CMSA

Solitonic Symmetry: Cohomology with TFT Coefficients
2026-03-03 16:15 - 18:00 (America/New_York) | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Sanjay Raman, Harvard
Title: Solitonic Symmetry: Cohomology with TFT Coefficients
Abstract: We review the formalism of https://arxiv.org/pdf/2307.00939, which develops the theory of solitonic symmetry in quantum field theory. The algebraic structure of solitonic symmetry is determined by the fusion of topological functionals in a given path-integral formulation of topological field theory, and acts generically on topological defects determined by homotopy classes of maps to a “space of fields.” We will argue that the structure of solitonic symmetry in a theory with field space (Y) assembles into what looks like the cohomology of (Y) with coefficients in TFTs. We study this formalism in examples and show in particular that the maximal invertible solitonic subsymmetry reduces to the expected result.

Fukaya categories and higher representation theory
2026-03-05 13:30 - 14:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Differential Geometry and Physics Seminar
Speaker: Vivek Shende (Syddansk Universitet & UC Berkeley)
Title: Fukaya categories and higher representation theory
Abstract: I will explain how Lagrangian Floer homology in certain monopole moduli spaces recovers the Khovanov homology and its relatives, by a description strikingly similar to the Oszvath-Szabo Heegard-Floer theory. I will also explain how the ‘sectorial descent’ of Fukaya categories can be used to construct Rouquier’s promised monoidal structure on the category of representations of the categorified “positive part” of the quantum group. This is joint work with Mina Aganagic, Elise LePage, and Peng Zhou.
Note: Location change to Harvard Science Center Room 300H

Freedman Seminar: Mattie Ji, Penn and Jeongwan Haah, Stanford
2026-03-06 15:00 - 17:15 | Hybrid
Freedman Seminar
Speakers: Mattie Ji (Penn) and Jeongwan Haah (Stanford)
Mattie Ji Title: Quantum Cellular Automata via Algebraic K-Theory
Abstract: Algebraic K-theory, on a very high level, is the study of how to break apart and assemble objects linearly, which makes the field amenable to classification questions. In this work, we apply this methodology to study the classification of quantum cellular automata (QCA). Over an arbitrary commutative ring R and a general class of metric spaces X, we construct a space of QCA that depends only on the large-scale (coarse) geometry of X. We explain how QCA classification groups (QCA modulo circuits) either arise naturally as or are refined by this space in most cases of interest.
Motivated by negative K-theory, we also show the classification of QCA on Euclidean lattices is given by an $Omega$-spectrum indexed by the dimension. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya R-algebras, whose negative homotopy groups are the QCA classification groups. When R is the complex numbers, our method can be adapted to yield an $Omega$-spectrum for QCA of $C^*$-algebras with unitary circuits. This talk is based on joint work with Bowen Yang.

Jeongwan Haah Title: Fermionic QCA in 2d are trivial
Abstract: We consider bounded spread automorphisms of Z/2-graded algebra (fermionic QCA) on the two-dimensional lattice and prove that every fQCA is a unitary circuit followed by fermionic shifts when stabilized by Majorana modes. This is an analog of a theorem by Freedman and Hastings for the case of ungraded algebras. The overall argument follows a similar line in that we show invertible subalgebras in 1d is trivial, but the stabilization is used crucially. By an existing argument, this triviality of fQCA in 2d implies that the 3d (bosonic) QCA that disentangles the Walker-Wang model with three-fermion theory is nontrivial. The latter was known to be nontrivial against Clifford gates but remained conjectural against more general unitary gates. To my knowledge, this gives the only example ungraded QCA that is proved to be nontrivial against general unitary circuits and shifts, and the only example ungraded invertible subalgebra that is not isomorphic to any tensor product algebra. I will explain elements new to the fermionic setting and give an overview of the nontriviality argument. (Based on an upcoming work with Jeffrey Kwan and David Long)

The categorical 't Hooft expansion
2026-03-09 15:00 - 16:00 | Virtual
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Davide Gaiotto (Perimeter Institute)
Title: The categorical ‘t Hooft expansion
Abstract: The ‘t Hooft expansion is the key structure underlying dualities between gauge theories of large matrices and string theories. I will review categorical aspects of the ‘t Hooft expansion, matching the formal deformation space of certain “fundamental modifications” of the gauge theory to the formal deformation space of a category of boundary conditions (“D-branes”) for a 2d dg-TFT, to be identified with the world-volume theory for the string theory. I will illustrate this by reviewing the holographic description of matrix models.

