CMSA Q&A Seminar: Laura DeMarco
2024-10-09 12:00 - 13:00 (America/New_York) | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Laura DeMarco, Harvard University
Topic: What is Teichmuller geometry and why is it important?

Dolbeault Virasoro algebra and M5 branes
2024-10-11 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Brian Williams, Boston University
Title: Dolbeault Virasoro algebra and M5 branes
Abstract: The worldvolume theory on a stack of M5 branes in M-theory is superconformal. We propose a conjecture that in the holomorphic twist of the theory on a stack of M5 branes an infinite-dimensional enhancement of the (twisted) superconformal algebra is a symmetry. This algebra is closely related to the exceptional infinite-dimensional Lie superalgebra called E(3|6). We show that under the usual AGT correspondence this enhanced algebra degenerates to the Virasoro algebra at a particular central charge.

Scattering Amplitude from a Twistor Point of View
2024-10-11 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Keyou Zeng
Title: Scattering Amplitude from a Twistor Point of View
Abstract: Scattering amplitude is a key quantity in quantum field theory. Although challenging to compute at higher loops and for large particle numbers, physicists have developed various tools to gain a deeper understanding of amplitudes. In this seminar, I will introduce a novel approach, initiated by K. Costello and N. Paquette, which makes use of twistor correspondence. This approach enables the computation of certain amplitudes using chiral algebra. I will also present an upcoming work that constructs a top-down holography model for computing tree and loop amplitudes of certain non-SUSY theories in various self-dual backgrounds.

Topological Modular Forms, its equivariant refinements and relation with supersymmetric quantum field theories
2024-10-15 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Mayuko Yamashita, Kyoto University
Title: Topological Modular Forms, its equivariant refinements and relation with supersymmetric quantum field theories
Abstract: This talk is about the Segal-Stolz-Teichner program, which is one of the most deep and interesting topics relating homotopy theory and physics. Mathematically, they propose a geometric model of TMF, the spectrum (in homotopy theory) of Topological Modular Forms, in terms of supersymmetric quantum field theories. Their proposal, although far from solid formulation or a proof, has been a guiding principle leading us to many new interesting ideas and discoveries in both mathematics and physics. In this talk, I will give an overview of this topic, as well as my current works using equivariant twisted TMF.

CMSA Q&A Seminar: Nazim Bouatta
2024-10-16 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Nazim Bouatta (HMS)
Topic: What are AlphaFold2 and OpenFold

From Word Prediction to Complex Skills: Data Flywheels for Mathematical Reasoning
2024-10-16 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Anirudh Goyal (University of Montreal)
Title: From Word Prediction to Complex Skills: Data Flywheels for Mathematical Reasoning
Abstract: This talk examines how large language models (LLMs) evolve from simple word prediction to complex skills, with a focus on mathematical problem solving. A major driver of AI products today is the fact that new skills emerge in language models when their parameter set and training corpora are scaled up. This phenomenon is poorly understood, and a mechanistic explanation via mathematical analysis of gradient-based training seems difficult. The first part of the talk focuses on analysing emergence using the famous (and empirical) Scaling Laws of LLMs. Then I talk about howc LLMs can verbalize these skills by assigning labels to problems and clustering them into interpretable categories. This metacognitive ability allows us to leverage skill-based prompting, significantly improving performance on mathematical reasoning. I then present a framework that combines LLMs with human oversight to generate challenging, out-of-distribution math questions. This process led to the creation of the MATH^2 dataset, which enhances both model and human performance, driving further advances in mathematical reasoning capabilities.

Bosonic and fermionic 1-form symmetries and anomaly matching
2024-10-18 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
*via Zoom only*
Speaker: Rajath Radhakrishnan (ICTP, Trieste)
Title: Bosonic and fermionic 1-form symmetries and anomaly matching
Abstract: In this talk, I will consider bosonic and fermionic (non-invertible) 1-form symmetries in 2+1d QFTs. These are 1-form symmetries implemented by topological line operators with real spins. I will present a classification of topological quantum field theories in which all line operators have real topological spins, and use this framework to classify the anomalies associated with these 1-form symmetries. Additionally, I will discuss the anomaly matching condition for these symmetries under an RG flow. I will illustrate this condition in concrete examples.

