CMSA Math-Science Literature Lecture: Homological (homotopical) algebra and moduli spaces in Topological Field theories
2021-02-23 09:00 - 10:30 (America/New_York) | Virtual
Kenji Fukaya (Simons Center for Geometry and Physics)
Title: Homological (homotopical) algebra and moduli spaces in Topological Field theories
Abstract: Moduli spaces of various gauge theory equations and of various versions of (pseudo) holomorphic curve equations have played important role in geometry in these 40 years. Started with Floer’s work people start to obtain more sophisticated object such as groups, rings, or categories from (system of) moduli spaces. I would like to survey some of those works and the methods to study family of moduli spaces systematically.
Talk chair: Peter Kronheimer
Slides | Video

CMSA Math-Science Literature Lecture: On the History of quantum cohomology and homological mirror symmetry
2021-03-30 09:00 - 22:30 | Virtual
Maxim Kontsevich (IHÉS)
Title: On the History of quantum cohomology and homological mirror symmetry
Abstract: About 30 years ago, string theorists made remarkable discoveries of hidden structures in algebraic geometry. First, the usual cup-product on the cohomology of a complex projective variety admits a canonical multi-parameter deformation to so-called quantum product, satisfying a nice system of differential equations (WDVV equations). The second discovery, even more striking, is Mirror Symmetry, a duality between families of Calabi-Yau varieties acting as a mirror reflection on the Hodge diamond.
Later it was realized that the quantum product belongs to the realm of symplectic geometry, and a half of mirror symmetry (called Homological Mirror Symmetry) is a duality between complex algebraic and symplectic varieties. The search of correct definitions and possible generalizations lead to great advances in many domains, giving mathematicians new glasses, through which they can see familiar objects in a completely new way.
I will review the history of major mathematical advances in the subject of HMS, and the swirl of ideas around it.
Talk chair: Paul Seidel
Video

CMSA Math-Science Literature Lecture: Isadore Singer’s Work on Analytic Torsion
2021-04-06 09:00 - 10:30 | Virtual
Edward Witten (IAS)
Title: Isadore Singer’s Work on Analytic Torsion
Abstract: I will review two famous papers of Ray and Singer on analytic torsion written approximately half a century ago. Then I will sketch the influence of analytic torsion in a variety of areas of physics including anomalies, topological field theory, and string theory.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer.
Talk chair: Cumrun Vafa
Slides | Video

Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer
2021-04-06 09:00 - 2021-04-27 10:30 | Virtual
In 2021, the CMSA hosted a lecture series on the literature of the mathematical sciences. This series highlights significant accomplishments in the intersection between mathematics and the sciences. Speakers include Edward Witten, Lydia Bieri, Simon Donaldson, Michael Freedman, Dan Freed, and many more.
Videos of these talks can be found in this Youtube playlist.

In the Spring 2021 semester, the CMSA hosted a sub-program on this series titled A Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer. Below is the schedule for talks in that subprogram
April 6, 2021 | 9:00 – 10:30am ET

Edward Witten (IAS)
Title: Isadore Singer’s Work on Analytic Torsion

April 13, 2021 | 9:00 – 10:30am ET

Claire Voisin (College de France)
Title: K-theory and characteristic classes in topology and complex geometry (a tribute to Atiyah and Hirzebruch)

April 20, 2021 | 9:00 – 10:30am ET

Dan Freed (the University of Texas at Austin)
Title: The Atiyah-Singer Index Theorem

April 27, 2021 | 9:00 – 10:30am ET

Frances Kirwan (University of Oxford)
Title: Moment maps and the Yang-Mills functional

CMSA Math-Science Literature Lecture: Quantum error correcting codes and fault tolerance
2021-04-08 09:00 - 10:30 | Virtual
Peter Shor (MIT)
Title: Quantum error correcting codes and fault tolerance
Abstract: We will go over the fundamentals of quantum error correction and fault tolerance and survey some of the recent developments in the field.

Talk chair: Zhengwei Liu
Video

CMSA Math-Science Literature Lecture: Deep Networks from First Principles
2021-04-16 13:00 - 14:30 | Virtual
Yi MaPhoto Copyright Noah Berger / 2019

Yi Ma (University of California, Berkeley)
Title: Deep Networks from First Principles
Abstract: In this talk, we offer an entirely “white box’’ interpretation of deep (convolution) networks from the perspective of data compression (and group invariance). In particular, we show how modern deep layered architectures, linear (convolution) operators and nonlinear activations, and even all parameters can be derived from the principle of maximizing rate reduction (with group invariance). All layers, operators, and parameters of the network are explicitly constructed via forward propagation, instead of learned via back propagation. All components of so-obtained network, called ReduNet, have precise optimization, geometric, and statistical interpretation. There are also several nice surprises from this principled approach: it reveals a fundamental tradeoff between invariance and sparsity for class separability; it reveals a fundamental connection between deep networks and Fourier transform for group invariance – the computational advantage in the spectral domain (why spiking neurons?); this approach also clarifies the mathematical role of forward propagation (optimization) and backward propagation (variation). In particular, the so-obtained ReduNet is amenable to fine-tuning via both forward and backward (stochastic) propagation, both for optimizing the same objective. This is joint work with students Yaodong Yu, Ryan Chan, Haozhi Qi of Berkeley, Dr. Chong You now at Google Research, and Professor John Wright of Columbia University.
Talk chair: Harry Shum
Slides | Video

