Topological Manifolds – The First 100 Years
2025-09-15 16:30 - 17:30 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Michael Freedman (Harvard CMSA and Logical Intelligence)
Title: Topological Manifolds – The First 100 Years
Abstract: I’ll review manifold topology in the topological category from its start with work of Rado (1925) and Kneser (1926) to the present. Work of Moise, Mazur, Kirby, Siebenmann, Sullivan, Kruskal, and the speaker will be discussed. In my view there is one pressing open question (the A-B slice problem). I will end with some thoughts on putting an AI to work on it.

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

Moduli spaces of 4d N=2 quantum field theories
2025-09-18 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Robert Moscrop, CMSA
Title: Moduli spaces of 4d N=2 quantum field theories
Abstract: Supersymmetry endows quantum field theories with several rich algebraic and geometric structures associated to their moduli space of vacua, providing powerful tools to study such theories non-perturbatively. For example, in four-dimensional theories with eight supercharges, the low energy dynamics of the theory is captured by an algebraic completely integrable system whose base is the Coulomb branch– a particular distinguished submanifold of the moduli space. This structure is so tightly constrained, that there is an ongoing program to classify such theories purely by understanding their Coulomb branch geometry. In this talk, I will give a gentle introduction to the geometry of the moduli spaces of 4d N=2 theories and, time permitting, discuss some recent results showcasing how the geometry of the Coulomb branch can be used to constrain certain physical quantities of the theory.

Top-Down Perspectives on Symmetry Theories
2025-09-19 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Max Hubner
Title: Top-Down Perspectives on Symmetry Theories
Abstract: I will review the construction and utility of symmetry theories for string constructed quantum field theories. Symmetry theories are extra-dimensional auxiliary theories separating aspects of a quantum field theory’s symmetries from many of its more messy features. For QFTs with extra-dimensional string constructions the symmetry theory derives directly from the extra-dimensional geometry. This perspective allows for the study of symmetries of famously string engineered systems, such as SCFTs in 5D and 6D, which we will discuss on an example by example basis.

Non-Supersymmetric Orbifolds, Quivers and Chen-Ruan Orbifold Cohomology
2025-09-22 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Max Hübner (Uppsala & CMSA)
Title: Non-Supersymmetric Orbifolds, Quivers and Chen-Ruan Orbifold Cohomology
Abstract: We consider D3-brane probes of non-supersymmetric orbifolds and IIA on the same class of non-supersymmetric orbifolds. Both setups are characterized, in part, by quivers (which in the latter case relate for example to D0-brane probes) from which symmetries constraining the scale-dependence and tachyonic instabilities of the two systems, respectively, can be derived. We demonstrate that these considerations can be matched via a geometric analysis of the asymptotic boundary of the relevant orbifolds, in all cases, via considerations centered on Chen-Ruan orbifold cohomology.

Turbulent Mixing and Antagonistic Microorganisms
2025-09-22 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: David Nelson, Harvard
Title: Turbulent Mixing and Antagonistic Microorganisms
Abstract: Unlike coffee and cream that homogenize when stirred, growing micro-organisms (e.g., bacteria and baker’s yeast) can actively kill each other and avoid mixing. How do such antagonistic interactions impact the growth and survival of competing strains, while being spatially advected by turbulent flows? By using analytic arguments and numerical simulations of a continuum model, we describe the dynamics of two antagonistic strains that are dispersed by both compressible and incompressible turbulent flows in two spatial dimensions. A key parameter is the ratio of the fluid transport time to that of biological reproduction, which determines the winning organism that ultimately takes over the whole population from an initial heterogeneous state, a process known as fixation. By quantifying the probability and mean time for fixation, we discuss how turbulence raises the threshold for biological nucleation and antagonism suppresses flow-induced mixing by depleting the population at interfaces. We highlight the unusual biological consequences of the interplay of turbulent fluid flows with antagonistic population dynamics, with potential implications for marine microbial ecology and origins of biological chirality.

Degeneration of Calabi-Yau 3-folds and 3-forms
2025-09-25 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Teng Fei, Rutgers
Title: Degeneration of Calabi-Yau 3-folds and 3-forms
Abstract: We study the geometries associated to various 3-forms on a symplectic 6-manifold of different orbital types. As an application, we demonstrate how this can be used to find Lagrangian foliations and other geometric structures of interest arising from certain degeneration of Calabi-Yau 3-folds.