Positive mass and rigidity theorems in Riemannian geometry
2024-10-18 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Puskar Mondal
Title: Positive mass and rigidity theorems in Riemannian geometry
Abstract: Positive mass theorem proved by Schoen-Yau, Witten, Taubes-Parker is one of the most important results in scalar curvature geometry in asymptotically flat settings. Since then several versions have been proven and generalized to other geometries such as asymptotically hyperbolic manifolds. The analogous theorem for strictly positive curvature geometries is absent. There have been counterexamples but a precise quantification does not exist.I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that $n+1~(ngeq 2)$ dimensional spherical caps with constant positive mean curvature totally umbilic boundaries are rigid under smooth perturbations, and such rigidity results fail for the hemisphere. The assertion of this result is based on the notion of a real Killing connection and solution of the boundary value problem associated with its Dirac operator. Additionally, an improved eigenvalue estimate for the Dirac operator on hypersurfaces in positively curved manifolds is obtained.

Foundation Seminar: Singularity Theorems, Part I
2024-10-21 09:30 - 10:30 | Black Hole Initiative, 20 Garden Street, Cambridge MA, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Foundation Seminar (Joint Seminar with BHI)
Location: BHI
Title: Singularity Theorems, Part I
Journal Club Discussion

Higher Vapnik–Chervonenkis theory
2024-10-21 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Artem Chernikov, University of Maryland
Title: Higher Vapnik–Chervonenkis theory
Abstract: Finite VC-dimension, a combinatorial property of families of sets, was discovered simultaneously by Vapnik and Chervonenkis in probabilistic learning theory, and by Shelah in model theory (where it is called NIP). It plays an important role in several areas including machine learning, combinatorics, mathematical logic, functional analysis and topological dynamics. We develop aspects of higher-order VC-theory, in particular establishing a generalization of the epsilon-net theorem for families of sets (and functions) on n-fold product spaces with bounded VC_n-dimension (i.e. there is a bound on the sizes of n-dimensional boxes that can be shattered). We obtain some applications in combinatorics and in model theory, including a strong version of Szemerdi’s regularity lemma for hypergraphs omitting a fixed finite n-partite n-hypergraph. Joint work with Henry Towsner.

Fusion 2-Categories and their Classification
2024-10-22 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Thibault Décoppet, Harvard University
Title: Fusion 2-Categories and their Classification
Abstract: Categorifying the classical notion of fusion (1-)category, fusion 2-categories were recently introduced. These objects have found many applications in Physics, most notably to the classification of topological orders, but also to the description of non-invertible symmetries in 2+1 dimensions. The first part of this talk will be devoted to reviewing the definition of a fusion 2-category and giving many examples. In the second half, I will present a remarkable result concerning the Morita theory of fusion 2-categories and explain how it can be used to give a homotopy coherent classification of fusion 2-categories.

CMSA Q&A Seminar: Dan Freed
2024-10-23 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Dan Freed, Harvard Mathematics & CMSA
Topic: What are topological phases of matter?

How Far Can Transformers Reason? The Globality Barrier and Inductive Scratchpad
2024-10-23 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Aryo Lotfi (EPFL)
Title: How Far Can Transformers Reason? The Globality Barrier and Inductive Scratchpad
Abstract: Can Transformers predict new syllogisms by composing established ones? More generally, what type of targets can be learned by such models from scratch? Recent works show that Transformers can be Turing-complete in terms of expressivity, but this does not address the learnability objective. This paper puts forward the notion of ‘globality degree’ of a target distribution to capture when weak learning is efficiently achievable by regular Transformers, where the latter measures the least number of tokens required in addition to the tokens histogram to correlate nontrivially with the target. As shown experimentally and theoretically under additional assumptions, distributions with high globality cannot be learned efficiently. In particular, syllogisms cannot be composed on long chains. Furthermore, we show that (i) an agnostic scratchpad cannot help to break the globality barrier, (ii) an educated scratchpad can help if it breaks the globality at each step, however not all such scratchpads can generalize to out-of-distribution (OOD) samples, (iii) a notion of ‘inductive scratchpad’, that composes the prior information more efficiently, can both break the globality barrier and improve the OOD generalization. In particular, some inductive scratchpads can achieve length generalizations of up to 6x for some arithmetic tasks depending on the input formatting.

