Conference on Geometry and Statistics
2025-11-17 09:00 - 2025-11-19 17:00 (America/New_York) | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Conference on Geometry and Statistics
Dates: November 17–19, 2025
Location: CMSA G10, 20 Garden Street, Cambridge MA & via Zoom

Speakers

Charles Fefferman, Princeton University
Stephan Huckemann, Georg-August Universität Göttingen
Sungkyu Jung, Seoul National University
Kei Kobayashi, Keio University
Clément Levrard, Université de Rennes
Ker-Chau Li, University of California, Los Angeles
Rong Ma, Harvard University
Steve Marron, University of North Carolina
Ezra Miller, Duke University
Hans-Georg Müller, University of California, Davis
Wilderich Tuschmann, Karlsruhe Institute of Technology
Melanie Weber, Harvard University
Andrew Wood, Australian National University
Horng-Tzer Yau, Harvard University

Organizer: Zhigang Yao, National University of Singapore

Youtube Playlist

SCHEDULE
download pdf
Monday, Nov. 17, 2025
9:00–9:25 am
Morning refreshments
9:25–9:30 am
Introductions
9:30–10:30 am
Speaker: Stephan Huckemann, Georg-August Universität Göttingen
Title: The Probability of the Cut Locus of a Fréchet Mean
Abstract: We show that the cut locus of a Fréchet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases (Le and Barden, 2014) and conjectured in general. The proof is based on first order and second order considerations, where the latter are based on a recent result by Générau (2020) on “Laplacians in the barrier sense”. This generalizes to Fréchet p-means for p > 2. The former allow also to rule out stickiness on Riemannian manifolds, and for generalization to 1 <= p < 2, with a conjecture. We close with discussing and conjecturing extensions to noncomplete manifolds and more general metric spaces. This is joint work with Alexander Lytchak.

Générau, F. (2020). Laplacian of the distance function on the cut locus on a Riemannian manifold. Nonlinearity 33(8), 3928.
Le, H. and D. Barden (2014). On the measure of the cut locus of a Fréchet mean. Bulletin of the London Mathematical Society 46(4), 698–708.
Lytchak, A. and S. F. Huckemann (2025). Zero mass at the cut locus of a Fréchet mean on a Riemannian manifold. arXiv preprint arXiv:2508.00747.

10:30–10:45 am
break
10:45 am–11:45 am
Speaker: Hans-Georg Müller, University of California, Davis
Title: Conformal Inference for Random Objects
Abstract: The underlying probability measure of random objects, i.e., metric-space-valued random variables, can be probed by distance profiles. These are one-dimensional distributions of probability mass falling into balls of increasing radius. In a regression setting with Euclidean covariates X and responses Y that are random objects, one can consider conditional Fréchet means that can be implemented with Fréchet regression and also conditional distance profiles, conditioning on X. Conditional distance profiles can then be leveraged to obtain conditional average transport costs, the expected cost for transporting a fixed conditional distance profile to a randomly selected conditional distance profile. The conditional average transport costs can then be utilized to obtain conditional conformity scores. In conjunction with the split conformal algorithm these scores lead to conditional prediction sets located in the object space with asymptotic conditional validity and attractive finite sample behavior. Based on joint work Hang Zhou (UNC).
11:45 am–1:15 pm
Lunch (Catered)
1:15–2:15 pm
Speaker: Horng-Tzer Yau, Harvard
Title: Ramanujan property of random regular graphs and delocalization of random band matrices
Abstract: In this lecture, we review recent works on random matrices. The first result is about the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $dgeq 3$ and denote its eigenvalues as $lambda_1=d/sqrt{d-1}geq lambda_2geqlambda_3cdotsgeq lambda_N$. We establish the edge universality for random $d$-regular graphs, namely, the distributions of $lambda_2$ and $-lambda_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69%$ of $d$-regular graphs on $N$ vertices.
are Ramanujan, meaning $max{lambda_2,|lambda_N|}leq 2$. This resolves a conjecture by Sarnak and Miller-Novikoff-Sabelli
The second result concerns $ N times N$ Hermitian $d$-dimensional random band matrices with band width $W$. In the bulk of the spectrum and in the large $ N $ limit, we prove that all $ L^2 $- normalized eigenvectors are delocalized in all dimensions under suitable conditions on $W$ and $N$. In addition, we proved that the eigenvalue statistics are given by those of the Gaussian unitary ensemble.
2:15–2:45 pm
break with refreshments
2:45–3:45 pm
Speaker: Clément Levrard, Université de Rennes
Title: Optimal reach estimation
Abstract: The reach of an embedded submanifold, a notion that dates back to the famous work Curvature measures of H. Federer, may be understood as a scale under which the submanifold is flat enough so that traditional Euclidean techniques in statistics locally apply, up to some approximation. I will expose several ways to estimate the reach from sample (on the submanifold), some of them being optimal from the point of view of minimax estimation theory. Along the way, intermediate estimation problems of local and global quantities will arise (curvature estimation, weak feature size estimation, distance estimation, etc.), for which various phenomenons can occur from a statistical point of view (different convergence rates, inconsistency). This will be an opportunity to provide a selective overview of the state of the art on these issues.
4:30–5:30 pm
CMSA Colloquium
Speaker: Zhigang Yao (National University of Singapore)
Title: Interaction of Statistics and Geometry: A New Landscape for Data Science
Abstract: Classical statistics views data as real numbers or vectors in Euclidean space, but modern challenges increasingly involve data with intrinsic geometric structures. A central problem in this direction is manifold fitting, with origins in H. Whitney’s work of the 1930s. The Geometric Whitney Problems ask: given a set, when can we construct a smooth 𝑑-dimensional manifold that approximates it, and how accurately can we estimate it?
In this talk, I will discuss recent progress on manifold fitting and its role in bridging geometry and data science. While many existing methods rely on restrictive assumptions, the manifold hypothesis—that data often lie near non-Euclidean structures—remains fundamental in modern statistical learning. I will highlight both theoretical insights and algorithmic challenges, drawing on recent works with, as well as ongoing research.
Youtube video

Tuesday, Nov. 18, 2025
9:00–9:30 am
Morning refreshments
9:30–10:30 am
Speaker: Charles Fefferman, Princeton University (via Zoom)
Title: Extrinsic and intrinsic manifold learning, old and new
Abstract: The talk will include an exposition of the old paper “Testing the manifold hypothesis”, joint work with S. Mitter and H. Narayanan, on extrinsic manifold learning (the manifold to be learned is assumed to be embedded in a high-dimensional Euclidean space). The talk will also include a new result on intrinsic manifold learning (the manifold to be learned is not assumed to be embedded, and the data consist of intrinsic distances corrupted by noise), provided the result is proven by the time of the conference.
10:30–10:45 am
break
10:45 am–11:45 am
Speaker: Steve Marron, University of North Carolina
Title: Data Integration Via Analysis of Manifolds (DIVAM)
Abstract: A major challenge in the age of Big Data is the integration of disparate data types into a single data analysis. That was tackled by Data Integration Via Analysis of Subspaces (DIVAS) in the context of data blocks measured on a common set of experimental cases. Joint variation was defined in terms of modes of variation having identical scores across data blocks. DIVAS allowed mathematically rigorous formulation of individual variation within each data block in terms of individual modes. The goal of DIVAM is to intrinsically extend the DIVAS approach to data objects lying in manifolds, such as shape data.
11:45 am–1:15 pm
Lunch Break
1:15–2:15 pm
Speaker: Ker-Chau Li, University of California, Los Angeles
Title: Investigation of Data clouds: From Galton’s Ellipses to Explainable AI (XAI), modeling or molding?
Abstract: Francis Galton’s seminal 1886 visualization of regression toward the mean in trait inheritance is arguably the first and most influential example of geometric thinking applied to statistical modeling. The pioneering geometric insight driving Galton’s use of elliptical contours to discover the bivariate normal distribution laid down the foundation for classic multivariate analysis (e.g., PCA, canonical correlation) and profoundly impacts modern methods like diffusion models.
Statistical models, particularly those based on parsimony, are effective for characterizing data distribution and facilitating scientific rule induction. However, the rise of unstructured big data (like images) has challenged these parsimonious approaches, necessitating the use of deep learning models. These models, containing billions of parameters, sacrifice transparency to excel in prediction. Seeking solutions to this “black-box” dilemma is now the heart of Explainable AI (XAI).
Leveraging the simplicity of elementary geometric concepts, this talk will present a new path toward interpretable and parsimonious XAI. Unstructured big data is highly plastic. Our approach moves beyond the standard data modeling perspective—which answers what the data is—and introduces a novel data molding perspective. This shift is key to unlocking the full potential of data’s plasticity, allowing us to effectively answer the crucial question: what the data can be used for.
I will first discuss a connection between manifold learning and my earlier works, helical confounding and liquid association. I will then turn to the data molding perspective and present two novel notions: mold-compliance and artificial-trait configurative-generation (ATCG). These notions guide our recent efforts in formulating novel algorithms for image data investigation, addressing issues like prediction validity and within-class heterogeneity. Data molding entails a dramatically different feature space extraction, which consequently shifts the subsequent investigation on the data clouds from out-of-distribution (OOD) to mold-violation, and from UMAP clustering to ATCG-induced hierarchical clustering.
2:15–2:45 pm
break with refreshments
2:45–3:45 pm
Speaker: Andrew Wood, Australian National University
Title: Empirical likelihood methods for Fréchet means on open books
Abstract: The open book is a simple example of a stratified space that captures some (but not all) of the properties of stratified spaces. Central limit theory for open books plus relevant background is given by Hotz et al. (2013, Annals of Applied Probability). In this talk I will describe some basic inference procedures for Fréchet means in open books based on empirical likelihood (Owen, book, 2001). Empirical likelihood (EL) is a type of nonparametric likelihood that can be useful for many types of data, including manifold-valued data and data from stratified spaces. An EL approach to basic inference for Fréchet means will be described. In particular, it will be shown how the non-regularity in the geometry of open books can result in non-regular behaviour in Wilks’s theorem (i.e. the large sample likelihood ratio test). The talk will also discuss difficulties in extending the EL inference theory from open books to more general stratified spaces, where the difference in dimension of adjacent strata can be 2 or more. For discussion of more general stratified spaces than open books, see the orthant spaces discussed in Barden and Le (2018, Proc of London Math Society) and the general stratified space setting considered by Mattingly et al. (2023, arxiv).
3:45–4:00 pm
break
4:00–5:00 pm
Speaker: Wilderich Tuschmann, Karlsruhe Institute of Technology
Title: A Spectator’s Perspective on the Manifold Hypothesis
Abstract: At its core, the Manifold Hypothesis asserts that real-world, high-dimensional data is not uniformly or randomly distributed throughout its high-dimensional “ambient” space, but concentrated on or near a low-dimensional manifold (or a collection of manifolds) embedded within that high-dimensional ambient space.
In my talk, I will discuss reasons and facts that speak for as well as against this hypothesis and also address geometric alternatives.