TQFTs do not detect the Milnor sphere
2026-03-10 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: TQFTs do not detect the Milnor sphere
Abstract: I will talk about the recent paper of Gripaios and Randall-Williams with that title, where they prove that a functorial TQFT (i.e. a symmetric monoidal functor from the n-dimensional bordism category into a suitably nice target category) assigns the same value to homotopy spheres that bound a parallelizable manifold, extending some no-go theorems of Reutter and Schommer-Pries.

Strongly adapted contact geometry of Anosov 3-flows
2026-03-12 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Surena Hozoori (Brandeis)
Title: Strongly adapted contact geometry of Anosov 3-flows
Abstract: We will discuss some recent developments in the contact geometric theory of Anosov 3-flows, whose roots go back to the works of Mitsumatsu and Eliashberg-Thurston in the mid 1990s. In particular, we provide a contact geometric characterization of Anosov 3-flows based on interactions with Reeb dynamics, as well as investigate the basic properties of the resulting geometries. Time permitting, we will discuss how these results allow one to re-approach some classical questions in Anosov dynamics.

Finite N indices from branes and negative branes
2026-03-16 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Kasia Budzik (Harvard University)
Title: Finite N indices from branes and negative branes
Abstract: Finite-N effects in large-N gauge theories, such as trace relations, are expected to be holographically dual to non-perturbative phenomena in string theory, such as Giant Graviton branes. A convenient setting to study these effects are supersymmetric indices of U(N) gauge theories. The finite-N indices can be reproduced by a series of corrections to the infinite-N result, known as the Giant Graviton expansion.
In this talk I will present a generalization of the Molien-Weyl formula computing generating functions of invariants of supergroups U(N|M), which arise as gauge groups of brane/negative brane systems in string theory. The formula leads to a new expansion relating finite-N and infinite-N indices of U(N) gauge theories. I will comment on its relation to Murthy’s Giant Graviton expansion and suggest a physical interpretation in terms of branes and negative branes. This talk is based on arXiv:2509.20451 and work in progress with Davide Gaiotto.

Dynamic reasoning
2026-03-18 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Emmanuel Abbé, EPFL, Institute of Mathematics and School of Computer and Communication Sciences & Apple
Title: Dynamic reasoning
Abstract: In the current AI landscape, reasoning is frequently equated with the generation of intermediate “thinking traces”. However, these traces are merely a mechanism, not the ultimate objective.
Relying solely on the presence of a trace can be deceptive, as models often learn to mimic the format of reasoning while effectively overfitting to specific training distributions.
To build more robust and versatile reasoners, we shift our focus to more specific structural properties of the thinking process, in particular compositionality (inductive CoT, AdaBack) and abstraction (AbstRaL).

Moduli of subcanonical points
2026-03-19 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Dawei Chen, Boston College
Title: Moduli of subcanonical points
Abstract: Subcanonical points are special Weierstrass points on smooth algebraic curves whose semigroups are symmetric. In this talk, I will explain the rich geometry of the moduli space of subcanonical points, with a focus on its connected components, birational geometry, topology, and the deformation theory of related monomial singularities.

Line operators in holomorphic QFT
2026-03-24 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar
Speaker: Keyou Zeng (CMSA)
Title: Line operators in holomorphic QFT
Abstract: We will discuss the recent work (arxiv:2508.11749), which defines the so called dg shifted Yangian. I will discuss their origin from the study of 3d holomorphic topological field theories, a variant of 3d TQFT. A dg shifted Yangian is supposed to control the fusion of line defect in the corresponding 3d HT theories. I will illustrate these discussions in examples.

Gauge theory on Hyperkähler manifolds
2026-03-26 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker. Emily Autumn Windes (New Uzbekistan University)
Title: Gauge theory on Hyperkähler manifolds
Abstract: In this talk, I describe various distinguished classes of connections on Hyperkähler manifolds and their dimensional reductions. Then, I describe a construction of new examples of Sp(2)-instantons, primitive HYM connections, and Spin(7)-instantons with symmetry on the manifold T*CP2. This talk is based on joint work with Jesse Madnick and Izar Alonso.

