Big Data Conference 2025
2025-09-11 09:00 - 2025-09-12 17:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Big Data Conference 2025
Dates: Sep. 11–12, 2025
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.
Invited Speakers

Markus J. Buehler, MIT
Yiling Chen, Harvard
Jordan Ellenberg, UW Madison
Yue M. Lu, Harvard
Pankaj Mehta, BU
Nick Patterson, Harvard
Gautam Reddy, Princeton
Trevor David Rhone, Rensselaer Polytechnic Institute
Tess Smidt, MIT

Organizers:
Michael M. Desai, Harvard OEB | Michael R. Douglas, Harvard CMSA | Yannai A. Gonczarowski, Harvard Economics | Efthimios Kaxiras, Harvard Physics | Melanie Weber, Harvard SEAS

Big Data Youtube Playlist

Schedule
Thursday, Sep. 11, 2025

9:00 am
Refreshments

9:30 am
Introductions

9:45–10:45 am
Gautam Reddy, Princeton
Title: Global epistasis in genotype-phenotype maps

10:45–11:00 am
Break

11:00 am –12:00 pm
Nick Patterson, Harvard
Title: The Origin of the Indo-Europeans
Abstract: Indo-European is the largest family of human languages, with very wide geographical distribution and more than 3 billion native speakers. How did this family arise and spread? This question has been discussed for nearly 250 years but with the advent of the availability of DNA from ancient fossils is now largely understood, at least in broad outlines. We will describe what we now know about the origins.

12:00–1:30 pm
Lunch break

1:30–2:30 pm
Markus Buehler, MIT
Title: Superintelligence for scientific discovery
Abstract: AI is moving beyond prediction to become a partner in invention. While today’s models excel at interpolating within known data, true discovery requires stepping outside existing truths. This talk introduces superintelligent discovery engines built on multi-agent swarms: diverse AI agents that interact, compete, and cooperate to generate structured novelty. Guided by Gödel’s insight that no closed system is complete, these swarms create gradients of difference – much like temperature gradients in thermodynamics – that sustain flow, invention, and surprise. Case studies in protein design and music composition show how swarms escape data biases, invent novel structures, and weave long-range coherence, producing creativity that rivals human processes. By moving from “big data” to “big insight”, these systems point toward a new era of AI that composes knowledge across science, engineering, and the arts.

2:30–2:45 pm
Break

2:45–3:45 pm
Jordan Ellenberg, UW Madison
Title: What does machine learning have to offer mathematics?

3:45–4:00 pm
Break

4:00–5:00 pm
Pankaj Mehta, Boston University
Title: Thinking about high-dimensional biological data in the age of AI
Abstract: The molecular biology revolution has transformed our view of living systems. Scientific explanations of biological phenomena are now synonymous with the identification of the genes and proteins. The preeminence of the molecular paradigm has only become more pronounced as new technologies allow us to make measurements at scale. Combining this wealth of data with new artificial intelligence (AI) techniques is widely viewed as the future of biology. Here, I will discuss the promise and perils of this approach. I will focus on our unpublished work with collaborators on two fronts: (i) transformer-based models for understanding genotype-to-phenotype maps, and (ii) LLM-based ‘foundational models’ for cellular identity, such as TranscriptFormer, which is trained on single-cell RNA sequencing (scRNAseq) data. While LLMs excel at capturing complex evolutionary and demographic structure in DNA sequence data, they are much less adept at elucidating the biology of cellular identity. We show that simple parameter-free models based on linear-algebra outperform TranscriptFormer on downstream tasks related to cellular identity, even though TranscriptFormer has nearly a billion parameters. If time permits, I will conclude by showing how we can combine ideas from linear algebra, bifurcation theory, and statistical physics to classify cell fate transitions using scRNAseq data.

Friday, Sep. 12, 2025

9:00-9:45 am
Refreshments

9:45–10:45 am
Yiling Chen, Harvard
Title: Data Reliability Scoring
Abstract: Imagine you are trying to make a data-driven decision, but the data at hand may be noisy, biased, or even strategically manipulated. Can you assess whether such a dataset is reliable—without access to ground truth?
We initiate the study of reliability scoring for datasets reported by potentially strategic data sources. While the true data remain unobservable, we assume access to auxiliary observations generated by an unknown statistical process that depends on the truth. We introduce the Gram Determinant Score, a reliability measure that evaluates how well the reported data align with the unobserved truth, using only the reported data and the auxiliary observations. The score comes with provable guarantees: it preserves several natural reliability orderings. Experimentally, it effectively captures data quality in settings with synthetic noise and contrastive learning embeddings.
This talk is based on joint work with Shi Feng, Fang-Yi Yu, and Paul Kattuman.