Sections of fibrations onto curves in characteristic p>0
2025-09-26 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Iacopo Brivio
Title: Sections of fibrations onto curves in characteristic p>0
Abstract: This talk is based on joint work in progress with Ben Church. Using symplectic geometry, Pieloch showed that every smooth fibration $fcolon Xto mathbb{P}^1$ of complex projective varieties always admits a section. I will explain how this theorem can be recovered using techniques from Hodge theory and the Minimal Model Program. An advantage of this approach is that it allows for a positive characteristic generalization, by replacing the Hodge theoretic input by a crystalline one. I will also give an example showing that Pieloch’s result can fail in characteristic p>0.

Graph integrals on Kahler manifolds
2025-09-29 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Minghao Wang, Boston University
Title: Graph integrals on Kahler manifolds
Abstract: I will talk about my recent work with Junrong Yan. We proved the convergence of Graph integrals on analytic Kahler manifolds in the sense of Cauchy principal values, which are originally from holomorphic quantum field theories. In particular, this allows us to construct geometric invariants of Calabi-Yau metrics. I will also talk about some potential applications of our results. References: arXiv:2507.09170, arXiv:2401.08113

Geometry and Quantum Theory Seminar
2025-09-30 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker 1: Max Hubner, CMSA
Title: On Topological Structures in String Theory
Abstract: Geometric engineering constructions in string theory often realize QFTs relative to an extra-dimensional geometry. This perspective parallels the symmetry TFT construction where a QFT is presented relative to its extra-dimensional symmetry quiche. Unsurprisingly, as we will discuss, these constructions are related. Topological features of the extra-dimensional geometry map onto the symmetry TFT. We discuss examples and generalization beyond purely geometric constructions in string theory.
Speaker 2: Bowen Yang, CMSA
Title: Bounded L theory
Abstract: Bounded L-groups arise in the intersection of algebraic L-theory and large-scale geometry, providing a framework for quadratic forms and automorphisms subject to uniform control conditions. These groups play a role in topology and surgery theory, especially in contexts where one needs to measure obstructions not just algebraically but also geometrically, with bounds on propagation or support. In this talk I will give a gentle introduction to the basic definitions, explain how bounded L-groups differ from classical L-groups, and outline an application to quantum many body invariants.

Tropicalized quantum field theory
2025-10-01 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Michael Borinsky, Perimeter Institute
Title: Tropicalized quantum field theory
Abstract: Quantum field theory (QFT) is one of the most accurate methods for making phenomenological predictions in physics, but it has a significant drawback: obtaining concrete predictions from it is computationally very demanding. The standard perturbative approach expands an interacting QFT around a free QFT, using Feynman diagrams. However, the number of these diagrams grows superexponentially, making the approach quickly infeasible.
I will talk about arXiv:2508.14263, which introduces an intermediate layer between free and interacting field theories: a tropicalized QFT. Often, this tropicalized QFT can be solved exactly. The exact solution manifests as a non-linear recursion equation fulfilled by the expansion coefficients of the quantum effective action. Geometrically, this recursion computes volumes of moduli spaces of metric graphs and is thereby analogous to Mirzakhani’s volume recursions on the moduli space of curves. Building on this exact solution, an algorithm can be constructed that samples points from the moduli space of graphs approximately proportional to their perturbative contribution. Via a standard Monte Carlo approach we can evaluate the original QFT using this algorithm. Remarkably, this algorithm requires only polynomial time and memory, suggesting that perturbative quantum field theory computations actually lie in the polynomial-time complexity class, while all known algorithms for evaluating individual Feynman integrals are at least exponential in time and memory. The (potential) capabilities of this approach are remarkable: For instance, we can compute perturbative expansions of massive scalar D=3 phi^3 and D=4 phi^4 quantum field theories up to loop orders between 20 and 50 using a basic proof-of-concept implementation. These perturbative orders are completely inaccessible using a naive approach.