CMSA Math-Science Literature Lecture: The Atiyah-Singer Index Theorem
2021-04-20 09:00 - 10:30 | Virtual
Dan Freed (The University of Texas at Austin)
Title: The Atiyah-Singer Index Theorem
Abstract: The story of the index theorem ties together the Gang of Four—Atiyah, Bott, Hirzebruch, and Singer—and lies at the intersection of analysis, geometry, and topology. In the first part of the talk I will recount high points in the early developments. Then I turn to subsequent variations and applications. Throughout I emphasize the role of the Dirac operator.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
Talk chair: Cumrun Vafa
Video

CMSA Math-Science Literature Lecture: Indistinguishability Obfuscation: How to Hide Secrets within Software
2021-04-23 16:00 - 18:30 | Virtual
Amit Sahai (UCLA)
Title: Indistinguishability Obfuscation: How to Hide Secrets within Software
Abstract: At least since the initial public proposal of public-key cryptography based on computational hardness conjectures (Diffie and Hellman, 1976), cryptographers have contemplated the possibility of a “one-way compiler” that translates computer programs into “incomprehensible” but equivalent forms. And yet, the search for such a “one-way compiler” remained elusive for decades.
In this talk, we look back at our community’s attempts to formalize the notion of such a compiler, culminating in our 2001 work with Barak, Goldreich, Impagliazzo, Rudich, Vadhan, and Yang, which proposed the notion of indistinguishability obfuscation (iO). Roughly speaking, iO requires that the compiled versions of any two equivalent programs (with the same size and running time) be indistinguishable to any efficient adversary. Leveraging the notion of punctured programming, introduced in our work with Waters in 2013, well over a hundred papers have explored the remarkable power of iO.
We’ll then discuss the intense effort that recently culminated in our 2020 work with Jain and Lin, finally showing how to construct iO in such a way that, for the first time, we can prove the security of our iO scheme based on well-studied computational hardness conjectures in cryptography.
Talk chair: Sergiy Verstyuk
Video

CMSA Math-Science Literature Lecture: Moment maps and the Yang-Mills functional
2021-04-27 09:00 - 10:30 | Virtual
Frances Kirwan (University of Oxford)
Title: Moment maps and the Yang-Mills functional
Abstract: In the early 1980s Michael Atiyah and Raoul Bott wrote two influential papers, ‘The Yang-Mills equations over Riemann surfaces’ and ‘The moment map and equivariant cohomology’, bringing together ideas ranging from algebraic and symplectic geometry through algebraic topology to mathematical physics and number theory. The aim of this talk is to explain their key insights and some of the new directions towards which these papers led.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
Talk chair: Peter Kronheimer
Video

Computational Biology Symposium
2021-05-03 10:00 - 16:00 | 20 Garden Street, Cambridge, MA 02138, MA, MA, 02138, United States
On May 3, 2021 the CMSA will be hosting a Computational Biology Symposium virtually on Zoom. This symposium will be organized by Vijay Kuchroo.
The symposium will begin at 10:00am ET. There will be a morning and afternoon session, with an hour break for lunch.
Videos of the talks can be found in this Youtube playlist. Links are also available in the schedule below.
Confirmed participants:

Uri Alon, Weizmann Institute
Elana Fertig, Johns Hopkins
Martin Hemberg, Brigham and Women’s Hospital
Peter Kharchenko, Harvard University
Smita Krishnaswamy, Yale University
John Marioni, EMBL-EBI
Eran Segal, Weizmann Institute
Meromit Singer, Harvard Medical School

Schedule:
PDF of the schedule Download

FRG Workshop on Geometric Methods for Analyzing Discrete Shapes
2021-05-07 11:00 - 2021-05-09 20:45 | Virtual
This workshop will take place May 7-9 (Friday-Sunday), 2021 virtually on Zoom
The aim of the workshop is to bring together a community of researchers in mathematics, computer science, and data science who develop theoretical and computational models to characterize shapes and analysis of image data.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The first half of the workshop will feature talks aimed at graduate students, newcomers, and a broad spectrum of audiences. Christopher Bishop (Stony Brook) and Keenan Crane (Carnegie Mellon) will each give two featured talks. The remaining part will have both background and research talks. There will also be organized discussions of open problems and potential applications.
For the discussions, we are soliciting open problems in mathematical theory and applications of shape analysis. You are encouraged to post problems by sending an email to geometricproblemsfrg@gmail.com.
We invite junior researchers to present a short talk in the workshop. The session will be held on Friday, May 7th or Saturday, May 8th at 4pm and are expected to be 15-20 minutes in length. It is a great opportunity to share your work and get to know others at the workshop. Depending on the number of contributed talks, the organizers will review the submissions and let you know if you have been selected. If you are interested please send your title and abstract to tianqi@cmsa.fas.harvard.edu by the end of May 2nd.

Organizers:

David Glickenstein, University of Arizona
Joel Hass, University of California, Davis
Patrice Koehl, University of California, Davis
Feng Luo, Rutgers University, New Brunswick
Tianqi Wu, Harvard University
Shing-Tung Yau, Harvard University

Featured lectures:

Christopher Bishop, Stony Brook
Keenan Crane, Carnegie Mellon

Speakers include:

Miri Ben-Chen, Technion – Israel Institute of Technology
Alexander Bobenko, Technische Universität Berlin, Germany
Ulrike Buecking, Free University, Germany
Nadav Dym, Duke University
Ivan Izmestiev, Vienna University of Technology
Yanwen Luo, Rutgers
Stephan Tillmann, The University of Sydney
Max Wardetzky, University of Goettingen
Xu Xu, Wuhan University

