CMSA

Higher current algebras and chiral algebras
2026-04-27 15:00 - 16:00 (America/New_York) | CMSA G102, 20 Garden Street, Cambridge, MA, 02138
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Charles Young (University of Hertfordshire)
Title: Higher current algebras and chiral algebras
Abstract: Vertex algebras capture physicists’ notion of OPEs in chiral CFTs, in complex dimension one. For various motivations, one would like to have analogs of vertex algebras in higher dimensions. Chiral algebras, in the sense of Beilinson-Drinfeld and Francis-Gaitsgory, provide a natural framework here, because they re-express the vertex algebra axioms (which are rather sui generis, and therefore hard to generalize) as something more recognizable (a chiral algebra is a Lie algebra, of a sort).
I will review this, and then go on to introduce a certain concrete model of the unit chiral algebra in higher dimensions. In higher dimensions one is forced to work up to coherent homotopy in some fashion; in this model it turns out to be in the mildest fashion one could hope for: namely, one moves from Lie algebras to their homotopy analogs, L-infinity algebras, and from chiral algebras to homotopy chiral algebras in a sense introduced by Malikov-Schechtman.
The main tool in the talk will be a strict cdga model — the polysimplicial model — of derived global sections of the structure sheaf on configuration space. The hope is that this model will prove well-adapted to doing concrete calculations, and in that direction, I will gesture towards a homotopy version of the usual Arnold/Orlik-Solomon relations for broken circuits.
This is joint work with Zhengping Gui and Laura Felder and is based largely on the preprint 2506.09728

Enacted collective cognition: Brainless problem-solving in weaver ants
2026-04-27 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Ofer Feinerman, Weizmann Institute of Science
Title: Enacted collective cognition: Brainless problem-solving in weaver ants
Abstract: Unlike most ants, weaver ants construct their nests by pulling together leaves. Because individual ants are small relative to the leaves, they assemble their bodies into temporary tools that bend the leaves into a hollow structure, later stabilized with larval silk. Remarkably, they achieve functional nests across a wide range of leaf shapes and configurations, suggesting that this distributed system is capable of solving complex, open-ended problems.
To understand how this is possible, we performed laboratory experiments using controlled leaf configurations. In simple cases, we show that ants can rely on a zipping heuristic that produces closed nests, and we use differential geometry to demonstrate how flexible leaves are transformed into rigid structures. Crucially, this zipping behavior forms a feedback loop in which ants continuously read and modify the evolving structure. In this sense, the nest itself functions as a shared physical information system.
This suggests that cognition in this system is not located within individual ants, but is enacted through the co-dynamics of the colony and the structure it builds. We present preliminary experiments with more complex leaf configurations, showing that this process can solve increasingly challenging construction problems. Together, these results point to a distributed, brainless, and enactive form of cognition.

The classifying space of a Morse flow category
2026-04-28 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar
Speaker: Lorenzo Riva, Harvard CMSA
Title: The classifying space of a Morse flow category

Abstract: Following a paper of Calle and Liu we show that, under suitable tameness assumptions, the classifying space of the Morse category associated to a manifold M with a Morse function recovers the homotopy type of M, thereby addressing a gap in a preprint of Cohen-Jones-Segal. We also give a counterexample when those assumptions are not satisfied.

Transcendental Epsilon Multiplicity via Divisor Volumes
2026-04-30 16:00 - 17:00 | CMSA Room G02, 20 Garden Street, Cambridge, MA, 02138
Algebra Seminar
Speaker: Sudipta Das, Tata Institute of Fundamental Research
Title: Transcendental Epsilon Multiplicity via Divisor Volumes
Abstract: In this talk, our goal is to establish a structural bridge between asymptotic commutative algebra and transcendence theory to show that there exists an ideal in a Noetherian local ring whose epsilon multiplicity is transcendental. By equating the local-cohomological definition of epsilon multiplicity to a global divisorial volume integral on a projective bundle, we apply Baker’s theorem on linear forms in logarithms to prove that the resulting arithmetic invariant falls strictly outside the field of algebraic numbers. This talk is based on collaborative work with Vinh Pham and Stephen Landsittel.

note room change to G02

Lifting F-split surfaces to the Witt vectors
2026-05-01 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Iacopo Brivio, CMSA
Title: Lifting F-split surfaces to the Witt vectors

Abstract: Algebraic varieties in positive characteristic are ill behaved compared to characteristic zero ones. Several important tools available over the complex numbers, such as the Hodge decomposition theorem, are either not available or straight-away false. There are two important classes of positive characteristic varieties which have better behavior: Witt liftable varieties and Frobenius split varieties. A folklore conjecture predicts that the latter class is contained in the former. In a joint work with Bernasconi, Kawakami and Witaszek we proved that this is the case for surfaces. I will give an overview of this result as well as some applications thereof.

