The Rozansky-Witten field theory in the functorial TQFT formalism
2025-10-10 12:00 - 13:00 (America/New_York) | 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.

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

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

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

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

CMSA Q&A Seminar: Thomas Grimm
2025-10-22 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Thomas Grimm, Utrecht & CMSA
Topic: What is o-minimality and what is it good for?

The Carleson project: A collaborative formalization
2025-10-22 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: María Inés de Frutos Fernández, Mathematical Institute, University of Bonn
Title: The Carleson project: A collaborative formalization
Abstract: A well-known result in Fourier analysis establishes that the partial Fourier sums of a smooth periodic function $f$ converge uniformly to $f$, but the situation is a lot more subtle for e.g. continuous functions. However, in 1966 Carleson proved that they do converge at almost all points for $L^2$ periodic functions on the real line. Carleson’s proof is famously hard to read, and there are no known easy proofs of this theorem. As a large collaborative project, we have formalized in Lean a generalization of Carleson’s theorem in the setting of doubling metric measure spaces (proven in 2023), and Carleson’s original result as a corollary. In this talk I will give an overview of the project, with a focus on how the collaboration was organized.

Character sheaves on the loop Lie algebra and its global implications
2025-10-23 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Zhiwei Yun, MIT
Title: Character sheaves on the loop Lie algebra and its global implications
Abstract: Motivated by Lusztig’s definition of character sheaves on a reductive Lie algebra, we propose a definition of character sheaves on the loop Lie algebra. The construction can be viewed as a p-adic analogue of the orbit method, and the resulting sheaves are closely related to characters of supercuspidal representations of J.K.Yu. Further, the construction suggests a definition of wildly ramified Hitchin moduli spaces suitable for the study of ramified geometric Langlands. This is joint work with Bao Chau Ngo.

On some new irrationality results
2025-10-23 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Ron Donagi, UPenn
Title: On some new irrationality results
Abstract: An algebraic variety X is rational if a Zariski-open subset of X is isomorphic to a Zariski-open subset of projective space. A weaker property is unirationality: X is unirational if a Zariski-open subset of projective space maps onto a Zariski-open subset of X. These properties are equivalent in dimensions 1 and 2. In the seventies it was discovered that they are not equivalent in dimension 3, as several different approaches succeeded in proving irrationality of some unirational varieties. The theory of Hodge atoms, recently developed by Katzarkov, Kontsevich, Pantev and Yu, uses ideas from mirror symmetry and quantum cohomology to exhibit new birational invariants capable of proving irrationality of some 4-dimensional unirational varieties. We illustrate the power of this new technique by applying it to the 4 dimensional intersection of quadrics in P^7.

Analytic Spread of Binomial Edge Ideals
2025-10-24 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Stephen Landsittel, CMSA
Title: Analytic Spread of Binomial Edge Ideals
Abstract: To an ideal J in a polynomial ring R over a field K we associate its analytic spread ell(J), which is the dimension of the fiber cone F(J) of J. When J is graded and generated in a single degree d, then F(J) is a finite type K-algebra.
To a graph G we associate its binomial edge ideal: J_G:= (x_i y_j – x_jy_i | {i,j} is an edge of G).
In this talk we will discuss recent work where sharp bounds are given for ell(J_G) and we compute the exact value when G is a pseudoforest. We accomplish this by computing the transcendence degree trdeg_{K} F(J), of the fiber cone over K.

de Rham Theory in Derived Differential Geometry
2025-10-27 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Grigorii Taroian, U Toronto
Title: de Rham Theory in Derived Differential Geometry
Abstract: In the talk, I will describe recent progress in building a version of de Rham theory for derived manifolds and derived differentiable stacks.
Derived differential geometry is a nascent field applying techniques from derived algebraic geometry to the study of spaces with smooth structures. As such, it serves as a natural home for studying objects arising in BV formalism. For instance, concepts such as critical loci of action functionals or their quotients by gauge actions can be naturally interpreted as derived differentiable stacks.
In our work, we build a version of de Rham theory for these spaces and prove a version of the de Rham isomorphism. Due to the highly singular nature of all objects involved, developing such a theory is significantly more challenging than in the usual differential geometry, and thus, we construct our formalism with inspiration from algebraic geometry rather than classical differential topology. As a main application of the developed theory, we obtain a version of the comparison morphism between de Rham and constant sheaf cohomology arising from the corresponding map of stacks. This should enable further developments, with a view towards a fully-fledged theory of shifted symplectic structures for derived differentiable stacks.
The talk is based on a preprint of the same name, arXiv:2505.03978.

