The Geometry of Machine Learning
2025-09-15 09:00 - 2025-09-18 17:00 (America/New_York) | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
The Geometry of Machine Learning
Dates: September 15–18, 2025
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA 02138
Despite the extraordinary progress in large language models, mathematicians suspect that other dimensions of intelligence must be defined and simulated to complete the picture. Geometric and symbolic reasoning are among these. In fact, there seems to be much to learn about existing ML by considering it from a geometric perspective, e.g. what is happening to the data manifold as it moves through a NN? How can geometric and symbolic tools be interfaced with LLMs? A more distant goal, one that seems only approachable through AIs, would be to gain some insight into the large-scale structure of mathematics as a whole: the geometry of math, rather than geometry as a subject within math. This conference is intended to begin a discussion on these topics.
Speakers

Maissam Barkeshli, University of Maryland
Eve Bodnia, Logical Intelligence
Adam Brown, Stanford
Bennett Chow, USCD & IAS
Michael Freedman, Harvard CMSA
Elliot Glazer, Epoch AI
James Halverson, Northeastern
Jesse Han, Math Inc.
Junehyuk Jung, Brown University
Alex Kontorovich, Rutgers University
Yann Lecun, New York University & META*
Jared Duker Lichtman, Stanford & Math Inc.
Brice Ménard, Johns Hopkins
Michael Mulligan, UCR & Logical Intelligence
Patrick Shafto, DARPA & Rutgers University

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

Geometry of Machine Learning Youtube Playlist

Schedule
Monday, Sep. 15, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
James Halverson, Northeastern
Title: Sparsity and Symbols with Kolmogorov-Arnold Networks
Abstract: In this talk I’ll review Kolmogorov-Arnold nets, as well as new theory and applications related to sparsity and symbolic regression, respectively. I’ll review essential results regarding KANs, show how sparsity masks relate deep nets and KANs, and how KANs can be utilized alongside multimodal language models for symbolic regression. Empirical results will necessitate a few slides, but the bulk will be chalk.

10:00–10:30 am
Break

10:30–11:30 am
Maissam Barkeshli, University of Maryland
Title: Transformers and random walks: from language to random graphs
Abstract: The stunning capabilities of large language models give rise to many questions about how they work and how much more capable they can possibly get. One way to gain additional insight is via synthetic models of data with tunable complexity, which can capture the basic relevant structures of real data. In recent work we have focused on sequences obtained from random walks on graphs, hypergraphs, and hierarchical graphical structures. I will present some recent empirical results for work in progress regarding how transformers learn sequences arising from random walks on graphs. The focus will be on neural scaling laws, unexpected temperature-dependent effects, and sample complexity.

11:30 am–12:00 pm
Break

12:00–1:00 pm
Adam Brown, Stanford
Title: LLMs, Reasoning, and the Future of Mathematical Sciences
Abstract: Over the last half decade, the mathematical capabilities of large language models (LLMs) have leapt from preschooler to undergraduate and now beyond. This talk reviews recent progress, and speculates as to what it will mean for the future of mathematical sciences if these trends continue.

Tuesday, Sep. 16, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Junehyuk Jung, Brown University
Title: AlphaGeometry: a step toward automated math reasoning
Abstract: Last summer, Google DeepMind’s AI systems made headlines by achieving Silver Medal level performance on the notoriously challenging International Mathematical Olympiad (IMO) problems. For instance, AlphaGeometry 2, one of these remarkable systems, solved the geometry problem in a mere 19 seconds!
In this talk, we will delve into the inner workings of AlphaGeometry, exploring the innovative techniques that enable it to tackle intricate geometric puzzles. We will uncover how this AI system combines the power of neural networks with symbolic reasoning to discover elegant solutions.

10:00–10:30 am
Break

10:30–11:30 am
Bennett Chow, USCD and IAS
Title: Ricci flow as a test for AI

11:30 am–12:00 pm
Break

12:00–1:00 pm
Jared Duker Lichtman, Stanford & Math Inc. and Jesse Han, Math Inc.
Title: Gauss – towards autoformalization for the working mathematician
Abstract: In this talk we’ll highlight some recent formalization progress using a new agent – Gauss. We’ll outline a recent Lean proof of the Prime Number Theorem in strong form, completing a challenge set in January 2024 by Alex Kontorovich and Terry Tao. We hope Gauss will help assist working mathematicians, especially those who do not write formal code themselves.