Wednesday, Nov. 19, 2025
9:00–9:30 am
Morning refreshments
9:30–10:30 am
Speaker: Melanie Weber, Harvard University
Title: Ricci Curvature, Ricci Flow, and the Geometry of Learning
Abstract: Geometric structure in data plays a crucial role in machine learning. In this talk, we study this observation through the lens of Ricci curvature and its associated Ricci flow. We start by reviewing a discrete notion of Ricci curvature introduced by Ollivier and the geometric flow that it induces. We further discuss the relationship between discrete Ricci curvature and its continuous counterpart via discrete-to-continuum consistency results, which imply that discrete Ricci curvature can provably characterize the geometry of a data manifold based on a finite sample. This provides a theoretical foundation for several applications of discrete Ricci curvature in machine learning, two of which we discuss in the remainder of this talk. First, we analyze learned feature representations in deep neural networks and show that they transform during training in ways that closely resemble a discrete Ricci flow. Our analysis reveals that nonlinear activations shape class separability and suggests geometry-informed training principles such as early stopping and depth selection. Second, we turn to deep learning on graphs, where we address representational limitations of state of the art graph neural networks through curvature-based data augmentations. We show that augmenting input graphs with geometric information provably increases the representational power of such models and yields performance gains in practice.
10:30–10:45 am
break
10:45 am–11:45 am
Speaker: Ezra Miller, Duke University
Title: Extracting bar lengths from multiparameter persistent homology
Abstract: Persistent homology in one parameter can be summarized using bar codes or persistence diagrams, which are elementary gadgets with many features amenable to vectorization and hence statistical analysis. For example, early work with Bendich, Marron, Pieloch, and Skwerer showed how to extract meaningful statistics from the top 100 bar lengths in persistent homology summaries of brain arteries. The story for persistent homology with multiple parameters, on the other hand, is still developing. Although it has the potential to be much more flexible and informative, multipersistence has structural issues that present fundamental mathematical challenges. There is no consensus on what might be meant by a “bar”, let alone “the top 100 bar lengths”. This talk recalls the basics of single and multiparameter persistent homology and discusses some of the mathematical issues, including obstacles and potential routes forward.
11:45 am–1:15 pm
Lunch Break
1:15–2:15 pm
Speaker: Kei Kobayashi, Keio University
Title: Metric Transformations of Data Spaces: Curvature Control and Related Developments
Abstract: We present our proposed method of increasing the accuracy of data analysis by means of two transformations of the metric of the data space. The first transformation is based on the curve length defined by the integral of the power of the density function, which can be computed approximately using an empirical graph; the second transformation can be interpreted as the extrinsic distance when the data space is embedded in a metric cone. The advantage of both distance transformations is that the hyperparameters allow the curvature to be monotonically transformed in a specific sense. Some statistical applications of these transformations and theoretical justifications are presented. Detailed analyses of the geodesics obtained by this method for several simple probability distributions will also be presented. The main part of this work is based on joint works with Henry P. Wynn.
2:15–2:45 pm
break with refreshments
2:45–3:45 pm
Speaker: Sungkyu Jung, Seoul National University
Title: Generalized Frechet means with random minimizing domains and its strong consistency
Abstract: In this talk, I will discuss a novel extension of Frechet means, referred to as generalized Frechet means, as a comprehensive framework for describing the characteristics of random elements. The generalized Frechet mean is defined as the minimizer of a cost function, and the framework encompasses various extensions of Frechet means that have appeared in the literature. The most distinctive feature of the proposed framework is that it allows the domain of minimization for the empirical generalized Frechet means to be random and different from that of its population counterpart. This flexibility broadens the applicability of the Frechet mean framework to various statistical scenarios, including sequential dimension reduction for non-Euclidean data. We establish a strong consistency theorem for generalized Frechet means. Applications such as verifying the consistency of principal geodesic analysis on the hypersphere, compositional principal component analysis on the composition space, and k-medoids clustering for data on a metric space will be discussed.
3:45–4:00 pm
break
4:00–5:00 pm
Speaker: Rong Ma, Harvard University
Title: Modern Nonlinear Embedding Methods Unpacked
Abstract: Learning and representing low-dimensional structures from noisy, high-dimensional data is a cornerstone of modern data science. Stochastic neighbor embedding algorithms, a family of nonlinear dimensionality reduction and data visualization methods, with t-SNE and UMAP as two leading examples, have become very popular in recent years. Yet despite their wide applications, these methods remain subject to points of debate, including limited theoretical understanding, ambiguous interpretations, and sensitivity to tuning parameters. In this talk, I will present our recent efforts to decipher and improve these nonlinear embedding approaches. Our key results include a rigorous theoretical framework that uncovers the intrinsic mechanisms, large-sample limits, and fundamental principles underlying these algorithms; a set of theory-informed practical guidelines for their principled use in trustworthy biological discovery; and a collection of new algorithms that address current limitations and improve performance in areas such as bias reduction and stability. Throughout the talk, I will highlight how these advances not only deepen our theoretical understanding but also open new avenues for scientific discovery.

Millennium Prize Problems Lecture - Madhu Sudan: P vs NP Problem
2025-12-03 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Pamphlet (pdf)
Slides (pdf)
Date: December 3, 2025
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall D, 1 Oxford St., Cambridge MA
Speaker: Madhu Sudan, Harvard University
Title: The P vs. NP problem: An Existential Question for Mathematics
At the beginning of the twentieth century, in response to questions raised by Hilbert, illustrious mathematicians such as Godel, Church and Turing formalized the notion of theorems and proofs. Proofs were automatically verifiable while theorems are logical propositions for which proofs exist. The formal definition of a computer, a definition that had strong influence on the later development of the technology, was a by-product of the effort to define the phrase “automatically verifiable”!
While the resulting theory had major implications already, one notion was however missing in the early definitions. Proofs were meant to be easily verifiable, while determining the truth of a proposition/conjecture (arguably a core task of mathematics) was not necessarily so. But what is “easiness” and how is it to be defined? While this was already hinted at by Godel in the 50s, the notion was finally formalized in seminal works of Cook, Levin and Karp in the early 70s. Central notions here included the adoption of the notion that polynomial time algorithms are (the only) tractable ones, and the realization that algorithms seeking to remove the existential quantifier in the definition of a “theorem” lead naively to exponential time algorithms. But are there no sophisticated algorithms to search for proofs? This is the profound “Is P = NP?” question.
In this talk we will introduce the question and explain implications of resolutions of this question to the modern computing infrastructure, to mathematics and other sciences. We will briefly describe the state of progress on this question and recent progress on weaker forms of this question. Finally we will also aim to connect this question, and why one may believe that P != NP (proof search can not be automated) even in the face of accumulating evidence on the ability of computers to solve more and more complex mathematical problems, which seem to implement brute force search in less than polynomial time.

Read more about the P vs NP Problem at the Clay Math website.

Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

Millennium Prize Problems Lecture - Barry Mazur: About the Birch and Swinnerton–Dyer Conjecture
2026-02-04 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Date: February 4, 2026
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall C, 1 Oxford St., Cambridge MA
Speaker: Barry Mazur, Harvard University
Title: About the Birch and Swinnerton–Dyer Conjecture
Abstract:
In the 1950s Bryan Birch and Peter Swinnerton–Dyer made computations that suggested a striking connection between a basic global invariant of an elliptic curve E over the field of rational numbers (namely, the rank of its group of rational points) and certain asymptotics of its local arithmetic invariants (i.e., the number of its rational points over finite fields).
This initial observation has evolved into their conjecture. My lecture will be an introduction to the general ideas behind its ever-expanding development.
Read more about the Birch and Swinnerton–Dyer Conjecture at the Clay Math website.

Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University
Barry Mazur joined the Harvard University faculty in 1959 as a Junior Fellow in the Society of Fellows and advanced through the ranks to become the Gerhard Gade University Professor of Mathematics, a position he has held since 1998. During his tenure at Harvard, he has mentored 60 doctoral students and served as a pivotal figure in bridging topology and number theory, notably through his classification of the possible torsion subgroups of elliptic curves over the rational numbers (Mazur’s torsion theorem), which identifies exactly 15 possible finite groups. This theorem, detailed in his 1977 paper “Modular curves and the Eisenstein ideal,” provided crucial insights into the Taniyama-Shimura conjecture and laid groundwork for Andrew Wiles’s 1994 proof of Fermat’s Last Theorem.
His broader research includes seminal works on étale homotopy theory (co-authored with Michael Artin in 1969), the arithmetic moduli of elliptic curves (with Nicholas M. Katz in 1985), and the Iwasawa main conjecture (proved with Andrew Wiles in 1984), as well as advancements in p-adic L-functions and the formulation of the Fontaine-Mazur conjecture on Galois representations. Mazur’s influence extends to public communication of mathematics; he has authored books like Imagining Numbers (2003), exploring historical perspectives on complex numbers.
Among his numerous honors, Mazur received the Cole Prize in Number Theory from the American Mathematical Society in 1982, the Chauvenet Prize in 1994 for expository writing, the Leroy P. Steele Prize for Lifetime Achievement in 2000, and election to the National Academy of Sciences in 1982. In 2011 (presented in 2013), he was awarded the National Medal of Science by President Barack Obama for his pioneering work in these fields.Most recently, in 2022, he received the Chern Medal from the International Mathematical Union, recognizing his profound discoveries and mentorship.

Millennium Prize Problems Lecture Series

2026 Ding Shum Lecture: Sanjeev Arora, Princeton
2026-03-04 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
2026 Ding Shum Lecture
Date: March 4, 2026
Time: 4:00 pm
Location: Harvard Science Center Hall D & via Zoom Webinar
Speaker: Sanjeev Arora, Princeton
Title: How could a Superhuman AI mathematician come about?

Abstract: Can AI systems exceed the capabilities of the human experts who provided their training data? The talk will examine the hypothesis of AI self‑improvement, involving mechanisms such as synthetic data generation, reinforcement learning, and tool‑augmented reasoning with formal verification loops.
I will also present recent work at Princeton, including the Gödel Prover V2 for Lean‑based theorem proving and a new inference pipeline that achieved state‑of‑the‑art performance (at the time of evaluation) on IMO‑ProofBench (Advanced) at moderate inference costs ($20–$30 per problem). These will illustrate how AI systems are sometimes able to escape “cognitive wells”—local optima in a model’s reasoning capabilities. While providing evidence for the feasibility of self‑improvement, they also highlight important hurdles and open questions.

Sanjeev Arora is Charles C. Fitzmorris Professor of Computer Science and Director of Princeton Language and Intelligence, a unit devoted to research and applications of large AI models. He got his Phd from UC Berkeley in 1994 and has been a faculty member at Princeton since then. He has been awarded the ACM Prize in Computing (2011), Fulkerson Prize in Discrete Mathematics (2012), Packard Fellowship, Sloan Fellowship, and the ACM Doctoral Dissertation Prize. He was a plenary speaker at the International Congress of Mathematicians in 2018 and is a member of the National Academy of Science and American Academy of Arts and Sciences.

This event is made possible by the generous funding of Ding Lei and Harry Shum.

CMSA/Tsinghua Math-Science Literature Lecture: Martin Bridson: Profinite rigidity: Chasing finite shadows of infinite groups
2026-03-10 08:45 - 10:15 | Massachusetts
CMSA/Tsinghua Math-Science Literature Lecture

Date: March 10, 2026
Time: 8:45 – 10:15 am ET
Location: Harvard Science Center Hall A, 1 Oxford Street, Cambridge MA & via Zoom Webinar
Speaker: Martin Bridson FRS is the Whitehead Professor of Pure Mathematics at Oxford and President of the Clay Mathematics Institute.
Title: Profinite rigidity: Chasing finite shadows of infinite groups
Abstract: There are many situations in geometry or elsewhere in mathematics where it is natural or convenient to explore infinite groups of symmetries via their actions on finite objects. But how hard is it to find these finite manifestations and to what extent does the collection of all such actions determine the infinite group?
In this talk, I will sketch some of the rich history of such problems and then describe some of the significant advances in recent years.
We’ll pay particular attention to groups that arise in 3-dimensional geometry and topology.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Millennium Prize Problems Lecture - Javier Gómez-Serrano: Navier-Stokes Existence or Breakdown
2026-03-11 17:00 - 18:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Date: March 11, 2026
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall C, 1 Oxford St., Cambridge MA & via Zoom Webinar
Speaker: Javier Gómez-Serrano, Brown University
Title: Navier-Stokes Existence or Breakdown
Abstract: The Navier-Stokes equations have been the cornerstone of fluid dynamics for over a century, accurately describing the motion of viscous fluids such as water and air. However, despite their fundamental importance to mathematics and physics, a profound question remains unanswered: do solutions to these equations always exist for all time, or can they break down and develop singularities (points where the equations lose their validity)? In this Millennium Prize Problems Lecture, I will explore the current mathematical landscape surrounding the Navier-Stokes and related equations. The talk will discuss the historical context, the ongoing search for global regularity versus finite-time blowup, and the latest analytical and computational breakthroughs pushing the boundaries of what we know about fluid behavior.
Read more about the Navier-Stokes Equation at the Clay Math website.

Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

Swampland and our Universe
2026-04-15 08:00 - 2026-04-16 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Swampland and our Universe
Dates: April 15–16, 2026
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA
The swampland program has inspired a range of new ideas in both cosmology and neutrino physics. This workshop brings together experts in neutrino physics, dark energy, dark matter, early-universe cosmology, and string theory to share insights on these developments and to discuss current and future experimental tests.
Speakers

Ignatios Antoniadis, IAS, Princeton
Alek Bedroya, Princeton
Mike Boylan-Kolchin, UT Austin
M.C. Gonzalez-Garcia, YITP Stony Brook & ICREA U. Barcelona
Mustapha Ishak-Boushaki, UT Dallas
Marc Kamionkowski, Johns Hopkins
Miguel Montero, Institute of Theoretical Physics, Madrid
Georges Obied, U Chicago
Matt Reece, Harvard
Tracy Slatyer, MIT

Organizers: Luis Anchordoqui (CUNY Lehman College), Sonia Paban (Harvard Physics), and Cumrun Vafa (Harvard Physics)