CMSA Math-Science Literature Lecture: Nonlinear stability of Kerr black holes for small angular momentum
2021-06-15 11:00 - 12:30 | Virtual
Sergiu Klainerman (Princeton University)
Title: Nonlinear stability of Kerr black holes for small angular momentum
Abstract: According to a well-known conjecture, initial data sets, for the Einstein vacuum equations, sufficiently close to a Kerr solution with parameters $a, m$, $|a|/m <1$, have maximal developments with complete future null infinity and with domain of outer communication (i.e complement of a future event horizon) which approaches (globally) a nearby Kerr solution. I will describe the main ideas in my recent joint work with Jeremie Szeftel concerning the resolution of the conjecture for small angular momentum, i.e. $, $|a|/m $ sufficiently small. The work, ArXiv:2104.11857v1, also depends on forthcoming work on solutions of nonlinear wave equations in realistic perturbations of Kerr, with Szeftel and Elena Giorgi, which I will also describe.
Talk chair: Lydia Bieri
Video

Big Data Conference 2021
2021-08-24 | Virtual
On August 24, 2021, the CMSA hosted our seventh annual Conference on Big Data. The Conference features many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The 2021 Big Data Conference took place virtually on Zoom.
Organizers:

Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University
Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business
Horng-Tzer Yau, Professor of Mathematics, Harvard University
Sergiy Verstyuk, CMSA, Harvard University

Speakers:

Andrew Blumberg, University of Texas at Austin
Moran Koren, Harvard CMSA
Hima Lakkaraju, Harvard University
Katrina Ligett, The Hebrew University of Jerusalem

Time (ET; Boston time)
Speaker
Title/Abstract

9:00AM
Conference Organizers
Introduction and Welcome

9:10AM – 9:55AM
Andrew Blumberg, University of Texas at Austin
Title: Robustness and stability for multidimensional persistent homology
Abstract: A basic principle in topological data analysis is to study the shape of data by looking at multiscale homological invariants. The idea is to filter the data using a scale parameter that reflects feature size. However, for many data sets, it is very natural to consider multiple filtrations, for example coming from feature scale and density. A key question that arises is how such invariants behave with respect to noise and outliers. This talk will describe a framework for understanding those questions and explore open problems in the area.

10:00AM – 10:45AM
Katrina Ligett, The Hebrew University of Jerusalem
Title: Privacy as Stability, for Generalization
Abstract: Many data analysis pipelines are adaptive: the choice of which analysis to run next depends on the outcome of previous analyses. Common examples include variable selection for regression problems and hyper-parameter optimization in large-scale machine learning problems: in both cases, common practice involves repeatedly evaluating a series of models on the same dataset. Unfortunately, this kind of adaptive re-use of data invalidates many traditional methods of avoiding overfitting and false discovery, and has been blamed in part for the recent flood of non-reproducible findings in the empirical sciences. An exciting line of work beginning with Dwork et al. in 2015 establishes the first formal model and first algorithmic results providing a general approach to mitigating the harms of adaptivity, via a connection to the notion of differential privacy. In this talk, we’ll explore the notion of differential privacy and gain some understanding of how and why it provides protection against adaptivity-driven overfitting. Many interesting questions in this space remain open.
Joint work with: Christopher Jung (UPenn), Seth Neel (Harvard), Aaron Roth (UPenn), Saeed Sharifi-Malvajerdi (UPenn), and Moshe Shenfeld (HUJI). This talk will draw on work that appeared at NeurIPS 2019 and ITCS 2020

10:50AM – 11:35AM
Hima Lakkaraju, Harvard University
Title: Towards Reliable and Robust Model Explanations
Abstract: As machine learning black boxes are increasingly being deployed in domains such as healthcare and criminal justice, there is growing emphasis on building tools and techniques for explaining these black boxes in an interpretable manner. Such explanations are being leveraged by domain experts to diagnose systematic errors and underlying biases of black boxes. In this talk, I will present some of our recent research that sheds light on the vulnerabilities of popular post hoc explanation techniques such as LIME and SHAP, and also introduce novel methods to address some of these vulnerabilities. More specifically, I will first demonstrate that these methods are brittle, unstable, and are vulnerable to a variety of adversarial attacks. Then, I will discuss two solutions to address some of the vulnerabilities of these methods – (i) a framework based on adversarial training that is designed to make post hoc explanations more stable and robust to shifts in the underlying data; (ii) a Bayesian framework that captures the uncertainty associated with post hoc explanations and in turn allows us to generate explanations with user specified levels of confidences. I will conclude the talk by discussing results from real world datasets to both demonstrate the vulnerabilities in post hoc explanation techniques as well as the efficacy of our aforementioned solutions.

11:40AM – 12:25PM
Moran Koren, Harvard CMSA
Title: A Gatekeeper’s Conundrum
Abstract: Many selection processes contain a “gatekeeper”. The gatekeeper’s goal is to examine an applicant’s suitability to a proposed position before both parties endure substantial costs. Intuitively, the introduction of a gatekeeper should reduce selection costs as unlikely applicants are sifted out. However, we show that this is not always the case as the gatekeeper’s introduction inadvertently reduces the applicant’s expected costs and thus interferes with her self-selection. We study the conditions under which the gatekeeper’s presence improves the system’s efficiency and those conditions under which the gatekeeper’s presence induces inefficiency. Additionally, we show that the gatekeeper can sometimes improve selection correctness by behaving strategically (i.e., ignore her private information with some probability).

12:25PM
Conference Organizers
Closing Remarks

Math Science Lectures in Honor of Raoul Bott
2021-10-04 02:51 - 2021-10-05 02:51 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On October 4th and October 5th, 2021, Harvard CMSA will host its annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker will be Michael Freedman (Microsoft). The lectures will take place from 11:00am – 12:15pm (ET) on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.

Lecture 1
October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Video

Lecture 2
October 5th, 11:00am (Boston time)
Title: Controlled Mather Thurston Theorems.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Video

Math Science Lectures in Honor of Raoul Bott: Michael Freedman
2021-10-04 11:00 - 12:30 | Virtual
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.
This will be the third annual lecture series held in honor of Raoul Bott.