5:00–6:00 pm
Special Lecture: Yann LeCun, Science Center Hall C

Wednesday, Sep. 17, 2025

8:30–9:00 am
Refreshments

9:00–10:00 am
Michael Mulligan, UCR and Logical Intelligence
Title: Spontaneous Kolmogorov-Arnold Geometry in Vanilla Fully-Connected Neural Networks
Abstract: The Kolmogorov-Arnold (KA) representation theorem constructs universal, but highly non-smooth inner functions (the first layer map) in a single (non-linear) hidden layer neural network. Such universal functions have a distinctive local geometry, a “texture,” which can be characterized by the inner function’s Jacobian, $J(mathbf{x})$, as $mathbf{x}$ varies over the data. It is natural to ask if this distinctive KA geometry emerges through conventional neural network optimization. We find that indeed KA geometry often does emerge through the process of training vanilla single hidden layer fully-connected neural networks (MLPs). We quantify KA geometry through the statistical properties of the exterior powers of $J(mathbf{x})$: number of zero rows and various observables for the minor statistics of $J(mathbf{x})$, which measure the scale and axis alignment of $J(mathbf{x})$. This leads to a rough phase diagram in the space of function complexity and model hyperparameters where KA geometry occurs. The motivation is first to understand how neural networks organically learn to prepare input data for later downstream processing and, second, to learn enough about the emergence of KA geometry to accelerate learning through a timely intervention in network hyperparameters. This research is the “flip side” of KA-Networks (KANs). We do not engineer KA into the neural network, but rather watch KA emerge in shallow MLPs.

10:00–10:30 am
Break

10:30–11:30 am
Eve Bodnia, Logical Intelligence
Title:
Abstract: We introduce a method of topological analysis on spiking correlation networks in neurological systems. This method explores the neural manifold as in the manifold hypothesis, which posits that information is often represented by a lower-dimensional manifold embedded in a higher-dimensional space. After collecting neuron activity from human and mouse organoids using a micro-electrode array, we extract connectivity using pairwise spike-timing time correlations, which are optimized for time delays introduced by synaptic delays. We then look at network topology to identify emergent structures and compare the results to two randomized models – constrained randomization and bootstrapping across datasets. In histograms of the persistence of topological features, we see that the features from the original dataset consistently exceed the variability of the null distributions, suggesting that the observed topological features reflect significant correlation patterns in the data rather than random fluctuations. In a study of network resiliency, we found that random removal of 10 % of nodes still yielded a network with a lesser but still significant number of topological features in the homology group H1 (counts 2-dimensional voids in the dataset) above the variability of our constrained randomization model; however, targeted removal of nodes in H1 features resulted in rapid topological collapse, indicating that the H1 cycles in these brain organoid networks are fragile and highly sensitive to perturbations. By applying topological analysis to neural data, we offer a new complementary framework to standard methods for understanding information processing across a variety of complex neural systems.

11:30 am–12:00 pm
Break

12:00–1:00 pm
Alex Kontorovich, Rutgers University
Title: The Shape of Math to Come
Abstract: We will discuss some ongoing experiments that may have meaningful impact on what working in research mathematics might look like in a decade (if not sooner).

5:00–6:00 pm
Mike Freedman Millennium Lecture: The Poincaré Conjecture and Mathematical Discovery (Science Center Hall D)

Thursday, Sep. 18, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Elliott Glazer, Epoch AI
Title: FrontierMath to Infinity
Abstract: I will discuss FrontierMath, a mathematical problem solving benchmark I developed over the past year, including its design philosophy and what we’ve learned about AI’s trajectory from it. I will then look much further out, speculate about what a “perfectly efficient” mathematical intelligence should be capable of, and discuss how high-ceiling math capability metrics can illuminate the path towards that ideal.

10:00–10:30 am
Break

10:30–11:30 am
Brice Ménard, Johns Hopkins
Title:Demystifying the over-parametrization of neural networks
Abstract: I will show how to estimate the dimensionality of neural encodings (learned weight structures) to assess how many parameters are effectively used by a neural network. I will then show how their scaling properties provide us with fundamental exponents on the learning process of a given task. I will comment on connections to thermodynamics.

11:30 am–12:00 pm
Break

12:00–12:30 pm
Patrick Shafto, Rutgers
Title: Math for AI and AI for Math
Abstract: I will briefly discuss two DARPA programs aiming to deepen connections between mathematics and AI, specifically through geometric and symbolic perspectives. The first aims for mathematical foundations for understanding the behavior and performance of modern AI systems such as Large Language Models and Diffusion models. The second aims to develop AI for pure mathematics through an understanding of abstraction, decomposition, and formalization. I will close with some thoughts on the coming convergence between AI and math.

12:30–12:45 pm
Break

12:45–2:00 pm
Mike Freedman, Harvard CMSA
Title: How to think about the shape of mathematics
Followed by group discussion

Support provided by Logical Intelligence.