Videos are available on the CMSA Youtube Swampland Playlist
Schedule
(download pdf)
Wednesday, Apr. 15, 2026
8:00–9:00 am
Breakfast
9:00–10:00 am
Marc Kamionkowski, Johns Hopkins: Dark-matter dynamics and new physics
Abstract: Galactic halos that are spherical, stationary, and composed of collisionless dark matter are easy to describe mathematically. If dark matter decays or interacts or there is some departure from equilibrium or time evolution of the system, all bets are off. In this case costly N-body simulations are required. If, however, one retains the assumption of spherical symmetry, these systems can be evolved numerically with a far simpler algorithm that is easily coded run in a matter of minutes on a laptop, rather than a day on a supercomputer. I will describe this approach and illustrate with simulations of self-interacting dark matter, decaying dark matter (with and without anisotropic velocity distributions, supermassive-black-hole growth, tidal stripping, mixed SIDM/CDM models. Come prepared with your own non-standard dark-matter model; we’ll see if we can simulate it during the talk!
10:00–10:30 am
Coffee Break
10:30–11:30 am
Tracy Slatyer, MIT: What (more) the CMB can teach us about dark matter
Abstract: Observations of the cosmic microwave background have already provided critical evidence for dark matter, but can also stringently constrain a range of dark matter properties. I will outline CMB constraints on dark matter properties based on purely gravitational effects, and then discuss in more detail how both CMB anisotropies and the blackbody spectrum can be used to test dark matter interactions with the Standard Model.
11:30 am–1:00 pm
Lunch Break (catered)
1:00–2:00 pm
Alek Bedroya, Princeton: How Quantum Gravity Constrains Physics on the Largest Length Scales
Abstract: I will review the hierarchy of energy scales in quantum gravity, from the Hubble scale in the IR to the quantum-gravity cutoff and the Planck scale in the UV, and emphasize the nontrivial UV/IR relations that connect them. I will discuss conjectures constraining scalar potentials and explain how they are related to the behavior of the other energy scales, together with bottom-up arguments based on general principles of quantum gravity such as holography. In particular, I will explain how well-motivated holographic assumptions lead to strong restrictions on scalar potentials, ruling out parametrically prolonged accelerated expansion for positive potentials and obstructing parametric scale separation for negative potentials associated with AdS vacua. Title: How Quantum Gravity Constrains Physics on the Largest Length Scales
Abstract: I will review the hierarchy of energy scales in quantum gravity, from the Hubble scale in the IR to the quantum-gravity cutoff and the Planck scale in the UV, and emphasize the nontrivial UV/IR relations that connect them. I will discuss conjectures constraining scalar potentials and explain how they are related to the behavior of the other energy scales, together with bottom-up arguments based on general principles of quantum gravity such as holography. In particular, I will explain how well-motivated holographic assumptions lead to strong restrictions on scalar potentials, ruling out parametrically prolonged accelerated expansion for positive potentials and obstructing parametric scale separation for negative potentials associated with AdS vacua.
2:00–2:30 pm
Coffee Break
2:30–3:30 pm
Mustapha Ishak-Boushaki, UT Dallas: Persistent and serious challenge to the ΛCDM throne: Evidence for dynamical dark energy rising from combinations of different types of datasets
Abstract: We derive multiple constraints on dark energy and compare dynamical dark energy models with a time-varying equation of state (w0waCDM) versus a cosmological constant model (LCDM). We use Baryon Acoustic Oscillation (BAO) from DESI and DES, Cosmic Microwave Background from Planck with and without lensing from Planck and ACT (noted CMBL and CMB, respectively), supernovae(SN), and cross-correlations between galaxy positions and galaxy lensing from DES. We use pairs or triplets of datasets where we exclude one type of dataset each time and categorize them as “NO SN”, “NO CMB” and “NO BAO” combinations. In all cases, we find that the combinations favor the w0waCDM model over LCDM, with significance ranging from 2.0 to 3.0-sigma. The persistence of this pattern across various dataset combinations even when any of the datasets is excluded supports an overall validation of this trending result regardless of any specific dataset. Next, we use larger combinations of these datasets after verifying their mutual consistency within the w0waCDM model. We find combinations that give robust significance levels, with DESI+DESY6BAO+CMBL+SN giving 3.4-sigma. In sum, while we need to remain cautious, the trend and pattern of these results beyond any single type of dataset and their associated systematics presents a compelling overall portrait not in favor of the LCDM and constitutes a serious challenge to the model’s reign. A few other cosmological results will be provided.
3:30–4:00 pm
Coffee Break
4:00–5:00 pm
Georges Obied, U Chicago: The Dark Dimension and its interplay with DESI data
Abstract: In this talk, I will discuss the motivation for considering an extra mesoscopic Dark Dimension of length l ~ 1 – 10 microns, taking into account theoretical and observational arguments. I will then talk about cosmological aspects of the Dark Dimension. In particular this scenario leads, by the universal coupling of the Standard Model sector to bulk gravitons, to massive spin 2 KK excitations of the graviton in the Dark Dimension (the “dark gravitons”) as an unavoidable dark matter candidate. Observations allow such an extra dimension of size in the micron range. Finally, I will discuss how this scenario can naturally accommodate features recently observed by the DESI survey such as an effective dark energy equation of state that is smaller than -1.

Thursday, Apr. 16, 2026
8:00–8:30 am
Breakfast
8:30–9:30 am
MC Gonzalez-Garcia, YITP Stony Brook & ICREA U. Barcelona: Massive Neutrinos in 2026: What we know, what we do not know (yet?), and what we do not understand
Abstract: In this talk I will present an update of the current understanding (and some not understanding) of the neutrino masses and the lepton mixing and some other minimal SM extensions as derived from direct scrutiny of the results of neutrino flavour oscillation experiments, some other laboratory probes, and the cosmos.
9:30–10:00 am
Coffee Break
10:00–11:00 am
Miguel Montero, IFT, Madrid: Neutrinos and B-L symmetry in the Dark Dimension scenario
Abstract: The Dark Dimension proposes the existe of a micrometer-sized large extra dimension, whose size is tied to the observed small vacuum energy. I will review the scenario, and then discuss how to embed the B-L global symmetry of the SM, focusing on one possibility which leads to an explanation of the observed coincidence between neutrino mass scale and the vacuum energy scale, while leading to 3 light species of right-handed neutrinos. I will also briefly discuss potential opportunities for detection of the resulting neutrino oscillations.
11:00–11:30 am
Coffee Break
11:30 am–12:30 pm
Ignatios Antoniadis, IAS, Princeton: Searching for the dark dimension in neutrino experiments
Abstract: Micron size extra dimensions offer a possibility to explain the smallness of neutrino masses if the right-handed neutrino propagates in the higher dimensional bulk. I will discuss the theoretical framework and the experimental signatures of this proposal in present and future experiments of KATRIN prototype, aiming to measure the magnitude of neutrino masses and to search for extra sterile-type species.
12:30–1:30 pm
Lunch Break (catered)
1:30–2:30 pm
Mike Boylan-Kolchin, UT Austin: Galaxies as Tracers of the Matter Density Field
Abstract: Galaxy formation is often (rightly) thought of as involving a complex interplay of messy astrophysical processes, but it also traces the nonlinear evolution of the matter density in the Universe. Remarkably, it appears that properties of this nonlinear field are intimately connected to properties of the initial linear fluctuations and some basic physics of dark matter interactions. I will explore some of these connections, with applications that include the surprisingly fast evolution of early galaxy formation as revealed by JWST and properties of the lowest-mass dark matter clumps capable of hosting galaxies in the local Universe.
2:30–3:00 pm
Coffee Break
3:00–4:00 pm
Matt Reece, Harvard: Axions from String Theory, and String Theory from Axions
Abstract: String theory compactifications contain the right ingredients to produce axion fields that might solve the Strong CP problem or contribute to dark matter or dynamical dark energy in our universe. After briefly reviewing some of these ingredients, I will frame the inverse question: suppose that an axion is discovered, and its decay constant is measured in an experiment. Could this help us to locate ourselves in the string landscape? In particular, I will discuss how an axion could give us clues about the fundamental string scale and the scale of supersymmetry breaking.

Millennium Prize Problems Lecture - Peter Sarnak: Riemann Hypothesis
2026-04-15 17:00 - 18:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Date: April 15, 2026
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall C, 1 Oxford St., Cambridge MA
Speaker: Peter Sarnak, Institute for Advanced Study
Title: The Riemann Hypothesis
Abstract: After reviewing the hypothesis as put forth by Riemann we discuss its generalizations and analogues. We highlight a few of their implications and workarounds, and probing their truths.
Read more about the Riemann Hypothesis at the Clay Math website.
Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

CMSA/Tsinghua Math-Science Literature Lecture: Nicolai Reshetikhin (Tsinghua): Asymptotic representation theory
2026-04-22 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA/Tsinghua Math-Science Literature Lecture
Date: April 22, 2026
Time: 9:00 – 10:30 am ET
Location: via Zoom Webinar
Speaker: Nicolai Reshetikhin, Yau Mathematical Sciences Center, Tsinghua University
Title: Asymptotic representation theory
Abstract: Loosely speaking asymptotic representation theory studies representations of “large” groups or algebras. One of the first results in this direction is the study of Plancherel measures on the symmetric group $S_N$ in the limit $Nto infty$ by Vershik and Kerov and Logan and Shepp. The first part of the talk will be an overview of results on statistics of irreducible representations in large tensor products. Then we focus on more modern results on statistics of tilting and projective modules in large tensor products and on how some problems in asymptotic representation theory are related to dimer models in statistical mechanics.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Sixth Annual Yip Lecture | Regina Barzilay, MIT: Can machine learning methods design drugs?
2026-04-23 16:00 - 17:00 | Massachusetts
Sixth Annual Yip Lecture
Date: April 23, 2026
Time: 4:00–5:00 pm ET
Location: Harvard Science Center Hall A & via Zoom Webinar
Speaker: Regina Barzilay, MIT
Title: Can ML methods design drugs?
Abstract: Today, life sciences are driven by prohibitively expensive wet lab experimentations, which limit the pace and scope of discovery. This talk focuses on AI algorithms that enable in-silico modeling of biological processes. Specifically, I will focus on algorithms for molecular and cellular modeling. I will highlight several successful examples where these algorithms have already transformed drug discovery. In the second part of the talk, I want to focus on problems where current methods fail to deliver as expected, motivating the need for algorithmic innovations.
Bio: Regina Barzilay is a School of Engineering Distinguished Professor for AI and Health in the Department of Electrical Engineering and Computer Science (EECS) at MIT. Since 2018, she has been the AI faculty lead for the MIT Jameel Clinic and a member of MIT CSAIL.
She is a member of three national academies, including the National Academy of Engineering, the National Academy of Medicine, and the American Academy of Arts & Sciences.
She is also a recipient of various awards, including a 2017 MacArthur fellowship “Genius Grant.” In 2020, she was awarded the Squirrel AI Award for Artificial Intelligence for the Benefit of Humanity. More recently, she has been recognized in the 2025 TIME100 AI List and awarded with the IEEE Frances E. Allen Medal for her development of innovative machine learning algorithms that have significantly advanced human language technology and transformed medical diagnostics and drug discovery.
She completed her PhD in Computer Science from Columbia University, and spent a year as a postdoc at Cornell University. Barzilay received her undergraduate degree from Ben-Gurion University of the Negev, Israel.
The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