Special Kähler geometry and collapsing
2025-10-02 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Valentino Tosatti, NYU Courant Institute
Title: Special Kähler geometry and collapsing
Abstract: Special Kähler geometry was first discovered in the context of N=2 supersymmetric 4D gauge theories, and it also plays a prominent role in mirror symmetry. A key observation of Donagi-Witten and Freed is that the base of every algebraic integrable system admits a special Kähler metric, while the total space admits a hyperkähler metric. In this talk I will consider compact hyperkähler manifolds with a an algebraic integrable system (i.e. a holomorphic Lagrangian torus fibration), and consider a family of hyperkähler metrics such that the volume of the torus fibers shrinks to zero. I will explain how the hyperkähler metrics must collapse to a special Kähler metric on the base (away from the discriminant locus), and what we can say about the metric completion of the limit.

Local Donaldson-Scaduto conjecture
2025-10-03 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Saman Habibi Esfahani
Title: Local Donaldson-Scaduto conjecture
Abstract: This talk is based on joint works with Gora Bera and Yang Li. Motivated by collapsing Calabi-Yau 3-folds and G2-manifolds with Lefschetz K3 fibrations in the adiabatic setting, Donaldson and Scaduto conjectured the existence and uniqueness of a special Lagrangian pair-of-pants in the Calabi-Yau 3-fold $ X times mathbb{C}$, where $X$ is either a hyperkähler K3 surface (global version) or an A2-type ALE hyperkähler 4-manifold (local version). After a brief introduction to the subject, we discuss the significance of this conjecture in the study of Calabi-Yau 3-folds and G2-manifolds, and then prove the local version of the conjecture.

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.

Non-perturbative aspects of self-dual gauge theory
2025-10-06 15:00 - 16:00 | Virtual
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Kevin Costello (Perimeter Institute)

Title: Non-perturbative aspects of self-dual gauge theory

Abstract: Self-dual gauge theory is conformal in perturbation theory, but has a non-trivial beta-function when instanton effects are included. I will give two computations of this beta-function, one based on the Grothendieck-Riemann-Roch formula and one using holography in the topological string. This leads to two new ways to compute the standard QCD beta-function at one loop, without using Feynman diagrams. If time permits, I will also discuss how instantons effect scattering amplitudes.

Geometry of dimer models
2025-10-06 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Alexei Borodin, MIT
Title: Geometry of dimer models
Abstract: Random dimer coverings of large planar graphs are known to exhibit unusual and visually apparent asymptotic phenomena that include formation of frozen regions and various phases in the unfrozen ones. For a specific family of subgraphs of the (periodically weighted) square lattice known as the Aztec diamonds, the asymptotic behavior of dimers admits a precise description in terms of geometry of underlying Riemann surfaces. The goal of the talk is to explain how the surface structure manifests itself through the statistics of dimers. Based on joint works with T. Berggren and M. Duits.

A Classifying Space for Phases of Matrix Product States
2025-10-07 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speakers: Daniel Spiegel, Harvard Math
Title: A Classifying Space for Phases of Matrix Product States
Abstract: Alexei Kitaev has conjectured that there should be a loop spectrum consisting of spaces of gapped invertible quantum spin systems, indexed by spatial dimension d of the lattice. Motivated by Kitaev’s conjecture, I will detail a concrete construction of a topological space B consisting of translation invariant injective matrix product states (MPS) of all physical and bond dimensions, which plays the role Kitaev’s space in dimension d = 1. Having such a space is a useful tool in the discussion of parametrized phases of MPS; in fact it allows us to define a parametrized phase as a homotopy class of maps into B. The space B is constructed as the quotient of a contractible space E of MPS tensors modulo gauge transformations. The projection map from E to B is a quasifibration, from which we can compute the homotopy groups of the classifying space B by a long exact sequence. In particular, B has the weak homotopy type K(Z, 2) x K(Z, 3), shedding light on Kitaev’s conjecture in the context of MPS.
Daniel Spiegel will speak for 60 minutes.
Sunghyuk Park (CMSA) will also speak for 15 minutes