Geometry of Machine Learning Special Lecture: Yann LeCun
2025-09-16 17:00 - 18:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Geometry of Machine Learning Special Lecture: Yann LeCun
Title: Self-Supervised Learning, JEPA, World Models, and the future of AI
Date: Tuesday, Sep. 16, 2025
Time: 5:00 pm ET
Location: Harvard Science Center, Hall C & via Zoom Webinar

Millennium Prize Problems Lecture - Michael Freedman: The Poincaré Conjecture and Mathematical Discovery
2025-09-17 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Millennium Prize Problems Lecture
Date: September 17, 2025
Location: Harvard Science Center Hall D & via Zoom Webinar
Time: 5:00–6:00 pm
Speaker: Michael Freedman, Harvard CMSA and Logical Intelligence
Title: The Poincaré Conjecture and Mathematical Discovery
Abstract: The AI age requires us to re-examine what mathematics is about. The Seven Millenium Problems provide an ideal lens for doing so. Five of the seven are core mathematical questions, two are meta-mathematical – asking about the scope of mathematics. The Poincare conjecture represents one of the core subjects, manifold topology. I’ll explain what it is about, its broader context, and why people cared so much about finding a solution, which ultimately arrived through the work of R. Hamilton and G. Perelman. Although stated in manifold topology, the proof requires vast developments in the theory of parabolic partial differential equations, some of which I will sketch. Like most powerful techniques, the methods survive their original objectives and are now deployed widely in both three- and four-dimensional manifold topology.

Read more about the Poincaré Conjecture 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

Mathematical foundations of AI
2025-10-06 09:00 - 2025-10-10 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Mathematical foundations of AI
Date: October 6–10, 2025
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA & via Zoom
Artificial intelligence (AI) has achieved unprecedented advances, yet our theoretical understanding lags significantly behind. This gap poses a significant obstacle to improving AI’s safety and reliability. Since the classical tools of learning theory have proven insufficient for understanding AI, researchers are now drawing insights from a vast array of fields—including functional analysis, probability theory, optimal transport, optimization, PDEs, information theory, geometry, statistics, electrical engineering, and ergodic theory. Those interdisciplinary efforts are gradually shedding light on the underlying principles governing modern AI. This workshop centers around these mathematical and interdisciplinary developments. It will feature a series of talks from people in various subfields. Open problem and small-group sessions will help foster new connections and new research avenues.

Registration required
In-person registration (This event is at capacity)
Zoom Webinar Registration

Speakers

Jason Altschuler, University of Pennsylvania
Guy Bresler, MIT
Sinho Chewi, Yale University
Lenaic Chizat, EPFL
Nabarun Deb, University of Chicago
Edgar Dobriban, University of Pennsylvania
Ahmed El Alaoui, Cornell University
Zhou Fan, Yale University
Boris Hanin, Princeton University
Jason Klusowski, Princeton University
Tengyu Ma, Stanford University
Alexander Rakhlin, MIT
Yuting Wei, University of Pennsylvania
Tijana Zrnic, Stanford University

Organizer: Morgane Austern, Harvard Statistics

Schedule
Monday, Oct. 6, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Yuting Wei, U Penn
To Intrinsic Dimension and Beyond: Efficient Sampling in Diffusion Models
The denoising diffusion probabilistic model (DDPM) has become a cornerstone of generative AI. While sharp convergence guarantees have been established for DDPM, the iteration complexity typically scales with the ambient data dimension of target distributions, leading to overly conservative theory that fails to explain its practical efficiency. This has sparked recent efforts to understand how DDPM can achieve sampling speed-ups through automatic exploitation of intrinsic low dimensionality of data. This talk explores two key scenarios: (1) For a broad class of data distributions with intrinsic dimension k, we prove that the iteration complexity of the DDPM scales nearly linearly with k, which is optimal under the KL divergence metric; (2) For mixtures of Gaussian distributions with k components, we show that DDPM learns the distribution with iteration complexity that grows only logarithmically in k. These results provide theoretical justification for the practical efficiency of diffusion models.

10:00–10:30 am
Break

10:30–11:30 am
Jason Klusowski, Princeton
The Value of Side Information in Unlabeled Data
Practitioners often work in settings with limited labeled data and abundant unlabeled data. During training, they may even have access to extra side information (some labeled, some not) that won’t be available once the model is deployed. When can this side information actually improve performance? I’ll present a simple framework where a rich-view model that sees the extra features generates pseudo-labels on the large unlabeled data, and a deployment model that only sees the standard features is trained on both real and pseudo-labels. The two are trained iteratively: each deployment model update calibrates the next round of pseudo-labels, and those refined pseudo-labels in turn guide the deployment model. Our theory shows that side information helps precisely when the rich-view and deployment models make different kinds of errors. We formalize this with a decorrelation score that quantifies how independent those errors are; the more independent, the greater the performance gains.