Lecture 1
October 4th, 11:00am (Boston time)
Title: The Universe from a single Particle
Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Video

Lecture 2
October 5th, 11:00am (Boston time)
Title: Controlled Mather Thurston Theorems.
Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Video

CMSA Math-Science Literature Lecture - Karen Uhlenbeck
2021-12-09 09:30 - 14:30 | Virtual
Karen Uhlenbeck (Institute for Advanced Study)
Title: The Noether Theorems in Geometry: Then and Now
Abstract: The 1918 Noether theorems were a product of the general search for energy and momentum conservation in Einstein’s newly formulated theory of general relativity. Although widely referred to as the connection between symmetry and conservation laws, the theorems themselves are often not understood properly and hence have not been as widely used as they might be. In the first part of the talk, I outline a brief history of the theorems, explain a bit of the language, translate the first theorem into coordinate invariant language and give a few examples. I will mention only briefly their importance in physics and integrable systems. In the second part of the talk, I describe why they are still relevant in geometric analysis: how they underlie standard techniques and why George Daskalopoulos and I came to be interested in them for our investigation into the best Lipschitz maps of Bill Thurston. Some applications to integrals on a domain a hyperbolic surface leave open possibilities for applications to integrals on domains which are locally symmetric spaces of higher dimension. The talk finishes with an example or two from the literature.
Talk Chair: Laura DeMarco
VIDEO

General Relativity Program
2022-01-24 09:00 - 2022-05-21 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses, a conference, and a workshop.
General Relativity Mincourses: March–May, 2022
General Relativity Conference: April 4–8, 2022
General Relativity Workshop: May 2–5, 2022

Program Visitors

Dan Lee, CMSA/CUNY, 1/24/22 – 5/20/22
Stefan Czimek, Brown, 2/27/22 – 3/3/22
Lan-Hsuan Huang, University of Connecticut, 3/13/22 – 3/19/222, 3/21/22 – 3/25/22, 4/17 /22– 4/23/22
Mu-Tao Wang, Columbia, 3/21/22 – 3/25/22, 5/7/22 – 5/9/22
Po-Ning Chen, University of California, Riverside, 3/21/22 – 3/25/22, 5/7/22–5/9/22
Marnie Smith, Imperial College London, 3/27/22 – 4/11/22
Christopher Stith, University of Michigan, 3/27/22 – 4/23/22
Martin Taylor, Imperial College London, 3/27/22 – 4/11/22
Marcelo Disconzi, Vanderbilt, 5/9/22 – 5/21/22
Lydia Bieri, University of Michigan, 5/5/22 – 5/9/22

General Relativity Program Minicourses
2022-03-01 10:00 - 2022-05-17 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Minicourses
General Relativity Program Minicourses

During the Spring 2022 semester, the CMSA hosted a program on General Relativity.
This semester-long program included four minicourses running in March, April, and May; a conference April 4–8, 2022; and a workshop from May 2–5, 2022.

Schedule
Speaker
Title
Abstract

March 1 – 3, 2022
10:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Dr. Stefan Czimek
Characteristic Gluing for the Einstein Equations
Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).

March 22 – 25, 2022
22nd & 23rd, 10:00 am – 11:30am ET
24th & 25th, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room, G-10.
Prof. Lan-Hsuan Huang
Existence of Static Metrics with Prescribed Bartnik Boundary Data
Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is
1. The conjecture and an overview of the results
2. Static regular: a sufficient condition for existence and local uniqueness
3. Convex boundary, isometric embedding, and static regular
4. Perturbations of any hypersurface are static regular
Video on Youtube: March 22, 2022

March 29 – April 1, 2022 10:00am – 12:00pm ET, each day
Location: Hybrid. CMSA main seminar room, G-10.
Prof. Martin Taylor
The nonlinear stability of the Schwarzschild family of black holes
Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.

April 19 & 21, 2022
10 am – 12 pm ET, each dayZoom only
Prof. Håkan Andréasson
Two topics for the Einstein-Vlasov system: Gravitational collapse and properties of static and stationary solutions.
Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole.
Video 4/19/2022
Video 4/22/2022

May 16 – 17, 2022
10:00 am – 1:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10.
Prof. Marcelo Disconzi
A brief overview of recent developments in relativistic fluids
Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.
1. Set-up, review of standard results, physical motivation.
2. The relativistic Euler equations: null structures and the problem of shocks.
3. The free-boundary relativistic Euler equations with a physical vacuum boundary.
4. Relativistic viscous fluids.
Video 5/16/2022
Video 5/17/2022

General Relativity Conference
2022-04-04 09:30 - 2022-04-08 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Schedule | April 4–8, 2022
Monday, April 4, 2022

Time (ET)
Speaker
Title/Abstract

9:30 am–10:30 am
Pieter Blue, University of Edinburgh, UK
(virtual)
Title: Linear stability of the Kerr spacetime in the outgoing radiation gauge
Abstract: This talk will discuss a new gauge condition (i.e. coordinate condition) for the Einstein equation, the linearisation of the Einstein equation in this gauge, and the decay of solutions to the linearised Einstein equation around Kerr black holes in this gauge. The stability of the family of Kerr black holes under the evolution generated by the Einstein equation is a long-standing problem in mathematical relativity. In 1972, Teukolsky discovered equations governing certain components of the linearised curvature that are invariant under linearised gague transformations. In 1975, Chrzanowski introduced the “outgoing radiation gauge”, a condition on the linearised metric that allows for the construction of the linearised metric from the linearised curvature. In 2019, we proved decay for the metric constructed using Chrzanowski’s outgoing radiation gauge. Recently, using a flow along null geodesics, we have constructed a new gauge such that, in this gauge, the Einstein equation is well posed and such that the linearisation is Chrzanowski’s outgoing radiation gauge.
This is joint work with Lars Andersson, Thomas Backdahl, and Siyuan Ma.