Workshop on Calabi-Yau metrics and optimal transport
2026-05-18 09:00 - 2026-05-22 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Workshop on Calabi-Yau metrics and optimal transport
Dates: May 18–22, 2026
Location: Harvard CMSA, 20 Garden Street, Cambridge MA
Recent advances in the study of Calabi-Yau metrics have revealed an interesting connection with optimal transport, and the regularity theory for optimal transport is expected to play an increasingly important role in the study of Kähler geometry. The goal of this workshop is to bring together the optimal transport and complex geometry communities to investigate problems arising from these exciting developments.
Videos of selected talks are now available on the Youtube Playlist
Minicourse Speakers

Robert McCann, University of Toronto
Yang Li, Cambridge University

Workshop Speakers

Rolf Andreasson, Chalmers University, Sweden
Benjy Firester, MIT
Jakob Hultgren, Umea University, Sweden
Young-Heon Kim, University of British Columbia
Nam Le, Indiana University
Jiakun Liu, University of Sydney
Arghya Rakshit, University of Toronto
Gabor Szekelyhidi, Northwestern University
Yueqiao Wu, Johns Hopkins University

Organizers:

Tristan Collins, University of Toronto
Mattias Jonsson, University of Michigan
Connor Mooney, University of California, Irvine
Freid Tong, University of Toronto

Schedule (pdf)
Monday, May 18, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Tutorial: Yang Li, Cambridge University (via Zoom Webinar)
Title: On the metric SYZ conjecture
Abstract: For a polarised degeneration family of Calabi-Yau manifolds near the large complex structure limit, the metric SYZ conjecture asks for a special Lagrangian torus fibration on the generic region of the Calabi-Yau manifolds. I will summarize the progress on the metric SYZ conjecture so far, emphasizing on some more recent progress.
10:45–11:15 am
Break
11:15 am–12:30 pm
Tutorial: Robert McCann, University of Toronto
Title: A geometric approach to apriori estimates for optimal transport maps
Abstract: A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma-Trudinger-Wang condition is the Pogorelov second derivative bound. This translates to an a priori interior modulus of the differential estimate for smooth optimal maps. We describe a new derivation of this estimate with Brendle, Leger and Rankin which relies in part on Kim, McCann, and Warren’s observation that the graph of an optimal map becomes a volume maximizing non-timelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost. This unexpected links optimal transport to the plateau problem in geometry with split signature, and shows the key difficulty is showing the maximizing non-timelike submanifold is in fact (uniformly) spacelike. J. Reine Angew. Math. 817 (2024) 251-266 doi.org/10.1515/crelle-2024-0071 arXiv 2311.10208
12:30–2:00 pm
Lunch (catered)
2:00–3:15 pm
Talk: Nam Le, Indiana University
Title: Variational approach to degenerate Monge-Ampère equations with mixed measures and monotonicity
Abstract: In this talk, we will discuss the solvability and uniqueness for several degenerate Monge-Ampère equations including the Monge-Ampère eigenvalue problem in real Euclidean spaces that involve singular Borel measures. Our approach systematically analyzes the Monge-Ampère energy from the variational point of view and appropriately exploits monotonicity arguments. We will examine several essential tools: the mixed Monge-Ampère measure, Aleksandrov-Blocki-Jerison type maximum principles, convex envelope, comparison principles for subcritical equations, and integration by parts whose failure leads to symmetry breaking and nonuniqueness phenomena.
3:15–3:45 pm
Break
3:45–5:00 pm
Talk: Yueqiao Wu, Johns Hopkins University
Title: Valuative aspects of complete Calabi-Yau metrics of Euclidean volume growth
Abstract: The search of a complete Calabi-Yau metric on an affine variety X amounts to solving a complex Monge-Ampère equation subject to nice “boundary conditions” at infinity. In the case where X is the complement of an SNC anticanonical divisor on a Fano manifold, generalizing the work of Tian-Yau, Collins-Li showed that such boundary data can be extracted from solutions to certain real Monge-Ampère equations. If we require the metric to have Euclidean volume growth, however, it is understood that the boundary conditions should come from prescribing a Calabi-Yau asymptotic cone at infinity. This is the same as giving the algebro-geometric data of a valuation which induces a degeneration of X to a K-stable affine cone. In this talk, we will explain that such valuations in fact always come from Fano type compactifications of X, similar to the ones considered by Tian-Yau and Collins-Li. In addition, K-semistability of the affine cone can be characterized intrinsically by a valuative criterion on X. Based on joint work with Mattias Jonsson.

Tuesday, May 19, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Tutorial: Robert McCann, University of Toronto
Title: Trading linearity for ellipticity: A low regularity Lorentzian splitting theorem
Abstract: While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. It is thus of great interest to study the theory in low regularity settings. In the lecture, we establish a low regularity splitting theorem by sacrificing linearity of the d’Alembertian to recover ellipticity. We exploit a negative homogeneity $p$-d’Alembert operator for this purpose. The same technique yields a simplified proof of Eschenberg (1988) Galloway (1989) and Newman’s (1990) confirmation of Yau’s (1982) conjecture, bringing all three Lorentzian splitting results into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry. Based on joint work with Mathias Braun, Nicola Gigli, Argam Ohanyan, and Clemens Saemann: 1) arXiv 2501.00702 2) arXiv 2408.15968 3) arXiv 2410.12632 4) arXiv 2507.06836
10:45–11:15 am
Break
11:15 am–12:30 pm
Tutorial: Yang Li, Cambridge University (via Zoom Webinar)
Title: On the metric SYZ conjecture
Abstract: For a polarised degeneration family of Calabi-Yau manifolds near the large complex structure limit, the metric SYZ conjecture asks for a special Lagrangian torus fibration on the generic region of the Calabi-Yau manifolds. I will summarize the progress on the metric SYZ conjecture so far, emphasizing on some more recent progress.
12:30–2:00 pm
Lunch Break
2:00–3:15 pm
Talk: Young-Heon Kim, University of British Columbia
Title: Trajectory Inference via Multi-marginal Schrödinger Bridges
Abstract: Trajectory inference arises in important scientific problems. In particular, biological development can be interpreted as a curve in the space of gene-expression distributions, and the goal is to infer this trajectory from observed data. There has been progress by using optimal transport (OT) as a way to interpolate between distributions. More recently, Schrödinger bridges, a stochastic generalization of OT, have been considered. In this talk, we discuss stability of such OT-based methods. This is joint work with Geoffrey Schiebinger and Rentian Yao.
3:15
Break
6:30 pm
Dinner

Wednesday, May 20, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Tutorial: Yang Li, Cambridge University (via Zoom Webinar)
Title: On the metric SYZ conjecture
Abstract: For a polarised degeneration family of Calabi-Yau manifolds near the large complex structure limit, the metric SYZ conjecture asks for a special Lagrangian torus fibration on the generic region of the Calabi-Yau manifolds. I will summarize the progress on the metric SYZ conjecture so far, emphasizing on some more recent progress.
10:45–11:15 am
Break
11:15 am–12:30 pm
Tutorial: Robert McCann, University of Toronto
Title: The monopolist’s free boundary problem in the plane: an excursion into the economic value of private information
Abstract: The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design. For multidimensional spaces of consumers (i.e. agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.
If the monopolists costs grow quadratically with the product type we show that a partially smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers. We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.
Based on works with Kelvin Shuangjian Zhang, Cale Rankin, and Lucas O’Brien in various combinations:
1) Math. Models Methods Appl. Sci. 34 (2024) 2351-2394; 2) J. Convex Anal. (Rockafellar 90 Issue), 32 (2) (2025) 579-584; 3) arXiv 2303.04937; 4) arxiv 2412.15505; 5) arXiv 2603.14100.