11:3 0am–12:00 pm
Break

12:00–1:00 pm
Guy Bresler, MIT
Global Minimizers of Sigmoid Contrastive Loss
The meta-task of obtaining and aligning representations through contrastive pre-training is steadily gaining importance since its introduction in CLIP and ALIGN. In this paper we theoretically explain the advantages of synchronizing with trainable inverse temperature and bias under the sigmoid loss, as implemented in the recent SigLIP models of Google DeepMind. Temperature and bias can drive the loss function to zero for a rich class of configurations that we call (m,b)-Constellations. (m,b)-Constellations are a novel combinatorial object related to spherical codes and are parametrized by a margin m and relative bias b. We use our characterization of constellations to theoretically justify the success of SigLIP on retrieval, to explain the modality gap present in SigLIP, and to identify the necessary dimension for producing high-quality representations. We also propose a reparameterization of the sigmoid loss with explicit relative bias, which appears to improve training dynamics. Joint work with Kiril Bangachev, Iliyas Noman, and Yury Polyanskiy.

Tuesday, Oct. 7, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Lénaïc Chizat, EPFL
The Hidden Width of Deep ResNets
We present a mathematical framework to analyze the training dynamics of deep ResNets that rigorously captures practical architectures (including Transformers) trained from standard random initializations. Our approach combines stochastic approximation of ODEs with propagation-of-chaos arguments. It yields three main insights:
– Depth begets width: infinite-depth ResNets of any hidden width behave throughout training as if they were infinitely wide;
– Unified phase diagram: the phase diagram of Transformers mirrors that of two-layer perceptrons, once the appropriate substitutions are made;
– Optimal shape scaling: for a given parameter budget P, a Transformer with optimal shape converges to its limiting dynamics at rate P^{-1/6}.
This is based on https://arxiv.org/abs/2509.10167

10:00–10:30 am
Break

10:30–11:30 am
Boris Hanin, Princeton
Kernel Learning on Manifolds
This talk concerns the L_2 risk of minimum norm interpolation with n samples in the RKHS of a kernel K. Unlike most prior work in this space our kernels will be defined on any close d-dimensional Riemannian manifold, and we require only that the kernels are trace class and elliptic. With these assumptions we get nearly sharp L_2 risk bounds with high probability over the data. Like prior work on round spheres our results essentially say that the number of samples n, the dimension of the manifold, and some details of the kernel determine a natural spectral cutoff lambda(n,d,K) and that minimal norm interpolation essentially learns exactly the projection of the data generating process onto the eigenfunctions of the Laplacian with frequency at most lambda(n,d,K). Joint work with Mengxuan Yang.

11:30–12:00
Break

12:00–1:00
Zhou Fan, Yale
Dynamical mean-field analysis of adaptive Langevin diffusions
In many applications of statistical estimation via sampling, one may wish to sample from a high-dimensional target distribution that is adaptively evolving to the samples already seen. We study an example of such dynamics, given by a Langevin diffusion for posterior sampling in a Bayesian linear regression model with i.i.d. regression design, whose prior continuously adapts to the Langevin trajectory via a maximum marginal-likelihood scheme. Using techniques of dynamical mean-field theory (DMFT), we provide a precise characterization of a high-dimensional asymptotic limit for the joint evolution of the prior parameter and law of the Langevin sample. We then carry out an analysis of the equations that describe this DMFT limit, under conditions of approximate time-translation-invariance which include, in particular, settings where the posterior law satisfies a log-Sobolev inequality. In such settings, we show that this adaptive Langevin trajectory converges on a dimension-independent time horizon to an equilibrium state that is characterized by a system of replica-symmetric fixed-point equations, and the associated prior parameter converges to a critical point of a replica-symmetric limit for the model free energy. We explore the nature of the free energy landscape and its critical points in a few simple examples, where such critical points may or may not be unique.

Wednesday, Oct. 8, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Jason Altschuler, U Penn
Negative Stepsizes Make Gradient-Descent-Ascent Converge
Solving min-max problems is a central question in optimization, games, learning, and controls. Arguably the most natural algorithm is Gradient-Descent-Ascent (GDA), however since the 1970s, conventional wisdom has argued that it fails to converge even on simple problems. This failure spurred the extensive literature on modifying GDA with extragradients, optimism, momentum, anchoring, etc. In contrast, we show that GDA converges in its original form by simply using a judicious choice of stepsizes. The key innovation is the proposal of unconventional stepsize schedules that are time-varying, asymmetric, and (most surprisingly) periodically negative. We show that all three properties are necessary for convergence, and that altogether this enables GDA to converge on the classical counterexamples (e.g., unconstrained convex-concave problems). The core intuition is that although negative stepsizes make backward progress, they de-synchronize the min/max variables (overcoming the cycling issue of GDA) and lead to a slingshot phenomenon in which the forward progress in the other iterations is overwhelmingly larger. This results in fast overall convergence. Geometrically, the slingshot dynamics leverage the non-reversibility of gradient flow: positive/negative steps cancel to first order, yielding a second-order net movement in a new direction that leads to convergence and is otherwise impossible for GDA to move in. Joint work with Henry Shugart.