Rigidity, expansion and polytopes
2025-10-27 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Eran Nevo (Hebrew University of Jerusalem)
Title: Rigidity, expansion and polytopes
Abstract: Given a graph G and an embedding of its vertices in R^d, what continuous motions of the vertices preserve all edge lengths? Clearly all motions induced by an isometry of R^d do, these are the trivial motions; are there any others? If the answer is NO for all (equivalently, for one) generic embedding, G is called d-rigid.
What are the d-rigid graphs?
This problem has been extensively studied since the 70s, and is still widely open for d≥3. It is studied mainly from algebraic geometry and combinatorial points of view. Variants of it, especially in dimensions 2 and 3, are of importance also beyond mathematics, e.g. in structural engineering, computational biology and more.
I will focus on a quantitative version of rigidity via spectral analysis of the related stiffness matrix, including the construction of “rigidity expanders”, generalizing expander graphs. Higher dimensional notions of rigidity and of stiffness matrices, and their relation to the study of polytopes, will be addressed too.

Geometry and Quantum Theory Seminar
2025-10-28 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speakers: (1) Elliott Gesteau, CMSA (60 min) and (2) Sanjay Raman, Harvard (30 min)
(1) Title: Holography for closed universes
Abstract: Recent advances in holography and black hole physics have prompted a number of new puzzles related to the physics of closed universes, which can be argued to have a one-dimensional Hilbert space of states from various perspectives. In this talk I will review this new set of puzzles, with an emphasis of the mathematical questions that they seem to raise.

(2) Title: Non-supersymmetric branes, bordism, and fractional charges
Abstract: We dive a little deeper into the non-supersymmetric brane configurations studied in Kaidi-Tachikawa-Yonekura (2024). We study the gravitational analogues of the branes in the aforementioned paper, relating them to relative defects predicted by the Cobordism Conjecture, and we argue for the existence of objects carrying fractional heterotic string and M2-brane charge.

Affine Springer fibers and representations
2025-10-30 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Roman Bezrukavnikov, MIT
Title: Affine Springer fibers and representations
Abstract: Relating representation categories of interest, such of modules over the quantum group, to topology of loop spaces has been an important theme in representation theory for some decades. I will describe a result of this sort involving a geometric object that has not appeared in this context until now, an affine Springer fiber, and mention its applications. Based on a joint work with Pablo Boixeda Alvarez, Michael McBreen and Zhiwei Yun.

RCD structures on singular Kahler varieties
2025-10-30 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Jian Song, Rutgers University
Title: RCD structures on singular Kahler varieties
Abstract: Let X be a 3-dimensional projective variety with klt singularities. We prove that every singular Kahler metric on X with bounded Nash entropy and Ricci curvature bounded below induces a unique compact RCD space homeomorphic to the projective variety X itself. In particular, singular Kahler- Einstein spaces of complex dimension 3 with bounded Nash entropy are compact RCD spaces topologically and holomorphically equivalent to the underlying projective variety. Such results establish connections among algebraic, geometric and analytic structures of klt singularities from birational geometry and provide abundant examples of RCD spaces from algebraic geometry via complex Monge-Ampere equations.

Skein remain the same
2025-10-31 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Sunghyuk Park, CMSA
Title: Skein remain the same
Abstract: The count of holomorphic curves in a Calabi-Yau 3-fold ending on a Lagrangian is famously not deformation invariant, but Ekholm and Shende have shown that it can be made invariant by counting in the skein. Given a 3-manifold M and a branched cover arising from the projection of a Lagrangian 3-manifold L in the cotangent bundle of M, we use the skein-valued curve count to construct a map from the skein of M to that of L. When M and L are products of surfaces and intervals, deforming L within the space of Lagrangians yields a skein-valued lift of the Kontsevich-Soibelman wall-crossing formula. After all, the skeins remain the same. Based on joint work (arXiv:2510.19041) with Tobias Ekholm, Pietro Longhi, and Vivek Shende.

A fully local extension of Chern-Simons theory
2025-11-04 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Dan Freed
Title: A fully local extension of Chern-Simons theory
Abstract: The Reshetikhin-Turaev-Witten invariants of 3-manifolds fit into a 3-dimensional topological field theory that also attaches invariants to 2-manifolds and 1-manifolds. It has long been asked to extend to invariants of 0-manifolds as well, a question that often takes the form: What does Chern-Simons theory attach to a point?. In joint work with Claudia Scheimbauer and Constantin Teleman we use the cobordism hypothesis to construct a fully local version of Chern-Simons theory. We also consider different versions with different tangential structures, as I will explain.