Thursday, May 21, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Talk: Gabor Szekelyhidi, Northwestern University
Title: Nondegenerate Neck Pinches along the 2d Lagrangian mean curvature flow
Abstract: The Thomas-Yau-Joyce conjecture predicts that the mean curvature flow can be used to decompose Lagrangian submanifolds in Calabi-Yau manifolds into special Lagrangian building blocks. The basic mechanism for this decomposition is given by neck pinches. I will discuss work on the behavior of such neck-pinch singularities, in particular the class of nondegenerate neck pinches, which satisfy certain properties conjectured by Joyce. The construction also relates to work of Neves on finite time singularity formation.
10:45–11:15 am
Break
11:15 am–12:30 pm
Talk: Rolf Andreasson, Chalmers University, Sweden
Title: Optimal transport between boundaries of dual reflexive polytope
Abstract: I will present an optimal transport problem between the boundaries of a pair of reflexive polytopes. Under a certain structural condition on its solution, this problem is related the study of metric degenerations of families of Calabi–Yau hypersurfaces in the corresponding toric Fano variety. A better understanding of such solutions and their regularity would shed light on several aspects of the degeneration and conjectural Gromov–Hausdorff limit, and I will present some open directions of research. This is based on joint work with Jakob Hultgren, Mattias Jonsson, Enrica Mazzon and Nicholas McCleerey.
12:30–2:00 pm
Lunch Break
2:00–3:15 pm
Talk: Jakob Hultgren, Umea University, Sweden
Title: Affine mondoromy, cost functions and real Monge-Ampère equations
Abstract: Recent results of Blum-Liu and Y. Li show that the metric SYZ conjecture holds. The solution hinges on the existence of valuatively independent bases for spaces of sections of the polarising bundle. These bases induce a cost function, providing a link between the non-Archimedean Monge-Ampère equation and optimal transport. The resulting SYZ-fibration is constructed on a large but non-explicit set. In order to better understand this set, more information about the cost function is arguably needed. We propose a cost function explicitly computable from the monodromy of an affine structure on the essential skeleton. In the case of the Fermat family of cubic curves, this cost function (as opposed to the one attained from the ambient projective space) can be shown to agree with the one attained from a valuatively independent basis. We conjecture that this equality of cost functions holds in general, and demonstrate in examples how the explicit cost function can be used to directly produce solutions to real Monge-Ampère equations on the essential skeleton. Joint work with Sohaib Khalid.
3:15–3:45 pm
Break
3:45–5:00 pm
Talk: Benjy Firester, MIT
Title: Free boundary Monge-Ampere equations with applications to optimal transport and Calabi-Yau geometry
Abstract: I will present a variational framework to solve a general class of free-boundary Monge-Ampère equations. This approach combines the classical first and second boundary value problems by imposing both the boundary data and the gradient image of the solution. I will explore applications to the Monge-Ampère eigenvalue problem, convex reconstruction theorems, and geometric problems including a hemispherical Minkowski problem, Calabi-Yau metrics, and free boundary toric Kähler–Einstein/Kähler-Ricci soliton metrics. I will also discuss the connection to the boundary regularity of optimal transport.

Friday, May 22, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Talk: Jiakun Liu, University of Sydney
Title: Free boundary problems in optimal transportation
Abstract: In this talk, I will present some recent results on the regularity of free boundaries in optimal transportation, including higher-order regularity, global regularity, and a model case involving multiple targets. These results are based on a series of joint works with Shibing Chen, Xianduo Wang, and Xu-Jia Wang.
10:45–11:15 am
Break
11:15 am–12:30 pm
Talk: Arghya Rakshit, University of Toronto
Title: Solutions to the Monge–Ampère equation with singular structures
Abstract: We construct examples of solutions to the Monge–Ampère equation with point masses exhibiting polyhedral singular structures. We further analyze the stability of these singular sets under small perturbations of the data. In addition, we construct solutions whose Monge–Ampère measure contains a singular component supported on lower-dimensional sets and we study the regularity of such solutions.

Introduction to First Proof: A conversation
2026-06-03 13:00 - 14:00 | Virtual
Introduction to First Proof: A conversation
Date: June 3, 2026
Time: 1:00–2:00 pm
Location: via Webinar
Harvard CMSA Director Dan Freed will lead a dialogue with First Proof Editors Mohammed Abouzaid (Stanford), Nikhil Srivastava (UC Berkeley), Rachel Ward (UT Austin), and Lauren Williams (Harvard) to explore the origins and goals of First Proof, sample some of their current work, and discuss their future plans. This hour-long conversation will give a window into the fast-moving and deep interaction of Mathematics and AI.

First Proof, Second Batch: Results
2026-06-10 13:00 - 14:00 | Virtual
First Proof, Second Batch: Results
Date: June 10, 2026
Time: 1:00–2:00 pm ET
Location: via Webinar
The First Proof Editors present the results of their Second Batch benchmark testing on AI systems.
First Proof Editors

Mohammed Abouzaid (Stanford)
Nikhil Srivastava (UC Berkeley)
Rachel Ward (UT Austin)
Lauren Williams (Harvard)

Big Data Conference 2026
2026-09-03 09:00 - 2026-09-04 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Big Data Conference 2026
Dates: Sep. 3–4, 2026
Location: Harvard University CMSA, 20 Garden Street, Cambridge MA & via Zoom
The Big Data Conference features 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.
Register to attend in person
Register for Zoom Webinar

Confirmed Speakers

SueYeon Chung, Harvard
Bailey Flanigan, MIT
Sergey Ovchinnikov, MIT
Ariel Procaccia, Harvard
Adit Radhakrishnan, MIT
Andrew Sutherland, MIT
Chris Wiggins, Columbia
Rex Ying, Yale
Wei Zhou, Harvard

Organizers

Michael Desai, Harvard
Michael R. Douglas, CMSA
Yannai Gonczarowski, Harvard
Melanie Weber, Harvard

Thursday, Sep. 3, 2026
8:45–9:10 am
Breakfast
9:10–9:15 am
Introductions
9:15–10:15 am
Rex Ying, Yale
A Riemannian Geometry Perspective on Foundation Models
Abstract: In development of foundation models and LLMs, Euclidean space has been the de facto geometric setting for machine learning architectures. However, at a large scale, real-world data often exhibit inherently non-Euclidean structures, such as multi-way relationships, hierarchies, symmetries, and non-isotropic scaling, in a variety of domains, such as languages, vision, and the natural sciences. It is challenging to effectively capture these structures within the constraints of Euclidean spaces. This talk will cover my research that moves beyond Euclidean geometry to maintain the scaling law for the next generation of foundation models. By adopting these geometries, foundation models could more efficiently leverage the structures. Task-aware adaptability that dynamically reconfigures embeddings to match the geometry of downstream applications, could further enhance efficiency and expressivity. This talk will demonstrate the advantages of hyperbolic geometry in building foundation models from individual non-Euclidean components, LLM pre-training / fine-tuning, retrieval-augmented generation, and mixture of expert models. We further present a comprehensive library that greatly simplifies the process of building and adapting foundation models in non-Euclidean. Finally, the talk will enumerate a few promising directions that are being actively explored in this domain.
10:15–10:30 am
Break
10:30–11:30 am
Adit Radhakrishnan, MIT
Toward universal steering and monitoring of AI models
Abstract: Artificial intelligence (AI) models contain much of human knowledge. Understanding the representation of this knowledge will lead to improvements in model capabilities and safeguards. Building on advances in feature learning, we developed an approach for extracting linear representations of semantic notions or concepts in AI models. We showed how these representations enabled model steering, through which we exposed vulnerabilities and improved model capabilities. We demonstrated that concept representations were transferable across languages and enabled multiconcept steering. Across hundreds of concepts, we found that larger models were more steerable and that steering improved model capabilities beyond prompting. We showed that concept representations were more effective for monitoring misaligned content than for using judge models. Our results illustrate the power of internal representations for advancing AI safety and model capabilities.
11:30 am–12:45 pm
Lunch
12:45–1:45 pm
Andrew Sutherland, MIT
Crowdsourcing mathematical databases
Abstract: The L-functions and Modular Forms Database (LMFDB) is one of the largest online repositories of mathematical research data. It contains comprehensive catalogs of mathematical objects that arise in the context of the Langlands program, including number fields, elliptic curves, modular forms, and higher-dimensional analogs of these objects. In cooperation with the Foundation for Science and AI Research (SAIR), we recently ran a competition aimed at solving the inverse Galois problem over Q in degree 24 (IGP24). This goal was achieved by collecting more than 50 million candidate number fields submitted by more than 100 teams, including examples that realize all 25,000 transitive permutation groups of degree 24 as Galois groups. AI tools and computer algebra systems played a central role in this project, both in building the infrastructure to run the contest and in enhancing the capabilities of participants.
1:45–2:00 pm
Break
2:00–3:00 pm
Bailey Flanigan, MIT
Algorithmic Tools for Trading Off Sortition Ideals
Abstract: Citizens’ assemblies and other deliberative minipublics — representative groups of everyday people convened to deliberate on a policy issue and then make recommendations — are now used by governments around the world. Choosing who sits on these panels is the problem of sortition: randomly selecting a small group of citizens that represents the broader population. Sortition has been the subject of substantial computer science research in recent years, and the resulting algorithms are now widely used in practice. This talk will describe the key challenges that arise in the practice of sortition, and the algorithmic tools that have been developed to navigate them optimally.
3:00–3:15 pm
Break
3:15–4:15 pm
Chris Wiggins, Columbia
Prescriptive Learning at Scale: Applied and Deployed Decision-Making
Abstract: Many applications in health and industry require making decisions while learning how the world responds to them: e.g., which article to recommend, which treatment to assign, or when to show a paywall. This problem is prescriptive, what to do rather than what is true. Statistical decision theory has taken up the problem repeatedly since Neyman’s interwar work; recent decades have brought a new wave of results, motivated in large part by the opportunity to make and evaluate decisions at scale online. I will highlight a thread of 21st-century results running through adaptive experimentation, contextual bandits, and off-policy evaluation: they combine advances in supervised learning with fundamental ideas from statistical sampling, and now shape digital products, from content recommendation to marketing, as well as adaptive interventions in health.