10:30 am–11:30 am
Peter Hintz, ETH Zürich
(virtual)
Title: Mode stability and shallow quasinormal modes of Kerr-de Sitter
black holesAbstract: The Kerr-de Sitter metric describes a rotating black hole with mass $m$ and specific angular momentum $a$ in a universe, such as our own, with cosmological constant $Lambda>0$. I will explain a proof of mode stability for the scalar wave equation on Kerr-de Sitter spacetimes in the following setting: fixing $Lambda$ and the ratio $|a/m|<1$ (related to the subextremality of the black hole in question), mode stability holds for sufficiently small black hole mass $m$. We also obtain estimates for the location of quasinormal modes (resonances) $sigma$ in any fixed half space $Imsigma>-C$. Our results imply that solutions of the wave equation decay exponentially in time to constants, with an explicit exponential rate. The proof is based on careful uniform estimates for the spectral family in the singular limit $mto 0$ in which, depending on the scaling, the Kerr-de Sitter spacetime limits to a Kerr or the de Sitter spacetime.

11:30 am–12:30 pm
Lars Andersson, Albert Einstein Institute, Germany
(virtual)
Title: Gravitational instantons and special geometry
Abstract: Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. In this talk, I will introduce some notions of special geometry, discuss known examples, and mention some open questions. The Chen-Teo gravitational instanton is an asymptotically flat, toric, Ricci flat family of metrics on $mathrm{CP}^2 setminus mathrm{S}^1$, that provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture. I will sketch a proof that the Chen-Teo Instanton is Hermitian and non-Kähler. Thus, all known examples of gravitational instantons are Hermitian. This talks is based on joint work with Steffen Aksteiner, cf. https://arxiv.org/abs/2112.11863.

12:30 pm–1:30 pm
break

1:30 pm–2:30 pm
Martin Taylor, Imperial College London
(virtual)
Title: The nonlinear stability of the Schwarzschild family of black holes
Abstract: I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

2:30 pm–3:30 pm
Po-Ning Chen, University of California, Riverside
(virtual)
Title: Angular momentum in general relativity

Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960s, due to the ‘supertranslation ambiguity’. In this talk, we will discuss how the mathematical theory of quasilocal mass and angular momentum leads to a new definition of angular momentum at null infinity that is free of any supertranslation ambiguity.This is based on joint work with Jordan Keller, Mu-Tao Wang, Ye-Kai Wang, and Shing-Tung Yau.

3:30 pm–4:00 pm
break

4:00 pm–5:00 pm
Dan Lee, Queens College (CUNY)
(hybrid: in person & virtual)
Title: Stability of the positive mass theorem
Abstract: We will discuss the problem of stability for the rigidity part of the Riemannian positive mass theorem, focusing on recent work with Kazaras and Khuri, in which we proved that if one assumes a lower Ricci curvature bound, then stability holds with respect to pointed Gromov-Hausdorff convergence.

Tuesday, April 5, 2022

Time (ET)
Speaker
Title/Abstract

9:30 am–10:30 am
Xinliang An, National University of Singapore
(virtual)
Title: Anisotropic dynamical horizons arising in gravitational collapse
Abstract: Black holes are predicted by Einstein’s theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being and about the structures of their inner spacetime singularities. In this talk, we will present several results in these directions. First, in a joint work with Qing Han, with tools from scale-critical hyperbolic method and non-perturbative elliptic techniques, with anisotropic characteristic initial data we prove that: in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This result extends the 2008 Christodoulou’s monumental work and it connects to black hole thermodynamics along the apparent horizon. Second, in joint works with Dejan Gajic and Ruixiang Zhang, for the spherically symmetric Einstein-scalar field system, we derive precise blow-up rates for various geometric quantities along the inner spacelike singularities. These rates obey polynomial blow-up upper bounds; and when it is close to timelike infinity, these rates are not limited to discrete finite choices and they are related to the Price’s law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism. If time permits, some results on fluid dynamics will also be addressed.

10:30 am–11:30 am
Sergiu Klainerman, Princeton
(virtual)
Title: Nonlinear stability of slowly rotating Kerr solutions
Abstract: I will talk about the status of the stability of Kerr conjecture in General Relativity based on recent results obtained in collaboration with Jeremie Szeftel and Elena Giorgi.

11:30 am–12:30 pm
Siyuan Ma, Sorbonne University
(virtual)
Title: Sharp decay for Teukolsky master equation
Abstract: I will talk about joint work with L. Zhang on deriving the late time dynamics of the spin $s$ components that satisfy the Teukolsky master equation in Kerr spacetimes.

12:30 pm–1:30 pm
Break

1:30 pm–2:30 pm
Jonathan Luk, Stanford
(virtual)
Title: A tale of two tails
Abstract: Motivated by the strong cosmic censorship conjecture, we introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes in odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, we show that the late-time tails are in general different from the stationary case in the presence of dynamical and/or nonlinear perturbations. This is a joint work with Sung-Jin Oh (Berkeley).

2:30 pm–3:30 pm
Gary Horowitz, University of California Santa Barbara
(virtual)
Title: A new type of extremal black hole
Abstract: I describe a family of four-dimensional, asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Surprisingly, the maximum charge for given mass is a nonsingular hairy black hole with a nondegenerate event horizon. Since the surface gravity is nonzero, if quantum matter is added, Hawking radiation does not appear to stop when this new extremal limit is reached. This raises the question of whether Hawking radiation will cause the black hole to turn into a naked singularity. I will argue that does not occur.