10:00–10:30 am
Break

10:30–11:30 am
Nabarun Deb, U Chicago
Generative Modeling via Parabolic Monge-Ampère PDEs
We introduce a novel generative modeling framework based on a discretized parabolic Monge-Ampère PDE, which emerges as a continuous limit of the Sinkhorn algorithm commonly used in optimal transport. Our method performs iterative refinement in the space of Brenier maps using a mirror gradient descent step. We establish theoretical guarantees for generative modeling through the lens of no-regret analysis, demonstrating that the iterates converge to the optimal Brenier map under a variety of step-size schedules. As a technical contribution, we derive a new Evolution Variational Inequality tailored to the parabolic Monge-Ampère PDE, connecting geometry, transportation cost, and regret. Our framework accommodates non-log-concave target distributions, constructs an optimal sampling process via the Brenier map, and integrates favorable learning techniques from generative adversarial networks and score-based diffusion models.

11:30–12:00
Break

12:00–1:00
Sinho Chewi, Yale
Discretization and distribution learning in diffusion models
First, I will review some literature on discretization of diffusion models, focusing on the use of randomized midpoints for deterministic vs. stochastic samplers. Then, I will argue that such sampling guarantees reduce distribution learning, in the form of learning to generate a sample, to score matching. To complement this result, we reduce other forms of distribution learning (parameter estimation and density estimation) to score matching as well. This leads to new consequences for diffusion models, such as asymptotic efficiency of a DDPM-based parameter estimator and algorithms for Gaussian mixture density estimation, as well as to a general approach for establishing cryptographic hardness results for score estimation.

Thursday, Oct. 9, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Ahmed El Alaoui, Cornell
How abundant are good interpolators?
We consider classifying labelled data in the interpolation regime where there exist linear classifiers (with possibly negative margin) correctly classifying all points in the dataset. Under the logistic model with gaussian features, we derive the large deviation rate function of the event that an interpolator chosen uniformly at random achieves a given generalization error. This describes the proportion of interpolators having any desired performance. We remark that in a wide regime of parameters, the vast majority of interpolators have inferior performance than the one found via a simple linear programming procedure, showing that the latter algorithm produces an atypically good classifier.
This is based on joint work with August Chen.

10:00–10:30 am
break

10:30–11:30 am
Tengyu Ma, Stanford
Self-play LLM Theorem Provers with Iterative Conjecturing and Proving
I will discuss some works on using RL for theorem proving, especially in the possible future regime where we ran out of high-quality training data. To keep improving the models with limited data, we draw inspiration from mathematicians, who continuously develop new results, partly by proposing novel conjectures or exercises (which are often variants of known results) and attempting to solve them. We design the Self-play Theorem Prover (STP) that simultaneously takes on two roles, conjecturer and prover, each providing training signals to the other. The model achieves state-of-the-art performance among whole-proof generation methods on miniF2F-test (65.0%, pass@3200), Proofnet-test (23.9%, pass@3200) and PutnamBench (8/644, pass@3200).

11:30–12:00
break

12:00–1:00
Edgar Dobriban, U Penn
Leveraging synthetic data in statistical inference
The rapid proliferation of high-quality synthetic data — generated by advanced AI models or collected as auxiliary data from related tasks — presents both opportunities and challenges for statistical inference. This paper introduces a GEneral Synthetic-Powered Inference (GESPI) framework that wraps around any statistical inference procedure to safely enhance sample efficiency by combining synthetic and real data. Our framework leverages high-quality synthetic data to boost statistical power, yet adaptively defaults to the standard inference method using only real data when synthetic data is of low quality. The error of our method remains below a user-specified bound without any distributional assumptions on the synthetic data, and decreases as the quality of the synthetic data improves. This flexibility enables seamless integration with conformal prediction, risk control, hypothesis testing, and multiple testing procedures, all without modifying the base inference method. We demonstrate the benefits of our method on challenging tasks with limited labeled data, including AlphaFold protein structure prediction, and comparing large reasoning models on complex math problems.

Friday, Oct. 10, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Tijana Zrnic, Stanford
Probably Approximately Correct Labels
Obtaining high-quality labeled datasets is often costly, requiring either extensive human annotation or expensive experiments. We propose a method that supplements such “expert” labels with AI predictions from pre-trained models to construct labeled datasets more cost-effectively. Our approach results in probably approximately correct labels: with high probability, the overall labeling error is small. This solution enables rigorous yet efficient dataset curation using modern AI models. We demonstrate the benefits of the methodology through text annotation with large language models, image labeling with pre-trained vision models, and protein folding analysis with AlphaFold. This is joint work with Emmanuel Candes and Andrew Ilyas.

10:00–10:30 am
Break

10:30–11:30 am
Alexander Rakhlin, MIT
Elements of Interactive Decision Making
Machine learning methods are increasingly deployed in interactive environments, ranging from dynamic treatment strategies in medicine to fine-tuning of LLMs using reinforcement learning. In these settings, the learning agent interacts with the environment to collect data and necessarily faces an exploration-exploitation dilemma. We present a general framework for interactive decision making that subsumes multi-armed bandits, contextual bandits, structured bandits, and reinforcement learning. We focus on both the statistical aspect of learning—aiming to develop a tight characterization of sample complexity in terms of properties of the class of models—and on the basic algorithmic primitives.