Twisted D-branes and TQFTs valued in Calabi-Yau categories
2026-05-04 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Surya Raghavendran, Yale University
Title: Twisted D-branes and TQFTs valued in Calabi-Yau categories
Abstract: Recently, Bozec–Calaque–Scherotzke have articulated a noncommutative version of the AKSZ construction, which associates to a smooth Calabi–Yau category a fully extended TQFT valued in a category of iterated Calabi–Yau cospans. In this talk, I will study a class of examples of such theories which arise in the context of conjectures of Costello and Li, which describe Type II strings in certain Ramond–Ramond backgrounds as topological strings. These TQFTs capture structural features of the BPS physics of D-branes that are universal in Chan–Paton factors. Conjecturally commutative limits of the values of such theories on closed manifolds can sometimes be geometrically quantized to yield algebraic structures with Hall-type products. Examples of this paradigm include CoHAs associated to complex 3-folds, CoHAs attached to local systems on 3-manifolds, and the categorified Hall algebras of Porta–Sala.

Dynamics as intersection problem
2026-05-04 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Nikita Nekrasov, Simons Center
Title: Dynamics as intersection problem
Abstract: Most classical and quantum field theories are based on an action principle. However, there are important exceptions to this — hydrodynamics and the theory of self-dual fields. In this talk we formulate the covariant relativistic fluid dynamics, with or without magnetic fields, as well as the theory of chiral boson in 1+1 dimensions, self-dual tensor in 1+5 dimensions, and self-dual four-form of IIB supergravity, in terms of intersection theory of an auxiliary phase space. This provides a common covariant geometric framework for systems without a conventional action, while at the same time laying the groundwork for quantization via the Kontsevich approach. Joint work with Paul Wiegmann.

New directions in synthetic data
2026-05-06 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Tatsunori Hashimoto, Stanford
Title: New directions in synthetic data
Abstract: Synthetic data has been an effective, if boring set of techniques: prompt some language model to restructure your corpus to match some downstream task, with occasionally some distillation. In this talk, we will take a more expansive view of synthetic data as a general algorithmic tool for generative modeling, arguing that the design space and possibilities of synthetic data are much bigger than it might seem. Through a few recent works, we will show that synthetic data has major benefits beyond transforming the data – improving in-domain perplexities, and enabling unique algorithmic primitives, such as neighborhood smoothing and concatenated ‘mega’ documents. With this broader view, we will point towards a nascent but interesting possibility of treating data itself as an algorithmic object to be engineered and optimized end-to-end.

From Poincaré/Koszul duality to (twisted) AdS/CFT correspondence
2026-05-08 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Keyou Zeng
Title: From Poincaré/Koszul duality to (twisted) AdS/CFT correspondence
Abstract: Poincaré duality is a fundamental result in the (co)homology theory of manifolds. It has many applications in topology and vast generalizations to other types of “spaces,” such as singular/stratified spaces and schemes. In this talk, I will discuss a variant of Poincaré duality for factorization algebras, also known as Koszul duality. At the end of the talk, I will relate this notion to a mathematical formulation of what physicists call the AdS/CFT correspondence, as proposed by Costello and Li.

When do anomalous finite symmetries in (3+1)d enforce gaplessness?
2026-05-11 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Matthew Yu (University of Oxford)
Title: When do anomalous finite symmetries in (3+1)d enforce gaplessness?
Abstract: I will explain a comprehensive framework for characterizing the infrared (IR) phases of a fermionic QFTs in (3+1)d, based on their quantum anomalies associated with a finite symmetry. We uncover a fundamental dichotomy among these anomalies: the first class of anomalies can always be realized by symmetric gapped states, while the second class can never be realized by gapped states without breaking the given symmetry, establishing the phenomenon of symmetry-enforced gaplessness in these settings. Using the construction of symmetry extension afforded to us by new developments in fusion 2-categories, we construct the candidate gapped states that theories with the first class of anomalies can flow to in the IR. As an application, I will provide examples of concrete predictions for the candidate IR phases of (3+1)d gauge theories based on our results.

Statistical Shape Analysis of Complex Natural Structures
2026-05-11 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Anuj Srivastava, Johns Hopkins University
Title: Statistical Shape Analysis of Complex Natural Structures
Abstract: Statistical modeling and analysis of structured data is a fast-growing field in Statistics and Data Science. Rapid advances in imaging techniques have led to tremendous amounts of data for analyzing imaged objects across several scientific disciplines. Examples include shapes of cancer cells, botanical trees, human biometrics, 3D genome, brain anatomical structures, crowd videos, nano-manufacturing, and so on. Shapes are relevant even in non-imaging data contexts, e.g., the shapes of COVID rate curves or the shapes of activity cycles in lifestyle data. Imposing statistical models and inferences on shapes seems daunting because the shape is an abstract notion and one requires precise mathematical representations to quantify shapes. This talk has two parts. In the first part, I will present some recent developments in “elastic representations” of structures such as functions, curves, surfaces, and graphs. In the second part, I will focus on statistical analyses: computing shape summaries, estimation under shape constraints, hypothesis testing, time-series models, and regression models involving shapes.