The Beilinson-Bloch conjecture for some non-isotrivial varieties over global function fields
2026-03-26 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Matt Broe, Boston University
Title: The Beilinson-Bloch conjecture for some non-isotrivial varieties over global function fields
Abstract: The Beilinson-Bloch conjecture is a generalization of the Birch and Swinnerton-Dyer conjecture, which relates the ranks of Chow groups of smooth projective varieties over global fields to the order of vanishing of L-functions. We prove the conjecture for certain classes of non-isotrivial varieties over Fq(t), including some cubic threefolds and fivefolds. We deduce the Birch and Swinnerton-Dyer conjecture for their intermediate Jacobians, and use it to establish new cases of the Tate conjecture over finite fields.

Exotic R^4's are unclassifiable
2026-03-27 15:00 - 17:15 | Virtual
Freedman Seminar
Speaker: Robert Gompf, UT Austin
Title: Exotic R^4’s are unclassifiable
Abstract: We will use descriptive set theory to show that there is a precise sense in which exotic R^4’s are unclassifiable. For other open manifolds, we can reach a much higher level of unclassifiability. This is work in progress with Aristotelis Panagiotopoulos.

K-theoretic stable envelopes, quantum loop groups and wall-crossings
2026-03-30 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Tianqin Zhu, Columbia University
Title: K-theoretic stable envelopes, quantum loop groups and wall-crossings
Abstract: The stable envelope is an important tool in both geometric representation theory and the enumerative geometry. One of the most important application is that it generates the geometric quantum loop group via the FRT formalism. In this talk, we will show that the geometric quantum loop group is isomorphic to the Drinfeld double given by the preprojective K-theoretic Hall algebra and the nilpotent K-theoretic Hall algebra. Moreover we will show a more refined result that the wall-crossing for the K-theoretic stable envelope is controlled by the universal R-matrix for the slope subalgebra of the Drinfeld double, which leads to the isomorphism between the wall subalgebra in geometric quantum loop groups and the slope subalgebras in the Drinfeld double. If time permits, I will talk about the recent progress of such isomorphism in the case of the critcial stable envelopes in both critical K-theory and critical cohomology. This is based on the work 2511.02161

1-shifted Yangians from 1-shifted Lie bialgebras
2026-03-31 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar
Speaker: Vasily Krylov, Harvard
Title: 1-shifted Yangians from 1-shifted Lie bialgebras
Abstract: In the previous talk, Keyou explained that the fusion of line defects in 3d holomorphic–topological field theories is expected to be governed by so-called DG-shifted Yangians. Following arXiv:2503.08770, I will present examples of such DG-shifted Yangians arising from 1-shifted Lie bialgebras. We will discuss both the classical and DG-shifted settings from a purely mathematical perspective.

CMSA Q&A Seminar: Lauren Williams, Harvard
2026-04-01 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Lauren Williams, Harvard
Topic: The First Proof Project

Failures of Holographic Emergence
2026-04-03 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Elliott Gesteau, CMSA
Title: Failures of Holographic Emergence
Abstract: Recent developments have taught us that some semiclassical spacetimes, in particular those containing closed universe components, cannot emerge from a holographic correspondence. In this talk, I will explain how one can get to this conclusion by using either quantum information theory or properties of the large N limit of AdS/CFT, and propose a criterion for detecting failures of spacetime emergence.

Twistorial constructions of higher genus integrability
2026-04-06 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Seraphim Jarov, Perimeter Institute
Title: Twistorial constructions of higher genus integrability
Abstract: I will present a new method to engineer integrable models in 4d with higher genus spectral parameters. The method has a twistorial origin – by working on a branched covering of twistor space, I show how one can derive deformations of holomorphic BF theory on twistor space which descend to elliptic and hyperelliptic models on R^4 via the Penrose transform. I show how one can bootstrap the Penrose transformed actions using symmetry and integrability to find deformations of self-dual Yang-Mills theory. I will also discuss some novel deformations of a BF type description of Hitchin’s equations. This is based on my paper: 2509.12486

Facets of link homology
2026-04-06 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Mikhail Khovanov, Johns Hopkins University
Title: Facets of link homology
Abstract: We will review some link homology theories of algebraic origin and their connections to representation theory and geometry.