10:45–11:00 am
Break

11:00 am –12:00 pm
Yue M. Lu, Harvard
Title: Nonlinear Random Matrices in High-Dimensional Estimation and Learning
Abstract: In recent years, new classes of structured random matrices have emerged in statistical estimation and machine learning. Understanding their spectral properties has become increasingly important, as these matrices are closely linked to key quantities such as the training and generalization performance of large neural networks and the fundamental limits of high-dimensional signal recovery. Unlike classical random matrix ensembles, these new matrices often involve nonlinear transformations, introducing additional structural dependencies that pose challenges for traditional analysis techniques.
In this talk, I will present a set of equivalence principles that establish asymptotic connections between various nonlinear random matrix ensembles and simpler linear models that are more tractable for analysis. I will then demonstrate how these principles can be applied to characterize the performance of kernel methods and random feature models across different scaling regimes and to provide insights into the in-context learning capabilities of attention-based Transformer networks.

12:00–1:30 pm
Lunch break

1:30–2:30 pm
Trevor David Rhone, Rensselaer Polytechnic Institute
Title: Accelerating the discovery of van der Waals quantum materials using AI
Abstract: van der Waals (vdW) materials are exciting platforms for studying emergent quantum phenomena, ranging from long-range magnetic order to topological order. A conservative estimate for the number of candidate vdW materials exceeds ~106 for monolayers and ~1012 for heterostructures. How can we accelerate the exploration of this entire space of materials? Can we design quantum materials with desirable properties, thereby advancing innovation in science and technology? A recent study showed that artificial intelligence (AI) can be harnessed to discover new vdW Heisenberg ferromagnets based on Cr2Ge2Te6 [1], [2] and magnetic vdW topological insulators based on MnBi2Te4 [3]. In this talk, we will harness AI to efficiently explore the large chemical space of vdW materials and to guide the discovery of vdW materials with desirable spin and charge properties. We will focus on crystal structures based on monolayer Cr2I6 of the form A2X6, which are studied using density functional theory (DFT) calculations and AI. Magnetic properties, such as the magnetic moment are determined. The formation energy is also calculated and used as a proxy for the chemical stability. We also investigate monolayers based on MnBi2Te4 of the form AB2X4 to identify novel topological materials. Further to this, we study heterostructures based on MnBi2Te4/Sb2Te3 stacks. We show that AI, combined with DFT, can provide a computationally efficient means to predict the thermodynamic and magnetic properties of vdW materials [4],[5]. This study paves the way for the rapid discovery of chemically stable vdW quantum materials with applications in spintronics, magnetic memory and novel quantum computing architectures.
[1] T. D. Rhone et al., “Data-driven studies of magnetic two-dimensional materials,” Sci. Rep., vol. 10, no. 1, p. 15795, 2020.
[2] Y. Xie, G. Tritsaris, O. Granas, and T. Rhone, “Data-Driven Studies of the Magnetic Anisotropy of Two-Dimensional Magnetic Materials,” J. Phys. Chem. Lett., vol. 12, no. 50, pp. 12048–12054.
[3] R. Bhattarai, P. Minch, and T. D. Rhone, “Investigating magnetic van der Waals materials using data-driven approaches,” J. Mater. Chem. C, vol. 11, p. 5601, 2023.
[4] T. D. Rhone et al., “Artificial Intelligence Guided Studies of van der Waals Magnets,” Adv. Theory Simulations, vol. 6, no. 6, p. 2300019, 2023.
[5] P. Minch, R. Bhattarai, K. Choudhary, and T. D. Rhone, “Predicting magnetic properties of van der Waals magnets using graph neural networks,” Phys. Rev. Mater., vol. 8, no. 11, p. 114002, Nov. 2024.
This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1548562. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract No. DE-AC02-06CH11357. This material is based on work supported by the National Science Foundation CAREER award under Grant No. 2044842.

2:30–2:45 pm
Break

2:45–3:45 pm
Tess Smidt, MIT
Title: Applications of Euclidean neural networks to understand and design atomistic systems
Abstract: Atomic systems (molecules, crystals, proteins, etc.) are naturally represented by a set of coordinates in 3D space labeled by atom type. This poses a challenge for machine learning due to the sensitivity of coordinates to 3D rotations, translations, and inversions (the symmetries of 3D Euclidean space). Euclidean symmetry-equivariant Neural Networks (E(3)NNs) are specifically designed to address this issue. They faithfully capture the symmetries of physical systems, handle 3D geometry, and operate on the scalar, vector, and tensor fields that characterize these systems.
E(3)NNs have achieved state-of-the-art results across atomistic benchmarks, including small-molecule property prediction, protein-ligand binding, force prediciton for crystals, molecules, and heterogeneous catalysis. By merging neural network design with group representation theory, they provide a principled way to embed physical symmetries directly into learning. In this talk, I will survey recent applications of E(3)NNs to materials design and highlight ongoing debates in the AI for atomistic sciences community: how to balance the incorporation of physical knowledge with the drive for engineering efficiency.

The Geometry of Machine Learning
2025-09-15 09:00 - 2025-09-18 17:00 | 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.