Millennium Prize Problems Lecture - Sourav Chatterjee: Yang-Mills and the foundations of quantum field theory
2025-10-15 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Millennium Prize Problems Lecture
Date: October 15, 2025
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall D, 1 Oxford St., Cambridge MA
Speaker: Sourav Chatterjee, Stanford University
Title: Yang-Mills and the foundations of quantum field theory
Abstract: Yang-Mills theories are the building blocks of the Standard Model of quantum mechanics, which is the best available model for our universe at the quantum scale. Yet, these theories do not have a rigorous mathematical foundation. Physical calculations are based on perturbation theory, but there are various phenomena that are believed to be out of the reach of perturbative arguments. Building a mathematical foundation is, therefore, important even from the physics point of view. A program with this objective, known as “constructive field theory”, was initiated in the 1960s. In spite of many successes, the program has not reached its original goal. Completing this program is the Clay Millennium Prize problem of Yang-Mills existence and mass gap. I will give a general introduction to the main questions, and an overview of exciting recent progress that has rejuvenated the quest for a solution in the last ten years.
Read more about the Yang-Mills Existence and Mass Gap 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

Math Science Lectures in Honor of Raoul Bott | Dennis Gaitsgory, MPIM | Function-theoretic implications of geometric Langlands
2025-10-20 16:30 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Two talks on Function-theoretic implications of geometric Langlands
Dates: October 20 & 21, 2025
Time: 4:30–5:30 pm
Location: Science Center Lecture Hall A and via Webinar

Speaker: Dennis Gaitsgory, Max Planck Institute for Mathematics
Abstract: The recently established geometric Langlands equivalence implies an explicit description of the space of (unramified) automorphic functions in terms of Langlands parameters. In these lectures, we will derive these description and explain how far we can go with it in order to deduce some expected properties of automorphic functions, e.g., Ramanujan and Arthur multiplicity conjectures. This is joint work with Vincent Lafforgue and Sam Raskin.

Lecture 1: Monday, October 20, 2025
From geometric to classical Langlands

Lecture 2: Tuesday, October 21, 2025
Analytic properties of automorphic functions as seen from algebraic geometry

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 | Dennis Gaitsgory, MPIM | Function-theoretic implications of geometric Langlands
2025-10-21 16:30 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Two talks on Function-theoretic implications of geometric Langlands
Dates: October 20 & 21, 2025
Time: 4:30–5:30 pm
Location: Science Center Lecture Hall A and via Webinar
Speaker: Dennis Gaitsgory, Max Planck Institute for Mathematics
Abstract: The recently established geometric Langlands equivalence implies an explicit description of the space of (unramified) automorphic functions in terms of Langlands parameters. In these lectures, we will derive these description and explain how far we can go with it in order to deduce some expected properties of automorphic functions, e.g., Ramanujan and Arthur multiplicity conjectures. This is joint work with Vincent Lafforgue and Sam Raskin.

Lecture 1: Monday, October 20, 2025
Function-theoretic implications of geometric Langlands: From geometric to classical Langlands

Lecture 2: Tuesday, October 21, 2025
Function-theoretic implications of geometric Langlands: Analytic properties of automorphic functions as seen from algebraic geometry

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.

Geometry Meets Physics: Finiteness, Tameness, and Complexity
2025-11-12 09:00 - 2025-11-14 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Geometry Meets Physics: Finiteness, Tameness, and Complexity
Dates: November 12–14, 2025
Location: CMSA G10, 20 Garden Street, Cambridge MA 02138
(note: this event is in-person only)
Finiteness is a fundamental property in consistent physical theories. From the earliest days of quantum field theory and string theory, the drive to eliminate unphysical infinities has been a guiding principle. More recently, finiteness has emerged as a key criterion for constraining effective theories that can be embedded in quantum gravity. Formulating and testing these constraints remains a central challenge in current research.
In parallel, mathematics has made remarkable advanced in addressing finiteness questions using tame geometry. Built on the framework of o-minimal structures, tame geometry offers a precise language for describing objects of finite geometric complexity. Recent developments, such as sharp o-minimality, go further by introducing a quantitative notion of complexity, opening new directions for analyzing finiteness in mathematics and physics alike.
This workshop brings together mathematicians and physicists to exchange ideas, explore new perspectives, and spark collaborations at the interface of geometry, logic, and fundamental physics.
Invited Speakers