3:30 pm–4:00 pm
Break

4:00 pm–5:00 pm
Lydia Bieri, University of Michigan
(virtual)
Title: Gravitational radiation in general spacetimes
Abstract: Studies of gravitational waves have been devoted mostly to sources such as binary black hole mergers or neutron star mergers, or generally sources that are stationary outside of a compact set. These systems are described by asymptotically-flat manifolds solving the Einstein equations with sufficiently fast decay of the gravitational field towards Minkowski spacetime far away from the source. Waves from such sources have been recorded by the LIGO/VIRGO collaboration since 2015. In this talk, I will present new results on gravitational radiation for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. A panorama of new gravitational effects opens up when delving deeper into these more general spacetimes. In particular, whereas the former sources produce memory effects that are finite and of purely electric parity, the latter in addition generate memory of magnetic type, and both types grow. These new effects emerge naturally from the Einstein equations both in the Einstein vacuum case and for neutrino radiation. The latter results are important for sources with extended neutrino halos.

Wednesday, April 6, 2022

Time (ET)
Speaker
Title/Abstract

9:30 am–10:30 am
Gerhard Huisken, Mathematisches Forschungsinstitut Oberwolfach
(virtual)
Title: Space-time versions of inverse mean curvature flow
Abstract: In order to extend the Penrose inequality from a time-symmetric setting to general asymptotically flat initial data sets several anisotropic generalisations of inverse mean curvature flow have been suggested that take the full space-time geometry into account. The lecture describes the properties of such flows and reports on recent joint work with Markus Wolff on inverse flow along the space-time mean curvature.

10:30 am–11:30 am
Carla Cederbaum, Universität Tübingen, Germany
(virtual)
Title: Coordinates are messy
Abstract: Asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $delta$ and to $0$ suitably fast, respectively. Provided their matter densities satisfy suitable integrability conditions, they have well-defined (ADM-)energy, (ADM-)linear momentum, and (ADM-)mass. This was proven by Bartnik using harmonic coordinates. To study their (ADM-)angular momentum and (BORT-)center of mass, one usually assumes the existence of Regge—Teitelboim coordinates on the initial data set $(M,g,K)$ in question. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge—Teitelboim coordinates We will also show that harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge—Teitelboim coordinates. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences these findings have for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.

11:30 am–12:30 pm
Stefanos Aretakis, University of Toronto
(virtual)
Title: Observational signatures for extremal black holes
Abstract: We will present results regarding the asymptotics of scalar perturbations on black hole backgrounds. We will then derive observational signatures for extremal black holes that are based on global or localized measurements on null infinity. This is based on joint work with Gajic-Angelopoulos and ongoing work with Khanna-Sabharwal.

12:30 pm–1:30 pm
Break

1:30 pm–2:30 pm
Jared Speck, Vanderbilt University
(virtual)
Title: The mathematical theory of shock waves in multi-dimensional relativistic and non-relativistic compressible Euler flow
Abstract: In the last two decades, there have been dramatic advances in the rigorous mathematical theory of shock waves in solutions to the relativistic Euler equations and their non-relativistic analog, the compressible Euler equations. A lot of the progress has relied on techniques that were developed to study Einstein’s equations. In this talk, I will provide an overview of the field and highlight some recent progress on problems without symmetry or irrotationality assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe various open problems, some of which are tied to the Einstein–Euler equations. Various aspects of this program are joint with L. Abbrescia, M. Disconzi, and J. Luk.

2:30 pm–3:30 pm
Lan-Hsuan Huang, University of Connecticut
(virtual)
Title: Null perfect fluids, improvability of dominant energy scalar, and Bartnik mass minimizers
Abstract: We introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability. In particular, we prove that a non-improvable initial data set without local symmetries must sit inside a null perfect fluid spacetime carrying a global Killing vector field. We also show that the dominant energy scalar is always almost improvable in a precise sense. Using these main results, we provide a characterization of Bartnik mass minimizing initial data sets which makes substantial progress toward Bartnik’s stationary conjecture.
Along the way we observe that in dimensions greater than eight there exist pp-wave counterexamples (without the optimal decay rate for asymptotically flatness) to the equality case of the spacetime positive mass theorem. As a consequence, we find counterexamples to Bartnik’s stationary and strict positivity conjectures in those dimensions. This talk is based on joint work with Dan A. Lee.

3:30 pm–4:00 pm
Break

4:00 pm–5:00 pm
Demetre Kazaras, Duke University
(virtual)
Title: Comparison geometry for scalar curvature and spacetime harmonic functions
Abstract: Comparison theorems are the basis for our geometric understanding of Riemannian manifolds satisfying a given curvature condition. A remarkable example is the Gromov-Lawson toric band inequality, which bounds the distance between the two sides of a Riemannian torus-cross-interval with positive scalar curvature by a sharp constant inversely proportional to the scalar curvature’s minimum. We will give a new qualitative version of this and similar band-type inequalities in dimension 3 using the notion of spacetime harmonic functions, which recently played the lead role in our recent proof of the positive mass theorem. This is joint work with Sven Hirsch, Marcus Khuri, and Yiyue Zhang.

Thursday, April 7, 2022

Time (ET)
Speaker
Title/Abstract

9:30 am–10:30 am
Piotr Chrusciel, Universitat Wien
(virtual)
Title: Maskit gluing and hyperbolic mass
Abstract: “Maskit gluing” is a gluing construction for asymptotically locally hyperbolic (ALH) manifolds with negative cosmological constant. I will present a formula for the mass of Maskit-glued ALH manifolds and describe how it can be used to construct general relativistic initial data with negative mass.