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
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
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
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

Schedule (download pdf)
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
Using AI for protein structure modeling and design
Abstract: In this talk, I’ll describe recent work in model interpretability, focusing on decomposing protein language models and structure prediction models like AlphaFold. The goal is to understand what they are learning, their limitations and how we can use this information to develop better models and design proteins.
1:45–2:00 pm
Break
2:00–3:00 pm
SueYeon Chung, Harvard/Flatiron Institute
Computing with Neural Manifolds: A Multi-Scale Framework for Understanding Biological and Artificial Neural Networks
Recent breakthroughs in experimental neuroscience and machine learning have opened new frontiers in understanding the computational principles governing neural circuits and artificial neural networks (ANNs). Both biological and artificial systems exhibit an astonishing degree of orchestrated information processing capabilities across multiple scales – from the microscopic responses of individual neurons to the emergent macroscopic phenomena of cognition and task functions. At the mesoscopic scale, the structures of neuron population activities manifest themselves as neural representations. Neural computation can be viewed as a series of transformations of these representations through various processing stages of the brain. The primary focus of my lab’s research is to develop theories of neural representations that describe the principles of neural coding and, importantly, capture the complex structure of real data from both biological and artificial systems.
In this talk, I will present three related approaches that leverage techniques from statistical physics, machine learning, and geometry to study the multi-scale nature of neural computation. First, I will introduce new theories based on statistical physics and convex geometry that connect complex geometric structures that arise from neural responses (i.e., neural manifolds) to the efficiency of neural representations in implementing a task. Second, I will employ these theories to analyze how these representations evolve across scales, shaped by the properties of single neurons, learning dynamics, and the transformations across distinct brain regions. Finally, I will show how these insights extend efficient coding principles beyond early sensory stages, linking representational geometry to efficient task implementations. This framework not only help interpret and compare models of brain data but also offers a principled approach to designing ANN models for higher-level vision. This perspective opens new opportunities for using neuroscience-inspired principles to guide the development of intelligent systems.

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 (via Zoom)
Mathew Vanherreweghe, Logical Intelligence
Sean Welleck, CMU (via Zoom)
Mattiew Wyart, JHU (via Zoom)

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
Compression is all you need: Modeling mathematics
Abstract: The mathematics humans discover and value (“human math”) is a vanishingly small subset of all valid deductions (“formal math”). I’ll argue that human math is distinguished by its compressibility through hierarchically nested definitions and theorems, like a polynomial-growth space rather than the exponential-growth space one might expect when proofs are viewed as strings of symbols. The argument combines toy monoid models with an empirical analysis of MathLib, a large Lean library of formalized mathematics we treat as a proxy for human math. I’ll close with how compression itself can serve as a measure of mathematical interest, giving agents a sense of direction toward where human math lives.
9:45–10:30 am
Boris Hanin, Princeton
10:30–11:00 am
Break
11:00–11:45 am
Slava Krushkal, University of Virginia
Using AI to study 4-manifold topology
Abstract: Central open questions in geometric classification theory of topological 4-manifolds have a reformulation in terms of the Round Handle Problem. It asks whether a given link in the 3-sphere is slice (bounds disjoint disks) in the 4-manifold obtained by attaching certain round handles to the 4-ball. I will discuss an algebraic-combinatorial formulation of the problem, ongoing AI-assisted work on it, and the results obtained to date.
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 (via Zoom)
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
Compiling Programs to Neurons
Abstract: Neural networks are ordinarily programmed indirectly: we specify architectures, objectives, and data, and rely on learning to discover a computation. I will focus on Cajal, a typed, higher-order linear programming language whose programs compile correctly to linear neurons and, with iteration, to recurrent neurons. This allows discrete programming structures such as conditionals and iteration to coexist with gradient-based learning. Experiments show that connecting compiled neurons with learned networks can improve learning speed and data efficiency. This work is led by PhD student Joey Velez-Ginorio and is joint with his advisors at UPenn, Konrad Kording and Steve Zdancewic. I will close with a complementary direction from my work: using language models together with formal verification to generate programs and proofs with machine-checkable guarantees. Together, these projects explore how programming languages can provide structure and control at the boundary between programs and learned systems.
9:45–10:30 am
Randall Balestriero, Brown University
Counterfactual World Models for Real World Deployment
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
Jared Duker Lichtman, Stanford

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
Kenneth Blakey, MIT
Ciprian Bonciocat, Stanford
Soham Chanda, USC
Octav Cornea, University of Montréal
Claire Dai, Harvard
Ilaria Di Dedda, University of Edinburgh
Tobias Ekholm, Uppsala University
Sheel Ganatra, USC
Sebastian Haney, Harvard
Andrew Hanlon, University of Oregon
Siu-Cheong Lau, Boston University
Yanki Lekili, Imperial College London
Elise LePage, Columbia
Yin Li, Uppsala University
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
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.