Understanding Optimization in Deep Learning with Central Flows
2025-10-08 14:00 - 15:00 | Hybrid – G10
New Technologies in Mathematics Seminar
Speaker: Alex Damian, Harvard
Title: Understanding Optimization in Deep Learning with Central Flows
Abstract: Traditional theories of optimization cannot describe the dynamics of optimization in deep learning, even in the simple setting of deterministic training. The challenge is that optimizers typically operate in a complex, oscillatory regime called the “edge of stability.” In this paper, we develop theory that can describe the dynamics of optimization in this regime. Our key insight is that while the *exact* trajectory of an oscillatory optimizer may be challenging to analyze, the *time-averaged* (i.e. smoothed) trajectory is often much more tractable. To analyze an optimizer, we derive a differential equation called a “central flow” that characterizes this time-averaged trajectory. We empirically show that these central flows can predict long-term optimization trajectories for generic neural networks with a high degree of numerical accuracy. By interpreting these central flows, we are able to understand how gradient descent makes progress even as the loss sometimes goes up; how adaptive optimizers “adapt” to the local loss landscape; and how adaptive optimizers implicitly navigate towards regions where they can take larger steps. Our results suggest that central flows can be a valuable theoretical tool for reasoning about optimization in deep learning.

Profinite tensor powers
2025-10-09 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: David Treumann (Boston College)
Title: Profinite tensor powers
Abstract: I’ll discuss the problem of defining a tensor product of profinitely many copies of a vector space V, and propose a definition $bigotimes_X^{mcc} V$ in the special situation that (1) V is finite-dimensional over F_2, and (2) the profinite X indexing the tensor factors is acted on with finitely many orbits by a pro-2-group. The “mcc” on the tensor sign stands for “magnetized and conditionally convergent.” A variant construction makes sense when V is a bimodule over a semisimple F_2-algebra, and the index set X has the profinite version of a cyclic order. The definition organizes some computations in Heegard Floer homology: it can be pitched as a computation of the HF of some pro-3-manifolds, though we do not know how to define such a thing. This is joint work with CM Michael Wong.

Symmetries and Moduli Spaces: Baby Steps beyond Calabi-Yau
2025-10-09 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Xingyang Yu, Virginia Tech
Title: Symmetries and Moduli Spaces: Baby Steps beyond Calabi-Yau
Abstract: I will explore the interplay between symmetries and moduli spaces in string compactifications, starting from the familiar Calabi–Yau case and then taking some baby steps toward more general settings. A classical benchmark is the line bundle over Calabi–Yau complex structure moduli space, whose physical counterpart corresponds to the Berry phase of the spectral flow operator in worldsheet SCFT. I will review this story and then discuss how it begins to change in c=1 theories with worldsheet anomalies, and in G_2 and Spin(7) compactifications where U(1)_R symmetry is absent. The goal is not a finished framework, but to highlight how anomalies and non-invertible symmetries may enter the picture and to raise open questions about what kinds of structures might live over moduli spaces beyond Calabi–Yau.

The Rozansky-Witten field theory in the functorial TQFT formalism
2025-10-10 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Lorenzo Riva
Title: The Rozansky-Witten field theory in the functorial TQFT formalism
Abstract: This will be a broad talk about the topic of my PhD thesis. We will discuss a particular example of a 3D field theory from physics called Rozansky-Witten which is interesting from both a physical and a mathematical point of view: its is connected with mirror symmetry, the A- and B-models, Calabi-Yau geometry, and the partition functions give finite-type invariants of 3-manifolds. In the rest of the talk we will try to formalize this field theory as a functor out of a certain cobordism 3-category (emphasis on “try”).

Geometry and Quantum Theory Seminar
2025-10-14 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Dylan Galt, Harvard
(60 minute talk)
Title: What is a “nonlinear” near-symplectic form?
Abstract: In this talk, I will explain how one might understand this question and why a possible answer can be found in the geometry of coassociative 4-folds, a special class of minimal submanifolds discovered by Harvey and Lawson.

Speaker: Keyou Zeng, CMSA
(30 minute talk)
Title: Cohomology of configuration space of points
Abstract: Configuration space of points is an interesting and important subject in mathematics and physics. I’ll review some classical results computing cohomology of configuration space of points. I’ll also introduce some recent progress in computing sheaf cohomology of configuration space of affine space.

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

Lech's inequality and stability of local rings
2025-10-16 14:00 - 15:00 | Virtual
Algebra Seminar
Speaker: Linquan Ma (Purdue University)
Title: Lech’s inequality and stability of local rings
Abstract: We explore Lech’s inequality relating the colength and multiplicity of m-primary ideals in a Noetherian local ring (R,m). We introduce a natural invariant that measures the sharpness of Lech’s inequality and show its connections with singularities of asymptotically semistable varieties and singularities arising from the MMP. We compute this invariant in various examples. This is joint work with Ilya Smirnov.

