yanwen Luo (Rutgers)
Peter Schroeder (Caltech)
Justin Solomon (MIT)
Tianqi Wu (Clark University)

Contributed Talk Speakers:

Oded Stein (MIT)
Bohan Zhou (Dartmouth)

Schedule
Schedule (PDF)
Friday, May 6, 2022

10:00–10:05 am

Welcome Opening

10:05–10:55 am
Anil N. Hirani
Title: Discrete vector bundles with connection
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.

11:10–12:00 pm

Problems and Application discussions

12:00–1:00 pm
Movie
“The Discrete Charm of Geometry”

Conference in Memory of Professor Masatake Kuranishi
2022-05-09 09:00 - 2022-05-12 12:30 | Science and Engineering Complex (SEC), 150 Western Ave, Allston, MA 02134, MA
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)
Videos are available on the conference playlist.

Speakers:
Charles Fefferman (Princeton University)
Teng Fei (Rutgers University)
Robert Friedman (Columbia University)
Kenji Fukaya (Simons Center, Stony Brook)
Akito Futaki (Tsinghua University)
Victor Guillemin (Massachusetts Institute of Technology)
Nigel Hitchin (Oxford University)
Blaine Lawson (Stony Brook University)
Yu-Shen Lin (Boston University)
Melissa C.C. Liu (Columbia University)
Takeo Ohsawa (Nagoya University)
Duong H. Phong (Columbia University)
Sebastien Picard (University of British Columbia)
Paul Seidel (Massachusetts Institute of Technology)
Gabor Szekelyhidi (University of Notre Dame)
Claire Voisin (Institut de Mathematiques, Jussieu, France)
Shing-Tung Yau (Harvard University)

Schedule (download pdf)

Monday, May 9, 2022

8:15 am
Light breakfast & coffee/tea

8:45–9:00 am
Opening Remarks

9:00–10:00 am
Kenji Fukaya
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.
Video

10:15–11:15 am
Nigel Hitchin (Zoom)
Title: Deformations: A personal perspective
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.
Video

11:30–12:30 pm
H. Blaine Lawson
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|
Video

12:30–2:30 pm
Lunch Break

2:30–3:30 pm
Gabor Szekelyhidi
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.
Video

3:30–4:00 pm
Coffee Break

4:00–5:00 pm
Takeo Ohsawa
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.
Video

Tuesday, May 10, 2022

8:15 am
Light breakfast & coffee/tea

9:00–10:00 am
Charles Fefferman (Zoom)
Title: Interpolation of Data by Smooth Functions
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.
Video

10:15–11:15 am
Claire Voisin
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.
Video

11:30– 2:30 pm
Victor Guillemin
Title: Semi-Classical Functions of Isotropic Type
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.
Video

12:30–2:30 pm
Lunch Break

2:30–3:30 pm
Teng Fei
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.
Video

Speakers Banquet

Wednesday, May 11, 2022

8:15 am
Light breakfast & coffee/tea

9:00–10:00 am
Shing-Tung Yau (Zoom)
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.
Video

10:15–11:15 am
Duong H. Phong
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.
Video

11:30–2:30 pm
Paul Seidel
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.
Video

12:30–2:30 pm
Lunch Break

2:30–3:30 pm
Melissa C.C. Liu
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.
Video

3:30–4:00 pm
Coffee Break

4:00–5:00 pm
Sebastien Picard
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.
Video

Thursday, May 12, 2022

8:15 am
Light breakfast & coffee/tea

9:00 am–10:00 am
Akito Futaki (Zoom)
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.
Video

11:30 am – 12:30 pm
Robert Friedman
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.
Video

12:30 pm
Concluding Remarks

Inflation and light Dark Matter constraints from the Swampland
2022-05-09 13:00 - 14:00 | Virtual
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.

Cosmology from the vacuum
2022-05-11 10:30 - 12:00
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.

Oblique Lessons from the W Mass Measurement at CDF II
2022-05-12 10:30 - 12:00
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.

Geometric Models for Sets of Probability Measures
2022-05-12 15:38 - 17:38
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.