A Mumford form on infinite Grassmannians
2026-04-07 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar
Speaker: Oswaldo Vázquez (Harvard)
Title: A Mumford form on infinite Grassmannians
Abstract: The Polyakov measure in bosonic string theory can be expressed in terms of the Mumford form, which is a trivializing section of a product of determinant line bundles over the moduli space of genus g curves. We will discuss the work in https://arxiv.org/abs/2412.18570, which reviews how the moduli space can be embedded in an infinite dimensional Grassmannian as a Virasoro orbit and generalizes the Mumford form to other such orbits. In particular, this “universal Mumford form” can be described in terms of coordinates in the Grassmannian, which are amenable to computation and thus could have implications to the evaluation of amplitudes.

Multiplicities of graded families of ideals on Noetherian local rings
2026-04-09 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Dale Cutkosky, University of Missouri
Title: Multiplicities of graded families of ideals on Noetherian local rings
Abstract: Let $R$ be an arbitrary $d$-dimensional Noetherian local ring with maximal ideal $m_R$. In this talk, we give a generalization of the multiplicity $e(I)$ of an $m_R$-primary ideal $I$ of $R$ to a multiplicity $e(mathcal I)$ of a graded family of $m_R$-primary ideals $mathcal I$ in $R$. This multiplicity gives the classical multiplicity $e(I)$ if $mathcal I={I^n}$ is the $I$-adic filtration, and agrees with the volume, $lim_{nrightarrow infty}d!frac{ell(R/I_n) }{n^d}$ for $R$ such that $dim N(hat R)>d$, the required condition for the volume of graded families of $m_R$-primary ideals to exist as a limit. We will show that many of the classical theorems for the multiplicity of an ideal generalize to this multiplicity, including mixed multiplicities, the Rees theorem and the Minkowski inequality and equality. We give proofs which are independent of the theory of volumes and Okounkov bodies for all of our results, with the one exception being the proof of the Minkowski equality. We do this by interpreting the multiplicity of graded families of $m_R$-primary ideals as an intersection product on the family of $R$-schemes which are obtained by blowing up $m_R$-primary ideals in $R$.

Quantum topology from dynamics
2026-04-10 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Sunghyuk Park, CMSA
Title: Quantum topology from dynamics
Abstract: Dynamics studies the long-term behavior of systems that evolve over time, such as the famous Lorenz system.
Quantum topology, by contrast, studies knots and low-dimensional manifolds through invariants that are usually constructed using representation-theoretic tools. In this talk, I will explain how quantum invariants of knots and 3-manifolds can be recovered from the dynamics of certain three-dimensional flows. Time permitting, I will also explain how this new bridge arises from ideas in topological string theory.

Abelian gauge fields and duality
2026-04-14 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar
Speaker: Dan Freed, Harvard Math & CMSA
Title: Abelian gauge fields and duality
Abstract: Motivated by the new paper arXiv:2603.19161, I will give a general talk about abelian gauge fields, including duality. I will start with classical Maxwell theory, then discuss various “finite” examples, the typical p-form abelian gauge field, and some exotic examples such as Ramond–Ramond fields and the B-field in superstring theory. I will then get to the new paper and, time permitting, will suggest a very general framework that covers all examples.

Interpolation for points in $mathbb{P}^N, Ngeq 2$
2026-04-16 16:00 - 17:00 | CMSA Room G02, 20 Garden Street, Cambridge, MA, 02138
Algebra Seminar
Speaker: Dipendranath Mahato, Tulane University
Title: Interpolation for points in $mathbb{P}^N, Ngeq 2$
Abstract: Interpolation problems study hypersurfaces in projective space passing through prescribed sets of points. Classically, one asks how many independent conditions a collection of points imposes on hypersurfaces of a fixed degree, a question that can be studied algebraically via homogeneous ideals and their Hilbert functions. In this talk, I will begin with the classical interpolation problem for reduced points and introduce the algebraic framework used to study it. I will then move to fat point schemes, where points are assigned multiplicities and hypersurfaces are required to vanish to higher order. In this setting, interpolation problems naturally lead to symbolic powers of ideals and containment relations between symbolic and ordinary powers. I will conclude by discussing open questions, including potential connections between interpolation problems and combinatorial structures such as matroids.