10:30 am–11:30 am
Greg Galloway, University of Miami (virtual)
Title: Initial data rigidity and applications
Abstract: We present a result from our work with Michael Eichmair and Abraão Mendes concerning initial data rigidity results (CMP, 2021), and look at some consequences. In a note with Piotr Chruściel (CQG 2021), we showed how to use this result, together with arguments from Chruściel and Delay’s proof of the their hyperbolic PMT result, to obtain a hyperbolic PMT result with boundary. This will also be discussed.

11:30 am–12:30 pm
Pengzi Miao, University of Miami
(virtual)
Title: Some remarks on mass and quasi-local mass
Abstract: In the first part of this talk, I will describe how to detect the mass of asymptotically flat and asymptotically hyperbolic manifolds via large Riemannian polyhedra. In the second part, I will discuss an estimate of the Bartnik quasi-local mass and its geometric implications. This talk is based on several joint works with A. Piubello, and with H.C. Jang.

12:30 pm–1:30 pm
Break

1:30 pm–2:30 pm
Yakov Shlapentokh Rothman, Princeton
(hybrid: in person & virtual)
Title: Self-Similarity and Naked Singularities for the Einstein Vacuum Equations
Abstract: We will start with an introduction to the problem of constructing naked singularities for the Einstein vacuum equations, and then explain our discovery of a fundamentally new type of self-similarity and show how this allows us to construct solutions corresponding to a naked singularity. This is joint work with Igor Rodnianski.

2:30 pm–3:30 pm
Marcelo Disconzi, Vanderbilt University
(virtual)
Title: General-relativistic viscous fluids.
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.

3:30 pm–4:00 pm
Break

4:00 pm–5:00 pm
Maxime van de Moortel, Princeton
(hybrid: in person & virtual)
Title: Black holes: the inside story of gravitational collapse
Abstract: What is inside a dynamical black hole? While the local region near time-like infinity is understood for various models, the global structure of the black hole interior has largely remained unexplored.
These questions are deeply connected to the nature of singularities in General Relativity and celebrated problems such as Penrose’s Strong Cosmic Censorship Conjecture.
I will present my recent resolution of these problems in spherical gravitational collapse, based on the discovery of a novel phenomenon: the breakdown of weak singularities and the dynamical formation of a strong singularity.

Friday, April 8, 2022

Time (ET)
Speaker
Title/Abstract

9:30 am–10:30 am
Ye-Kai Wang, National Cheng Kun University, Taiwan
(virtual)
Title: Supertranslation invariance of angular momentum at null infinity in double null gauge
Abstract: This talk accompanies Po-Ning Chen’s talk on Monday with the results described in the double null gauge rather than Bondi-Sachs coordinates. Besides discussing
how Chen-Wang-Yau angular momentum resolves the supertranslation ambiguity, we also review the definition of angular momentum defined by A. Rizzi. The talk is based on the joint work with Po-Ning Chen, Jordan Keller, Mu-Tao Wang, and Shing-Tung Yau.

10:30 am–11:30 am
Zoe Wyatt, King’s College London
(virtual)
Title: Global Stability of Spacetimes with Supersymmetric Compactifications
Abstract: Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in
supergravity and string theory. In this talk I will discuss a recent work with Lars Andersson, Pieter Blue and Shing-Tung Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory. Our proof uses the intersection of methods for quasilinear wave and Klein-Gordon equations, and so towards the end of the talk I will also comment more generally on coupled wave–Klein-Gordon equations.

11:30 am–12:30 pm
Elena Giorgi, Columbia University
(hybrid: in person & virtual)
Title: The stability of charged black holes
Abstract: Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

12:30 pm–1:30 pm
Break

1:30 pm–2:30 pm
Marcus Khuri, Stony Brook University
(virtual)
Title: The mass-angular momentum inequality for multiple black holes

Abstract: Consider a complete 3-dimensional initial data set for the Einstein equations which has multiple asymptotically flat or asymptotically cylindrical ends. If it is simply connected, axisymmetric, maximal, and satisfies the appropriate energy condition then the ADM mass of any of the asymptotically flat ends is bounded below by the square root of the total angular momentum. This generalizes previous work of Dain, Chrusciel-Li-Weinstein, and Schoen-Zhou which treated either the single black hole case or the multiple black hole case without an explicit lower bound. The proof relies on an analysis of the asymptotics of singular harmonic maps from
R^3 Gamma –>H^2 where Gamma is a coordinate axis. This is joint work with Q. Han, G. Weinstein, and J. Xiong.

2:30 pm–3:30 pm
Martin Lesourd, Harvard
(hybrid: in person & virtual)
Title: A Snippet on Mass and the Topology and Geometry of Positive Scalar Curvature
Abstract: I will talk about a small corner of the study of Positive Scalar Curvature (PSC) and questions which are most closely related to the Positive Mass Theorem. The classic questions are ”which topologies allow for PSC?” and ”what is the geometry of manifolds with PSC?”. This is based on joint work with Prof. S-T. Yau, Prof. D. A. Lee, and R. Unger.

3:30 pm–4:00 pm
Break

4:00 pm–5:00 pm
Georgios Moschidis, Princeton
(virtual)
Title: Weak turbulence for the Einstein–scalar field system.
Abstract: In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. In the presence of a negative cosmological constant, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time.
In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams.

Second Annual Yip Lecture: Extraterrestrial Life
2022-04-04 19:00 - 20:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Harvard CMSA hosted the second annual Yip Lecture on April 4, 2022.
The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.
This year’s speaker was Avi Loeb (Harvard).