Cobordism and Deformation Class of the Standard Model and Beyond: Proton Stability and Neutrino Mass
2022-05-13 09:30 - 11:00
Member Seminar
Speaker: Juven Wang
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.

SMaSH: Symposium for Mathematical Sciences at Harvard
2022-05-17 09:00 - 18:00 | Science and Engineering Complex (SEC), 150 Western Ave, Allston, MA 02134, MA
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)
Xihong Lin, Biostatistics (HSPH) & Statistics (FAS)
Yue Lu, Electrical Engineering (SEAS)
Susan Murphy, Statistics (FAS) & Computer Science (SEAS)
Lisa Randall, Physics (SEAS)
Eugene Shakhnovich, Chemistry (FAS)
Salil Vadhan, Computer Science (SEAS)
Horng-Tzer Yau, Mathematics (FAS)

This event was held in-person at the Science and Engineering Complex (SEC) at 150 Western Ave, Allston, MA 02134, and streamed on Zoom.
Harvard graduate students and postdocs presented Poster Sessions.

Venue: Science and Engineering Complex (SEC)

Speakers

Anurag Anshu, Computer Science (SEAS)
Morgane Austern, Statistics (FAS)
Demba Ba, Electrical Engineering & Bioengineering (SEAS)
Michael Brenner, Applied Mathematics (SEAS)
Rui Duan, Biostatistics (HSPH)
Yannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS)
Kosuke Imai, Government & Statistics (FAS)
Sham M. Kakade, Computer Science (SEAS) & Statistics (FAS)
Seth Neel, Technology & Operations Management (HBS)
Melanie Matchett Wood, Mathematics (FAS)

Schedule PDF
Schedule

9:00–9:30 am
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.

Demba Ba, Electrical Engineering & Bioengineering (SEAS)
Title: Geometry, AI, and the Brain
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.

Seth Neel, Technology & Operations Management (HBS)
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.

Hypergraph Matchings Avoiding Forbidden Submatchings
2022-05-17 09:30 - 10:30
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.

Statistical Mechanics of Mutilated Sheets and Shells
2022-05-18 09:30 - 10:30
Speaker: David Nelson, Harvard University
Title: Statistical Mechanics of Mutilated Sheets and Shells
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.

The Generalized Landau Paradigm (a review of generalized symmetries in condensed matter)
2022-05-18 16:00 - 17:30 | Virtual
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

Boundary conditions and LSM anomalies of conformal field theories in 1+1 dimensions
2022-05-18 16:00 - 17:30 | Virtual
Speaker: Linhao Li (ISSP, U Tokyo)
Title: Boundary conditions and LSM anomalies of conformal field theories in 1+1 dimensions
Abstract: In this talk, we will study a relationship between conformally invariant boundary conditions and anomalies of conformal field theories (CFTs) in 1+1 dimensions. For a given CFT with a global symmetry, we consider symmetric gapping potentials which are relevant perturbations to the CFT. If a gapping potential is introduced only in a subregion of the system, it provides a certain boundary condition to the CFT. From this equivalence, if there exists a Cardy boundary state which is invariant under a symmetry, then the CFT can be gapped with a unique ground state by adding the corresponding gapping potential. This means that the symmetry of the CFT is anomaly free. Using this approach, we will systematically deduce the anomaly-free conditions for various types of CFTs with several different symmetries. When the symmetry of the CFT is anomalous, it implies a Lieb-Schultz-Mattis type ingappability of the system. Our results are consistent with, where available, known results in the literature. Moreover, we extend the discussion to other symmetries including spin groups and generalized time-reversal symmetries. As an application, we propose 1d LSM theorem involving magnetic space group symmetries on the lattice. The extended LSM theorems apply to systems with a broader class of spin interactions, such as Dzyaloshinskii-Moriya interactions and chiral three-spin interactions.

The geometry of conditional independence models with hidden variables
2022-05-19 09:00 - 10:00
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.
The papers are available on arxiv: https://arxiv.org/pdf/2011.02450
and https://arxiv.org/pdf/2103.16550.pdf