Vijay Balasubramanian (UPenn)
Gregorio Baldi (CNRS, IMJ-PRG & IAS)
Gal Binyamini (Weizmann Institute & IAS)
Raf Cluckers (Lille, France)
Matilda Delgado (Max Planck Institute Munich)
Bruno Klingler (Humboldt University, Berlin & IAS)
Adele Padgett (Vienna)
David Prieto (Utrecht)
Washington Taylor (MIT)
David Urbanik (IHES, France & IAS)
Cumrun Vafa (Harvard)
Mick van Vliet (Utrecht)
Benny Zak (Weizmann Institute & IAS)

Organizers: Thomas Grimm, Harvard CMSA & Utrecht University | Gal Binyamini, Weizmann Institute & IAS | Bruno Klingler, Humboldt University, Berlin & IAS

Schedule
(download pdf)
Wednesday Nov. 12, 2025
8:30–8:55 am
Morning refreshments (Common Room)
8:55–9:00 am
Introductions
9:00–10:30 am
Lecture
Speaker: Gal Binyamini, Weizmann Institute & IAS
Title: O-minimality: finiteness and complexity
Abstract: O-minimality is a mathematical formalism of “tame geometry”: a geometry where every set has finite geometric complexity. I will give an introduction to o-minimality in general, and to quantitative variants where one measures the complexity of sets in terms of some natural parameters. I’ll try to focus on the main examples that potentially come up in the interaction with physics, and describe the state of the art and some conjectures.
10:30–11:00 am
Break
11:00 am–12:00 pm
Speaker: Benny Zak, Weizmann Institute & IAS
Title: Analytic tameness – complex cells
Abstract: Complex cells are a complex anayltic version of cells from o-minimality, invented by Binyamini and Novikov. We aim to introduce complex cells, and demonstrate their usefullness in quantifying the analytic information present in a complex set. If time permits, we will discuss applications of this theory.
12:00–1:00 pm
Catered Lunch (Common Room)
1:00–2:30 pm
Lecture
Speakers: David Prieto and Mick van Vliet, Utrecht
Title: Tameness and Complexity in Physical Theories
Abstract: We give an introductory overview of recent applications of o-minimality to physics, focusing on quantum field theories and quantum gravity. In the first part of the lecture we explain how o-minimality makes a first appearance in physical theories when considering amplitudes in quantum field theory. In the second part, we concentrate on a class of theories where finiteness principles seem to be essential, namely the quantum field theories which are consistent with quantum gravity. We review some of these finiteness principles and interpret them through the lens of the o-minimal framework. Along the way, we highlight recent progress in this direction, as well as open questions to explore in the future.
2:30–3:00 pm
Break with refreshments (Common Room)
3:00–4:00 pm
Speaker: Matilda Delgado, Max Planck Institute Munich
Title: Dualities and the Compactifiability of Moduli Space
Abstract: After introducing (self-)dualities in string theory and their action on the field content & spectrum of the theory, I will present the notion of compactifiability for the moduli space of massless fields as the condition that its volume is finite or grows no faster than Euclidean space. I will argue that compactifiability generically implies the existence of non-trivial dualities by providing evidence from string theory. Moreover, I will explain how one can connect compactifiability to the condition that the spectrum of objects charged under the duality group transform in a semisimple representation. Finally, I will provide a bottom-up argument for compactifiability, and argue that it (at least in supersymmetric cases) can be explained by the finiteness of the number of massless states upon compactification to 1D. Based on arXiv:2412.03640.
5:00 PM
Millennium Lecture and Reception: Pierre Deligne (IAS) (Science Center Hall D)
Title: What is the Hodge conjecture?

Thursday, Nov. 13, 2025
8:30–9:00 am
Morning refreshments (Common Room)
9:00–10:30 am
Lecture
Speaker: Bruno Klingler, Humboldt University, Berlin & IAS
Title: Tame geometry and Hodge theory
Abstract: I will give an introduction to applications of o-minimality in complex geometry, in particular in Hodge theory.
10:30–11:00 am
Break
11:00 am–12:00 pm
Speaker: Cumrun Vafa, Harvard
Title: The Swampland Program
12:00–1:30 pm
Catered Lunch (Common Room)
1:30–2:30 pm
Speaker: Gregorio Baldi, CNRS, IMJ-PRG & IAS
Title: The Hodge locus
Abstract: We will survey various recent results around the distribution of the Hodge locus of a (mixed) variation of Hodge structures. Various concrete applications to moduli spaces will also be presented.
2:30–3:00 pm
Break with refreshments (Common Room)
3:00–4:00 pm
Speaker: Vijay Balasubramanian, U Penn
Title: Chaos and complexity in quantum dynamics
4:30–5:30
Discussion/Q&A session
6:30 PM
Dinner: Changsho Restaurant, 1712 Massachusetts Ave., Cambridge, MA 02138