Polynomial invariants of conjugation over finite fields
2026-05-14 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Aryaman Maithani, University of Utah
Title: Polynomial invariants of conjugation over finite fields

Abstract: Consider the conjugation action of GL₂(K) on the polynomial ring K[X₂ₓ₂].
When K is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant.
We describe the ring of invariants when K is a finite field, and show that it is a hypersurface.

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.

Abelian duality via derived geometry
2026-05-18 15:00 - 16:00 | CMSA Room G02, 20 Garden Street, Cambridge, MA, 02138
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Owen Gwilliam, UMass Amherst
Title: Abelian duality via derived geometry
Abstract: We discuss how to synthesize differential cohomology and the BV formalism to describe generalized Maxwell theories (or abelian p-form gauge theories), and how this framework allows a succinct formulation of abelian duality. Given time, we will discuss how these methods apply to the 6d self-dual 2-form gauge theory that appears as part of the 6d N=(1,0) and (2,0) superconformal theories known as abelian tensor multiplets. This is joint work in progress with Chris Elliott, Ingmar Saberi, and Brian Williams.

Geometry and Quantum Theory Seminar
2026-05-19 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar
Speaker: Vasily Krylov, Harvard

Separation of timescales controls feature learning and overfitting in large neural networks
2026-05-20 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Pierfrancesco Urbani, Universite Paris-Saclay, CNRS, CEA, Institut de physique theorique
Title: Separation of timescales controls feature learning and overfitting in large neural networks
Abstract: To understand the inductive bias and generalization capabilities of large, overparameterized machine learning models, it is essential to analyze the dynamics of their training algorithms. Using dynamical mean field theory we investigate the learning dynamics of large two-layer neural networks. Our findings reveal that, for networks with a large width, the training process exhibits a separation of timescales phenomenon. This leads to several key observations:
1. The emergence of a slow timescale linked to the growth in Gaussian/Rademacher complexity of the network;
2. An inductive bias favoring low complexity when the initial model complexity is sufficiently small;
3. A dynamical decoupling between feature learning and overfitting phases;
4. A non-monotonic trend in test error, characterized by a “feature unlearning” regime at later stages of training.
Joint work with Andrea Montanari.

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

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

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

Big Data Conference 2026
2026-09-03 09:00 - 2026-09-04 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Big Data Conference 2026
Dates: Sep. 3–4, 2026
Location: Harvard University CMSA, 20 Garden Street, Cambridge MA & via Zoom
The Big Data Conference features speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
Register to attend in person
Register for Zoom Webinar

Confirmed Speakers

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

Organizers

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

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

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

Colloquium
2026-09-07 16:30 - 17:30, 2026-09-14 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

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

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

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

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

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

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

Support provided by Logical Intelligence.

Geometry and Quantum Theory Seminar
2026-09-08 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar

CMSA Q&A Seminar
2026-09-09 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

New Technologies in Mathematics Seminar
2026-09-09 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: tba

AI for the Working Mathematician
2026-09-09 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AI for the Working Mathematician
Speaker: Daniel Litt, University of Toronto
Title: TBA

Geometry and Mathematical Physics Seminar
2026-09-10 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Geometry and Mathematical Physics Seminar
Speaker: Brian Williams, Boston University

Member Seminar
2026-09-11 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

Lagrangian Floer Theory and Applications Program
2026-09-15 09:00 - 2026-11-21 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Lagrangian Floer theory and Applications Program
Dates: September 15–November 21, 2026
Location: CMSA G10, 20 Garden St., Cambridge MA 02138
This thematic program will focus on recent developments in Lagrangian Floer theory and applications. These include the development of family Floer theory, degeneration techniques, Floer homotopy theory in the Lagrangian case, Floer theory in prime characteristic, and applications to problems in singularity theory, geometry and dynamics.
A one-week workshop will be held near the start of the program (September 28 – October 2, 2026), with the same title as the thematic program.
Organizers: Denis Auroux (Harvard), Jonny Evans (Lancaster), and Chris Woodward (Rutgers)

To express interest in attending the workshop or program, and to apply for funding if available, please fill out the form.
Funding from the National Science Foundation may be available for US-based graduate students and postdoctoral fellows.

image: Water Sky Garden, Vancouver
Date: 31 January 2010, 12:57:46
Source: Studio Echelman
Author: Peter Vanderwarker, Studio Echelman