Higgs and Coulomb branches: Geometry and Representation Theory
2026-04-17 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Vasily Krylov
Title: Higgs and Coulomb branches: Geometry and Representation Theory
Abstract: Higgs and Coulomb branches of quiver gauge theories form two important families of Poisson varieties that are expected to be exchanged under so-called 3D mirror symmetry. Quantized Coulomb branches are associative algebras deforming the algebras of functions on Coulomb branches. They are closely related to many important representation-theoretic structures, such as Yangians, quantum groups, and Hecke algebras. In this talk, I will discuss how 3D mirror symmetry, together with other insights motivated by physics, yields very explicit answers to purely representation-theoretic questions about representations of some of these quantum groups. Talk is based on joint works with Dinkins, Karpov, Klyuev, and Lance.

Zigzags, adjoints, and bordisms
2026-04-20 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Lorenzo Riva (Harvard CMSA)
Title: Zigzags, adjoints, and bordisms
Abstract: We will learn how to freely add adjoints to a category using a combinatorial procedure which involves drawing zigzags of squares. Peculiarly, we can use these to draw some bordisms. This is joint work with Martina Rovelli.

Introduction to A-infty categories
2026-04-21 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar
Speaker: Ahsan Khan, Harvard CMSA
Title: Introduction to A-infty categories

Abstract: I will give a pedagogical account of A-infty categories and the various places they show up in mathematical physics.

Boundedness for K-trivial varieties with fibrations
2026-04-23 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: François Greer, MSU
Title: Boundedness for K-trivial varieties with fibrations
Abstract: According to the Beauville-Bogomolov decomposition theorem, any smooth K-trivial variety admits a finite cover by a product of (1) abelian varieties, (2) strict Calabi-Yau varieties, and (3) irreducible holomorphic symplectic varieties (IHSV). In a fixed dimension, all abelian varieties are diffeomorphic, and indeed deformation equivalent through non-algebraic complex tori. The corresponding question remains largely open for cases (2) and (3). If we assume that a variety of class (2) or (3) admits a non-trivial fibration structure, then much more can be shown. In particular, fibered Calabi-Yau threefolds have bounded moduli problem, and IHSV of a fixed dimension with a Lagrangian fibration have bounded moduli. This is based on joint work with Engel, Filipazzi, Mauri, and Svaldi.

The Intermingling of Symmetry and Parametrization in Matrix Product States
2026-04-24 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Daniel Spiegel
Title: The Intermingling of Symmetry and Parametrization in Matrix Product States
Abstract: In the study of quantum spin systems, it is by now well-known that interesting phases of quantum matter can arise from gapped ground states when the system is invariant under a symmetry group G or when the system varies continuously with a parameter in a topological space X. In these cases, phases are characterized by indices taking values in group cohomology of G or the cohomology of X, respectively. The situation where one has both a symmetry and a parametrization is much less studied but can lead to interesting phases even when both the aforementioned indices are trivial. In this talk, I will discuss work in progress on a simple construction for general index for symmetry protected parametrized systems of matrix product states and will show some illustrative examples.

Compression Is All You Need: Modeling Mathematics
2026-04-24 14:00 - 16:30 | Virtual
Freedman Seminar
Speaker: Mike Freedman, Harvard CMSA
Title: Compression Is All You Need: Modeling Mathematics
Abstract: The talk will exposit a recent eponymous arXiv posting with coauthors Vitaly Aksenov, Eve Bodnia, and Mike Mulligan. The approach is to think like a physicist and model a seemingly complex bit of reality: mathematics, by a simple toy model where exact computations can be carried out and then compared with observation. The models are finitely generated monoids and the data is derived from MathLib a large Lean-based repository. The hierarchical nature of definitions and lemmas in math is modeled by adding redundant generators to the monoids – think of the powers of 10 within the natural numbers which support place notation. Place notation confers an exponential compression of how we describe numbers; exploration of MathLib shows that this theme persists to (human) mathematics writ large. We hope that the observables we describe will help our agents navigate to interesting mathematical destinations.