Differential Geometry and Physics Seminar
2025-10-16 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Andy Neitzke, Yale
Title: Abelianization of tau functions
Abstract: The symplectic and hyperkahler geometry of moduli spaces of flat connections over Riemann surfaces is in a sense quantized by the theory of isomonodromic tau functions. These functions in turn arise as partition functions in the conformal field theory of twisted free fermions. I will describe a new scheme for computing these tau functions via abelianization, and what it produces in one simple example, related to the Painleve I equation. This scheme is joint work with Qianyu Hao. The talk is intended to be self-contained (you don’t have to know in advance what a tau function or a conformal field theory is).

DMFT, Two Point Correlations of Resolvents, and Applications to Machine Learning Theory
2025-10-17 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Blake Bordelon
Title: DMFT, Two Point Correlations of Resolvents, and Applications to Machine Learning Theory
Abstract: Machine learning algorithms evolve the parameters of a model in a high dimensional and disordered loss landscape. To characterize the effects of random initialization of model parameters, randomly sampled training data, and the effect of SGD noise, it often is useful to invoke ideas from random matrix theory and the physics of disordered systems. In this seminar, I describe a general idea, known as dynamical mean field theory (DMFT) which describes the evolution of a disordered dynamical system in infinite dimensions. I will briefly describe simple examples of interest to theoretical neuroscientists and machine learning theorists. For linear dynamical systems, I will show that this method characterizes the typical case trajectory in terms of two point correlations of resolvent matrices evaluated at different frequencies. This bispectral object can account for puzzling effects such as late time divergence of gradient descent at the interpolation threshold (when parameters = dataset size) despite the Jacobian of the dynamics having real and non-positive eigenvalues. I will then describe a novel two point correlation result for general free products of the form M = O B O^T A for O sampled from the Haar measure. I will use this result to characterize the exact asymptotics of the performance of a linear transformer trained to perform in-context linear regression on “generic” (randomly rotated) covariance matrices.

Freedman Seminar: Michael Freedman, CMSA & Bowen Yang, CMSA
2025-10-17 13:00 - 16:00 | Virtual
Freedman Seminar
Speaker: Michael Freedman, Harvard CMSA
Title: Sullivan’s work on Lipschitz structures
Abstract: I’ll begin with an elementary, but now little known, piece of PL topology: engulfing. John Stalling used it to give an alternative proof of the high dimensional Poincare conjecture. Then I’ll explain Dennis Sullivan’s enhancement of Kirby’s torus trick (which relies on engulfing.) I’ll note an open question regarding Lipschitz structures on 4-manifolds.

Speaker: Bowen Yang, CMSA
Title: Quantum Cellular Automata and Algebraic L-Theory
Abstract: Quantum cellular automata (QCAs) are models of reversible quantum dynamics that preserve locality; they can be thought of as quantum analogues of classical cellular automata, but with much richer structure. I will describe a classification of the Clifford subclass of QCAs using methods from algebraic L-theory. The main result identifies the group of Clifford QCAs, up to natural equivalences, with L-theory homology of the underlying space. This gives a conceptual explanation of previously observed periodic patterns in lattice models and extends the picture to more general spaces. I will outline the ideas behind the construction and indicate how the framework connects topology, operator algebras, and quantum information. If time permits, I will also comment on what is known — and unknown — about the general (non-Clifford) case.

Categorical 't Hooft expansion and Chiral Algebras
2025-10-20 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Adrian López-Raven, Perimeter
Title: Categorical ‘t Hooft expansion and Chiral Algebras
Abstract: In https://arxiv.org/abs/2411.00760, we show how holographic dual B-model backgrounds can be systematically derived from the ‘t Hooft expansion of specific families of chiral algebras. The resulting holographic dual backgrounds are typically non-commutative and appear to be novel. In this talk I’ll review certain aspects of our construction. In particular, we’ll review how to build a category of D-branes for the String Theory dual, starting from the planar limit of the chiral algebra. Given its generality, I’ll emphasize the potential utility of the construction in the study of weak coupling holography for general theories with a large N limit.

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.
