CMSA

Discovery of unstable singularity with machine precision
2025-11-05 14:00 - 15:00 (America/New_York) | 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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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?

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.

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.

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.

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.

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.

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).

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.

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

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.

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”).

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.

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.

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.

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

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.

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.