Friday Nov. 14, 2025
8:30–9:00 am
Morning refreshments (Common Room)
9:00–10:00 am
Speaker: Washington Taylor, MIT
Title: Finiteness, connectivity, and the power of fibrations in the Calabi-Yau landscape
10:00–10:30am
Break
10:30–11:30 am
Speaker: Adele Padgett, Vienna
Title: Tameness of multisummable series
Abstract: There are sophisticated theories of summability that map divergent series solutions of differential or functional equations to solutions that are holomorphic in sector-like domains. Van den Dries and Speissegger proved that functions obtained from real multisummable power series have tame geometric behavior when restricted to the real numbers. It would be desirable to know that these functions are also tame on their whole sector-like domains, but recently Speissegger and I proved that these functions are in general only tame on part of their domains. I will present this result and discuss the domains on which some examples are tame, including the Stirling series which appears in the asymptotic expansion of the Gamma function. In this talk, “tame” means definable in an o-minimal structure.
11:30 am–1:00 pm
Catered Lunch (Common Room)
1:00–2:00 pm
Speaker: Raf Cluckers, Lille, France
Title: Finiteness and tameness in (non-archimedean) geometry
Abstract: Non-archimedean geometry work with orders of magnitude rather than with precise measurements. The former works for example with orders of vanishing of functions, and the latter typically works with real or complex numbers. I will discuss recent progress on non-archimedean tame geometry. I will present analogues of o-minimality, of Pila-Wilkie’s o-minimal counting results, and of other finiteness results, in non-archimedean settings.
2:00–2:30 pm
Break with refreshments (Common Room)
2:30–3:30 pm
Speaker: David Urbanik, IHES, France & IAS
Title: Degrees of Hodge Loci

Millennium Prize Problems Lecture - Pierre Deligne: What is the Hodge conjecture?
2025-11-12 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Date: November 12, 2025
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall D, 1 Oxford St., Cambridge MA
Speaker: Pierre Deligne, Institute for Advanced Study
Title: What is the Hodge conjecture?
Abstract: The Hodge conjecture is about projective non-singular complex algebraic varieties. It characterizes the cohomology classes coming from algebraic cycles. I will explain these terms, tell why the conjecture is so hard to attack, and why we care.

Series Pamphlet (pdf)
Read more about the Hodge Conjecture 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

Conference on Geometry and Statistics
2025-11-17 09:00 - 2025-11-19 17:00 | 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
Date: March 4, 2026
Time: 4:00 pm
Location: Harvard Science Center Hall D & via Zoom Webinar
Speaker: Sanjeev Arora, Princeton
Title and Abstract TBA

Registration required
In-person registration
Zoom Webinar Registration

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 09:00 - 10:30 | MA
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 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.

Registration is required.
In-person registration
Zoom Webinar registration

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 Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Date: March 11, 2026
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall C, 1 Oxford St., Cambridge MA
Speaker: Javier Gómez-Serrano, Brown University
Topic: Navier-Stokes Existence or Breakdown

Register to attend in-person.
Register for the Zoom Webinar.

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.
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Speakers

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

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

Millennium Prize Problems Lecture - Peter Sarnak: Riemann Hypothesis
2026-04-15 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
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
Topic: Riemann Hypothesis

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Register for the Zoom Webinar.

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, Yau Mathematical Sciences Center, Tsinghua University
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
Speaker: Nicolai Reshetikhin, Yau Mathematical Sciences Center, Tsinghua University
Details TBA

Registration is required.
Zoom webinar registration link

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
2026-04-23 16:00 - 17:00 | MA
Sixth Annual Yip Lecture
Date: April 23, 2026
Time: 4:00-5:00 pm ET
Location: Harvard Science Center Hall A
Speaker: Regina Barzilay, MIT

Details TBA

In-person registration
Webinar registration

Regina Barzilay is a Delta Electronics professor in the Department of Electrical Engineering and Computer Science and a member of the Computer Science and Artificial Intelligence Laboratory at the Massachusetts Institute of Technology. Her research interests are in natural language processing, applications of deep learning to chemistry and oncology. She is a recipient of various awards including the NSF Career Award, the MIT Technology Review TR-35 Award, Microsoft Faculty Fellowship and several Best Paper Awards at NAACL and ACL. In 2017, she received a MacArthur fellowship, an ACL fellowship and an AAAI fellowship. She received her Ph.D. in Computer Science from Columbia University, and spent a year as a postdoc at Cornell University.
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.
Limited support may be available for approved postdocs and early career applicants. The application form can be found at: https://forms.gle/1zxTEKhZyz4TPfSY6

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Minicourse Speakers

Robert McCann, University of Toronto
Yang Li, Cambridge University

Workshop Speakers

Rolf Andreasson, Chalmers University, Sweden
Jakob Hultgren, Umea University, Sweden
Young-Heon Kim, University of British Columbia
Nam Le, Indiana University
Jiakun Liu, University of Sydney
Duong H. Phong, Columbia University
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

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 & 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.
Details TBA

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

Details TBA

Support provided by Logical Intelligence.

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

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

details tba