Friday, Sep. 4, 2026
8:45–9:15 am
Breakfast
9:15–10:15 am
Ariel Procaccia, Harvard
No Generation Without Representation
Abstract: AI systems and democratic processes are confronting similar challenges around representation. I examine two related questions that cut across both domains. First, how can AI enable democratic processes that handle vast spaces of opinions or statements while ensuring proportional representation of a population’s views? Second, when AI systems themselves provide normative guidance, whose viewpoints do they reflect, and can we make this precise? Drawing on social choice theory, I present formal frameworks and algorithms for both problems, showing that meaningful representation guarantees are feasible and practical.
10:15–10:30 am
Break
10:30–11:30 am
Wei Zhou, Harvard
Biobank-scale genetic discovery: from association testing to global meta-analysis
Abstract: Biobanks linking genomic data with electronic health records provide unprecedented opportunities for genetic discovery for complex human diseases, but they also pose analytical challenges that extend well beyond sample size. Within a biobank, association studies must account for population structure and relatedness, highly unbalanced case–control ratios, rare genetic variants, longitudinal and censored outcomes, and the computational demands of analyzing hundreds of thousands of individuals and millions of genetic variants. Across biobanks, additional challenges arise from differences in ancestry, phenotype definitions, recruitment strategies and genetic effects.In this talk, I will discuss statistical and computational methods developed to address these challenges at successive stages of biobank analysis. These include scalable generalized linear mixed models for binary traits, survival mixed models for censored time-to-event outcomes, and gene- and region-based tests that aggregate rare variants. I will describe statistical approximations and computational strategies that make these analyses feasible at biobank scale while maintaining calibration in the presence of relatedness and highly unbalanced phenotypes. I will then introduce the Global Biobank Meta-analysis Initiative (GBMI) and describe how genetic evidence can be combined across biobanks without sharing individual-level data. Examples from GBMI will illustrate how combining evidence across biobanks can increase statistical power through larger sample sizes and broaden genetic discovery through greater ancestral diversity.
11:30 am–12:45 pm
Lunch
12:45–1:45 pm
Sergey Ovchinnikov, MIT
1:45–2:00 pm
Break
2:00–3:00 pm
SueYeon Chung, Harvard

Fall CMSA Welcome Event
2026-09-03 16:15 - 17:30 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Fall CMSA Welcome Event
Date: September 3, 2026
Time: 4:15 pm
Location: CMSA Common Room, 20 Garden Street, Cambridge MA

All CMSA and Math affiliates are invited.

The Geometry of Machine Learning 2026
2026-09-08 09:00 - 2026-09-11 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
The Geometry of Machine Learning 2026
Dates: September 8–11, 2026
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA 02138 & via Zoom Webinar
Register to attend in person
Register for Zoom Webinar
Large language models are presently, and will increasingly, be complemented by other dimensions of intelligence: formal verification and energy-based optimizers, becoming parts of larger ecosystems. Can AIs reason geometrically and can we use geometry to reveal how data is currently processed in NNs? Can AIs reveal the geometry of mathematics, as well as studying geometry as a subject within math. This conference is intended to continue the discussion of these topics.
Confirmed Speakers:

Nada Amin, Harvard
Randall Balestriero, Brown
Michael Brenner, Harvard and Google
Bennet Chow, UCSD
Surya Ganguli, Stanford
Boris Hanin, Princeton
Roi Holtzman, Oxford
Robert Koirala, UCSD
Dmitry Krotov, Dynamical Mind
Slava Krushkal, Virginia
Jared Duker Lichtman, Stanford
Mike Mulligan, UCR, Logical Intelligence
Luca Pesce, Harvard
Gabriel Poesia, U Michigan
Mathew Vanherreweghe, Logical Intelligence
Sean Welleck, CMU (via Zoom)
Mattiew Wyart, JHU

Organizers: Michael R. Douglas (CMSA) and Mike Freedman (CMSA)

Schedule
Tuesday, Sep. 8, 2026
8:15–8:45 am
Breakfast
8:45–9:30 am
Mike Mulligan, UCR, Logical Intelligence
9:45–10:30 am
Boris Hanin, Princeton
10:30–11:00 am
Break
11:00–11:45 am
Slava Krushkal, University of Virginia
12:00–12:45 pm
Robert Koirala and Bennett Chow, UCSD
AI for Ricci flow: Discovery and Formalization

Wednesday, Sep. 9, 2026
8:15–8:45 am
Breakfast
8:45–9:30 am
Surya Ganguli, Stanford
9:45–10:30 am
Mathew Vanherreweghe, Logical Intelligence
10:30–11:00 am
Break
11:00–11:45 am
Michael Brenner, Harvard and Google
12:00–12:45 pm
Mattiew Wyart, JHU
Learn from your own latents, not from tokens
Abstract: Language models need more than a hundred thousand times the data a child does. One explanation is that predicting raw tokens is simply the wrong level: methods like data2vec and JEPA instead train a network to predict its own internal representations, with strong empirical results but no theory of why. Using a hierarchical grammar that models that language and images have a hidden hierarchical structure, we quantify the gain exactly. Token-level learning needs a number of examples growing exponentially with the depth of the hierarchy; latent prediction needs a number independent of it. We also show data2vec performs this hierarchical prediction implicitly, which suggests that explicitly stacking levels – as in H-JEPA – yields little.

Thursday, Sep. 10, 2026
8:15–8:45 am
Breakfast
8:45–9:30 am
Dmitry Krotov, Dynamical Mind
Dense Associative Memory: Physical systems for novel AI architectures
Abstract: Dense Associative Memories are recurrent neural networks with fixed-point attractor states that are described by an energy function. In contrast to conventional Hopfield Networks, which were popular in the 1980s, Dense Associative Memories have a very large information storage capacity, making them appealing tools for many problems in AI. In this talk, I will provide an intuitive understanding and mathematical framework for this class of models and give examples of problems in AI that can be tackled using these new ideas. Specifically, I will explore the relationship between Dense Associative Memories and transformers. I will present a neural network called the Energy Transformer, which unifies energy-based modeling, associative memories, and transformers in a single architecture. I will demonstrate how Energy Transformers can be used for challenging tasks in image processing, solve partial differential equations, and serve as computational modules for energy-based language modeling. I will also discuss an exciting possibility of mapping these models onto analog hardware accelerators, which could enable much more energy-efficient inference compared to GPUs.
9:45–10:30 am
Roi Holtzman, Oxford
10:30–11:00 am
Break
11:00–11:45 am
Sean Welleck, CMU (via Zoom)
12:00–12:45 pm
Gabriel Poesia, University of Michigan (via Zoom)
Friday, Sep. 11, 2026
8:15–8:45 am
Breakfast
8:45–9:30 am
Nada Amin, Harvard
9:45–10:30 am
Randall Balestriero, Brown University
10:30–11:00 am
Break
11:00–11:45 am
Luca Pesce, Harvard CMSA
A spiked perspective on feature learning with gradient-based methods
Abstract: Depth is widely believed to give neural networks a clear computational advantage over shallow models, and making this belief precise is a central problem in learning theory. We study a controlled high-dimensional setting where this can be done. The targets are hierarchical: the relevant structure is distributed across latent subspaces of decreasing dimension, so that a shallow model must resolve all of it simultaneously, while a deep network does not. Each layer forms an intermediate representation, allowing learning to proceed in stages, with every stage reducing the effective dimension of the remaining problem. We analyze this staged mechanism through the gradient descent dynamics of a deep network yielding a sharp separation in sample complexity between shallow and deep architectures. The result is a concrete account of why depth allows such functions to be learned from substantially fewer samples than shallow methods require.
12:00–12:45 pm
Talk tba

Support provided by Logical Intelligence.

Ahlfors Lectures: Mohammed Abouzaid
2026-09-16 - 2026-09-17 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Ahlfors Lectures
Dates: September 16 & 17, 2026
Time: 4:00–5:00 pm
Location: Science Center Hall A
Speaker: Mohammed Abouzaid (Stanford)
Register to attend in person or online at: https://www.math.harvard.edu/event/ahlfors-lecture-series-mohammed-abouzaid/

September 16, 2026
Title: Framed bordism and nearby Lagrangians
Abstract: The problem of understanding Lagrangian embeddings in symplectic manifolds remains a mystery. The simplest class of symplectic manifolds are the cotangent bundles of smooth manifolds, which is the setting for the most prominent open problem in the subject: Arnold’s nearby Lagrangian conjecture which asserts the connectedness of the space of Lagrangians for which the relative class of the symplectic form vanishes. In particular, the conjecture asserts that all such submanifolds are diffeomorphic. In view of recent work with Alvarez-Gavela, Courte, and Kragh, as well as work of Procelli and Smith, I will use the question as a motivation for joint work with Andrew Blumberg on a bordism-theoretic approach to define bordism-valued refinements of Floer homology.

September 17, 2026
Title: Complex bordism and Hamiltonian fibrations
Abstract: As a consequence of his theory of weights, Deligne proved that smooth complex projective varieties, carrying a smooth morphism to the projective line, have rational cohomology groups that are isomorphic to the tensor product of the cohomology of the fibre with that of the base. In joint work with McLean and Smith, we should that the result not only holds for symplectic manifolds, but is in fact a consequence of an analogous result that holds for complex cobordism. The proof identifies bordism of “stably complex derived orbifolds” as the correct framework in which to formulate these results, and I will discuss joint work with Shaoyun Bai to understand these bordism groups.
The Ahlfors Lecture Series is presented by the Harvard University Math Department in memory of Professor Lars Ahlfors.