Extraterrestrial Life
Abstract: Are we alone? It would be arrogant to think that we are, given that a quarter of all stars host a habitable Earth-size planet. Upcoming searches will aim to detect markers of life in the atmospheres of planets outside the Solar System. We also have unprecedented technologies to detect signs of intelligent civilizations through industrial pollution of planetary atmospheres, space archaeology of debris from dead civilizations or artifacts such as photovoltaic cells that are used to re-distribute light and heat on the surface of a planet or giant megastructures. Our own civilization is starting to explore interstellar travel. Essential information may also arrive as a “message in a bottle”, implying that we should examine carefully any unusual object that arrives to our vicinity from outside the Solar System, such as `Oumuamua.

Abraham (Avi) Loeb is the Frank B. Baird, Jr., Professor of Science at Harvard University and a bestselling author (in lists of the New York Times, Wall Street Journal, Publishers Weekly, Die Zeit, Der Spiegel, L’Express and more). He received a PhD in Physics from the Hebrew University of Jerusalem in Israel at age 24 (1980–1986), led the first international project supported by the Strategic Defense Initiative (1983–1988), and was subsequently a long-term member of the Institute for Advanced Study at Princeton (1988–1993). Loeb has written 8 books, including most recently, Extraterrestrial (Houghton Mifflin Harcourt, 2021), and nearly a thousand papers (with an h-index of 118) on a wide range of topics, including black holes, the first stars, the search for extraterrestrial life, and the future of the Universe. Loeb is the head of the Galileo Project in search for extraterrestrial intelligence, the Director of the Institute for Theory and Computation (2007–present) within the Harvard-Smithsonian Center for Astrophysics, and also serves as the Head of the Galileo Project (2021–present). He had been the longest serving Chair of Harvard’s Department of Astronomy (2011–2020) and the Founding Director of Harvard’s Black Hole Initiative (2016–2021). He is an elected fellow of the American Academy of Arts & Sciences, the American Physical Society, and the International Academy of Astronautics. Loeb is a former member of the President’s Council of Advisors on Science and Technology (PCAST) at the White House, a former chair of the Board on Physics and Astronomy of the National Academies (2018–2021) and a current member of the Advisory Board for “Einstein: Visualize the Impossible” of the Hebrew University. He also chairs the Advisory Committee for the Breakthrough Starshot Initiative (2016–present) and serves as the Science Theory Director for all Initiatives of the Breakthrough Prize Foundation. In 2012, TIME magazine selected Loeb as one of the 25 most influential people in space and in 2020 Loeb was selected among the 14 most inspiring Israelis of the last decade.
Click here for Loeb’s commentaries on innovation and diversity.
Website: https://www.cfa.harvard.edu/~loeb/
See the Harvard Gazette article featuring Avi Loeb: “Oh, if I could talk to the aliens” published March 8, 2022.
Prof. Loeb’s books:
Extraterrestrial: The First Sign of Intelligent Life Beyond Earth (2021)
Life in the Cosmos: From Biosignatures to Technosignatures (2021)
Avil Loeb is the head of the Galileo Project at Harvard.

The previous Yip Lecture featured Peter Galison (Harvard), who spoke on the EHT’s hunt for an objective image of a black hole.

Workshop on Machine Learning and Mathematical Conjecture
2022-04-15 09:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On April 15, 2022, the CMSA will hold a one-day workshop, Machine Learning and Mathematical Conjecture, related to the New Technologies in Mathematics Seminar Series.
Location: Room G10, 20 Garden Street, Cambridge, MA 02138.
Organizers: Michael R. Douglas (CMSA/Stony Brook/IAIFI) and Peter Chin (CMSA/BU).
Machine learning has driven many exciting recent scientific advances. It has enabled progress on long-standing challenges such as protein folding, and it has helped mathematicians and mathematical physicists create new conjectures and theorems in knot theory, algebraic geometry, and representation theory.
At this workshop, we will bring together mathematicians, theoretical physicists, and machine learning researchers to review the state of the art in machine learning, discuss how ML results can be used to inspire, test and refine precise conjectures, and identify mathematical questions which may be suitable for this approach.
Speakers:

James Halverson, Northeastern University Dept. of Physics and IAIFI
Fabian Ruehle, Northeastern University Dept. of Physics and Mathematics and IAIFI
Andrew Sutherland, MIT Department of Mathematics

CMSA/Tsinghua Math-Science Literature Lecture: Three Introductory Lectures on Game Theory for Mathematicians: Game Theory Basics and Classical Existence Theorems
2022-04-18 09:30 - 11:00 | Virtual
Eric Maskin (Harvard University) Three Introductory Lectures on Game Theory for Mathematicians
April 18, 2022 | 9:30 – 11:00 am ET
Title: Game Theory Basics and Classical Existence Theorems
Abstract: Games in extensive and normal form. Equilibrium existence theorems by Nash, von Neumann, and Zermelo
Talk chairs: Scott Kominers, Sergiy Verstyuk
SLIDES | VIDEO

CMSA/Tsinghua Math-Science Literature Lecture: Three Introductory Lectures on Game Theory for Mathematicians: Mechanism Design
2022-04-20 09:30 - 11:00 | Virtual
Eric Maskin (Harvard University) Three Introductory Lectures on Game Theory for Mathematicians
April 20, 2022 | 9:30 – 11:00 am ET
Title: Mechanism Design
Abstract: Given a social goal, under what circumstances can we design a game to achieve that goal?
Talk chairs: Scott Kominers, Sergiy Verstyuk
SLIDES | VIDEO

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

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

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.

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

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.

2022 Summer Introduction to Mathematical Research
2022-05-21 09:00 - 2022-06-12 17:00 | Virtual
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