CMSA Q&A Seminar: Paul Seidel, MIT
2025-11-05 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Paul Seidel, MIT
Topic: Fukaya categories of Landau-Ginzburg models

Discovery of unstable singularity with machine precision
2025-11-05 14:00 - 15:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
New Technologies in Mathematics Seminar
Speaker: Yongji Wang, NYU Courant Institute of Mathematical Sciences
Title: Discovery of unstable singularity with machine precision
Abstract: Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial conditions. Historically, numerical approaches have primarily identified stable singularities. However, these are not expected to exist for key open problems, such as the boundary-free Euler and Navier-Stokes cases, namely the Millennium Prize problem. For these problems, the true challenge lies in finding unstable singularities, which are exceptionally elusive, as any tiny perturbation can divert the system from its blow-up trajectory.
In this talk, I will present a new computational framework which has led to the first systematic discovery of new families of unstable singularities in various fluid equations. Our approach merges curated machine learning architectures with a multi-stage training scheme and a high-precision Gauss-Newton optimizer, creating a powerful tool for navigating the complex landscape of nonlinear PDEs. Beyond discovering these singularities, the precision of this method is another key breakthrough, achieving unprecedented accuracies on the order of $O(10^{-13})$—a level constrained only by the round-off errors of the GPU hardware. This level of precision meets the stringent requirements for rigorous mathematical validation of the discovered solution via computer-assisted proofs, offering a new pathway to resolving long-standing challenges in mathematical physics.

Dimension Reduction and Adiabatic Limits of Generalized ASD Instantons
2025-11-06 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Dylan Galt, Harvard
Title: Dimension Reduction and Adiabatic Limits of Generalized ASD Instantons
Abstract: I will begin by explaining a dimension reduction result for Tian’s generalized ASD instantons over product manifolds, which generalizes work of Yuanqi Wang on codimension-one reduction and includes the cases of G2 and Spin(7) instantons as well as degree zero Hermitian-Yang-Mills connections. The proof is inspired by a simple observation about the Yang-Mills energy and suggests a natural adiabatic picture for these generalized ASD connections. I will describe ongoing work towards establishing such an adiabatic scheme, emphasizing some of the complications that arise in the general case and explaining what can be said for G2 instantons. Everything I will talk about is joint work with my collaborator Langte Ma.

Classification of 2D Stabilizer States
2025-11-07 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Bowen Yang
Title: Classification of 2D Stabilizer States
Abstract: I will explain how translation-invariant two-dimensional stabilizer states are completely classified by finite abelian groups with nondegenerate quadratic forms—that is, by abelian anyon theories. The proof uses the algebraic structure of stabilizer codes as modules over Laurent polynomial rings, revealing how their physical features reflect in this module-theoretic framework. arXiv:2509.10418

The Moyal bracket and the BV cohomology of the spinning particle
2025-11-10 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Ezra Getzler, Northwestern
Title: The Moyal bracket and the BV cohomology of the spinning particle
Abstract: The spinning particle is the one-dimensional reduction of the Neveu-Schwartz-Ramond superstring. It consists of a supersymmetric particle moving in a one-dimensional supergravity background, and its quantization is the Hilbert superspace of harmonic spinors. (These models are classified by N, the number of copies of fermionic fields. In this talk, N=1. The extension to N=2 is work in progress with Ivo.) It is actually an AKSZ model (so a generalization of one-dimensional Chern-Simons), and so associated to a differential graded symplectic supermanifold, by which we mean a pair (ω,Q), where ω is a(n exact) symplectic form and Q is an odd function of degree 1. The cohomology of the ring of functions of this supermanifold with differential the Poisson bracket with Q determines the classical BV cohomology of the spinning particle, so is important for understanding perturbative BV quantization of this model. I calculated this cohomology in earlier work for N=1, and showed that it is somewhat bizarre, with two series of cohomology classes in arbitrary negative degrees, each a copy of the functions on the target manifold.
In the study of quantum BFV, we should instead consider the Moyal bracket on the target, and lift Q to an element Q satisfying [Q,Q]=0. The cohomology of the differential [Q,-] is the Moyal cohomology of the differential graded symplectic supermanifold. (This lift corresponds to the choice of a Spinc structure on the target manifold.) In this talk, I prove that the Moyal cohomology, unlike the Poisson cohomology, is well-behaved: in the spectral sequence from Poisson to Moyal cohomology, the extra cohomology classes of negative degree cancel each other pairwise at the E1 page.

Khovanov homology from KLRW algebras
2025-11-11 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Sunghyuk Park, CMSA
Title: Khovanov homology from KLRW algebras
Abstract: This is the first in a four-part series, organized together with Vasily Krylov, Sebastian Haney, and Lorenzo Riva, aimed at understanding Aganagic’s categorification of quantum link invariants through Coulomb branches and homological mirror symmetry. In this first talk, I will lay the algebraic background for this story by explaining how Khovanov homology can be understood in terms of higher representation theory via Khovanov-Lauda-Rouquier-Webster (KLRW) algebras.

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

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

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

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

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

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

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

Series Pamphlet (pdf)
Read more about the Hodge Conjecture at the Clay Math website.
Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series