CMSA/Tsinghua Math-Science Literature Lecture: Robert Gompf: On classifying smoothings of R^4–from the beginning to the present
2026-09-16 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA/Tsinghua Math-Science Literature Lecture
Date: September 16, 2026
Time: 9:00 – 10:30 am ET
Location: CMSA G10, 20 Garden Street & via Zoom Webinar

Please register to attend
In-person registration

Zoom Webinar registration

Speaker: Robert E. Gompf, University of Texas, Austin
Title: On classifying smoothings of R^4–from the beginning to the present
Abstract: The field of 4-manifold topology was born around 1981 with two independent revolutionary breakthroughs, by Fields medalists Michael Freedman and Simon Donaldson. Freedman showed that topological 4-manifolds obeyed the well-developed theory of higher-dimensional manifolds, while Donaldson used nonlinear analysis (gauge theory) to show that smooth 4-manifolds are quite different from their high-dimensional counterparts. The clash between the two theories makes 4-manifold topology uniquely intricate within manifold theory. One of the most startling consequences is that unlike in other dimensions, Euclidean 4-space R^4 admits exotic smoothings. Equivalently, there are smooth 4-manifolds that are homeomorphic, but not diffeomorphic, to R^4. We will trace the classification theory of these manifolds from their discovery to the most recent development (joint with Aristotelis Panagiotopoulos) that they cannot be classified by any countable family of numerical invariants.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Workshop on Lagrangian Floer theory and applications
2026-09-28 08:00 - 2026-10-02 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Workshop on Lagrangian Floer Theory and Applications
Dates: September 28 – October 2, 2026
Location: CMSA G10, 20 Garden St., Cambridge MA 02138
This workshop is part of the Lagrangian Floer Theory and Applications Program
Registration Form
Speakers will include:

Mohammed Abouzaid, Stanford University
Octav Cornea, University of Montréal
Tobias Ekholm, Uppsala University
Sheel Ganatra, USC
Andrew Hanlon, University of Oregon
Yanki Lekili, Imperial College London
Georgios Dimitroglou Rizell, Uppsala University
Paul Seidel, MIT
Egor Shelukhin, University of Montréal
Vivek Shende, Berkeley/Syddansk

Organizers: Denis Auroux (Harvard), Jonny Evans (Lancaster), and Chris Woodward (Rutgers)

Schedule outline
Monday 9/28/26

9:00 – 9:30
Breakfast

9:30 – 10:30
Talk 1

10:30 – 11:00
Break

11:00 – 12:00
Talk 2

12:00 – 1:30
Lunch

1:30 – 2:30
Talk 3

2:30 – 3:00
Break

3:00 – 4:00
Talk 4

Tuesday 9/29/26

9:00 – 9:30
Breakfast

9:30 – 10:30
Talk 5

10:30 – 11:00
Break

11:00 – 12:00
Talk 6

12:00 – 1:30
Lunch

1:30 – 2:30
Talk 7

2:30 – 3:00
Break

3:00 – 4:00
Talk 8

Wednesday 9/30/26

9:00 – 9:30
Breakfast

9:30 – 10:30
Talk 9

10:30 – 11:00
Break

11:00 – 12:00
Talk 10

12:00 – 1:30
Lunch

1:30 – 2:30
Talk 11

2:30 – 3:00
Break

3:00 – 4:00
Talk 12

Thursday 10/1/26

9:00 – 9:30
Breakfast

9:30 – 10:30
Talk 13

10:30 – 11:00
Break

11:00 – 12:00
Talk 14

12:00 – 1:30
Lunch

1:30 – 2:30
Talk 15

2:30 – 3:00
Break

3:00 – 4:00
Talk 16

6:30
Dinner

Friday 10/2/26

9:00 – 9:30
Breakfast

9:30 – 10:30
Talk 17

10:30 – 11:00
Break

11:00 – 12:00
Talk 18

12:00 – 1:30
Lunch

1:30 – 2:30
Talk 19

2:30 – 3:00
Break

3:00 – 4:00
Talk 20

Math Science Lectures in Honor of Raoul Bott | Daniel Litt, U Toronto
2026-10-19 16:00 - 17:00 | Massachusetts
Dates: October 19 & 21, 2026
Time: 4:00–5:00 pm ET
Location: Science Center Lecture Hall A and via Webinar
In-person registration
Webinar Registration
Speaker: Daniel Litt, University of Toronto & Benedict H. Gross Distinguished Visitor, Harvard Mathematics Department
Lecture 1: The arithmetic of differential equations
Monday, October 19, 2026, followed by a reception in the Math Dept Austine & Chilton McDonnell Common Room, 4th floor

Lecture 2: Non-abelian motives
Wednesday, October 21, 2026

Harvard Mathematics Professor Raoul Bott (1923 – 2005), was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem.

Math Science Lectures in Honor of Raoul Bott | Daniel Litt, Benedict H. Gross Distinguished Visitor, Harvard Mathematics Department
2026-10-21 16:00 - 17:00 | Massachusetts

CMSA/Tsinghua Math-Science Literature Lecture: Peng Shan, Tsinghua University
2026-10-29 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA/Tsinghua Math-Science Literature Lecture
Date: Thursday, October 29, 2026
Time: 9:00 – 10:30 am ET
Location: online via Zoom Webinar

Please register to attend
Zoom Webinar registration

Speaker: Peng Shan, Tsinghua University
Title TBA

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Northeast Conference on Categorical Methods
2026-11-06 09:00 - 2026-11-07 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Northeast Conference on Categorical Methods
Dates: November 6–​7, 2026
Location: CMSA G10, 20 Garden St., Cambridge MA 02138

This conference is intended to bring together researchers in all areas of mathematics and mathematical physics whose work involves the use of methods from categorical algebra, abstract homotopy theory, and higher category theory. Our goal is to showcase new, exciting research and offer an avenue for researchers from the Northeast US to discuss their work and collaborate with people in this ever-growing community.

There will be time for 4 short talks by early-career mathematicians.

REGISTRATION FORM (link) Deadline: September 20, 2026
Zoom Webinar Registration

Invited Speakers

Andrew Blumberg, Columbia University
Soren Galatius, Columbia University
Tony Pantev, University of Pennsylvania
Nick Rozenblyum, University of Toronto
Tomer Schlank, University of Chicago/Harvard University
Brian Williams, Boston University
Inna Zakharevich, Cornell University

Organized by Dan Freed, Harvard Math & CMSA; Owen Gwilliam, UMass Amherst; and Lorenzo Riva, Harvard CMSA

Stable Homotopy Theory and Arithmetic Geometry Boot Camp
2027-02-22 09:00 - 2027-02-26 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Stable Homotopy Theory and Arithmetic Geometry Boot Camp
Dates: February 22-26, 2027

Stable Homotopy Theory and Arithmetic Geometry Workshop I
2027-03-01 09:00 - 2027-03-05 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Stable Homotopy Theory and Arithmetic Geometry Workshop I
Dates: March 1–5, 2027

Seventh Annual Yip Lecture | L. Mahadevan, Harvard
2027-03-03 16:00 - 17:00 | Massachusetts
Sixth Annual Yip Lecture
Date: Wednesday, March 3, 2027
Time: 4:00–5:00 pm
Location: Harvard Science Center Hall D & via Zoom Webinar
Speaker: L. Mahadevan, Harvard

The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

2027 Ding Shum Lecture: Avi Wigderson, IAS
2027-03-31 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
2027 Ding Shum Lecture
Date: Wednesday, March 31, 2027
Time: 4:00 pm
Location: Harvard Science Center Hall D & via Zoom Webinar

Speaker: Avi Wigderson, IAS

A reception will follow the talk in the Mathematics Department Austine & Chilton McDonnell Common Room, 4th floor, Science Center.

This event is made possible by the generous funding of Ding Lei and Harry Shum.

The Conformal Bootstrap and Mathematics
2027-04-05 - 2027-04-09 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
The Conformal Bootstrap and Mathematics
The conformal bootstrap is a method originating in physics that seeks to define and study conformal quantum field theories in two and higher dimensions. In recent years, surprising connections have emerged between the conformal bootstrap and several areas of mathematics, including the spectral theory of automorphic forms, the sphere packing problem, and related questions in geometry, analysis, and number theory. These developments suggest both that the conformal bootstrap “knows” a remarkable amount of mathematics, and that these mathematical structures can offer new ways of understanding quantum field theory.
This workshop will bring together experts from conformal field theory, number theory, spectral geometry, discrete geometry, and related fields to explore these emerging connections. One goal is to investigate how ideas and techniques from the conformal bootstrap can lead to new advances in mathematics. A complementary goal is to use insights from these mathematical disciplines to sharpen our understanding of conformal field theory and quantum field theory more broadly.
Organizers: James Bonifacio, University of Mississippi; Henry Cohn, MIT; Dalimil Mazáč, Institute of Theoretical Physics, Saclay; and Peter Sarnak, IAS

Stable Homotopy Theory and Arithmetic Geometry Workshop II
2027-05-03 09:00 - 2027-05-07 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Stable Homotopy Theory and Arithmetic Geometry Workshop II
Dates: May 3–7, 2027