The spin-statistics theorem for TFTs
2024-10-25 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Luuk Stehouwer, Dalhousie University
Title: The spin-statistics theorem for TFTs
Abstract: In quantum field theory (QFT) the spin-statistics theorem says that in a unitary QFT, a particle has half-integer spin if and only if it is a fermion. I show how to phrase this statement in the language of functorial field theories. More precisely, I explain when a functorial field theory “has fermions” and “has spinors” and when they are “related”. I will then restrict to topological field theories (TFTs) and define unitary TFTs. There are counterexamples of the spin-statistics theorem for non-unitary TFTs. I will prove that every unitary TFT satisfies the spin-statistics theorem.

Formality Theorem and Webs
2024-10-25 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Ahsan Khan
Title: Formality Theorem and Webs
Abstract: The “formality theorem” of Kontsevich was a key result that implies that every Poisson manifold admits a deformation quantization. I will review the ideas behind the formality theorem and discuss a potentially novel viewpoint on it involving webs and twisted masses.

Freedman CMSA Seminar
2024-10-25 14:30 - 17:30 | Virtual
Freedman CMSA Seminar
*Note: via Zoom only*
2:00-3:30 pm ET
Speaker: Matt Hastings, Microsoft Quantum Program
Title: Invertible Phases of Matter and Quantum Cellular Automata: Dimensions One to Three
Abstract: A Quantum Cellular Automaton (QCA) is a *-automorphism of the algebra of local operators. While local quantum circuits provide one example of QCA, we are most interested in nontrivial QCA which are those which cannot be written as conjugation by a local quantum circuit. For systems in one and two spatial dimensions, all nontrivial QCA are shifts (i.e., translations by some amount), up to conjugation by a quantum circuit, but in three and higher dimensions, other examples are known. I’ll explain the relation between QCA and a certain “boundary algebra” of operators in one lower spatial dimension, and also the relation to invertible phases of matter on the boundary, and use this to explain and motivate some of these results in dimensions one through three.

3:30-4:00 pm ET
Break/Discussion

4:00-5:30 pm ET
Speaker: Lukasz Fidkowski, U Washington, Physics
Title: Invertible Phases of Matter and Quantum Cellular Automata: Higher dimensions
Abstract: We discuss the explicit construction of a non-trivial QCA in 3 dimensions, one which takes the form of multiplication by a discrete Chern-Simons functional in an appropriate basis for the Hilbert space. We relate the non-trivialness of the QCA to the fact that the Chern-Simons action is not the integral of a gauge invariant local quantity. One property of this QCA is that it creates a specific non-trivial time reversal symmetry protected topological (SPT) phase when acting on a non-trivial tensor product state. Motivated by this, we construct a general class of QCA in arbitrary dimensions based on time reversal protected SPTs, and conjecture a general correspondence between unoriented cobordism (which classifies such SPTs) and QCA.

Foundation Seminar: Singularity Theorems, Part II
2024-10-28 09:30 - 10:30 | 20 Garden Street, Cambridge, MA 02138, MA, MA, 02138, United States
Foundation Seminar (Joint Seminar with BHI)
Location: BHI
Title: Singularity Theorems, Part II
Journal Club Discussion

Boundaries and duality for 3d gauge theories
2024-10-29 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Ben Gammage, Harvard University
Title: Boundaries and duality for 3d gauge theories
Abstract: 3d N=4 supersymmetric gauge theory has a pair of topological twists, the A-model and B-model, the latter of which is also known as Rozansky-Witten theory. Conjecturally, boundary conditions for these TFTs ought to admit descriptions in terms of (microlocal) perverse or coherent sheaves of categories, respectively. Unfortunately, neither of these admits a general mathematical definition; nevertheless, in some cases these are well-defined 2-categories. We will survey these situations and the duality, known as 3d mirror symmetry, which relates the A- and B-models of different theories, together with its relation to the relative Langlands duality of Ben-Zvi–Sakellaridis-Venkatesh.

CMSA Q&A Seminar: Michael Freedman
2024-10-30 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Michael Freedman, Harvard CMSA
Topic: Broad perspective on manifolds: all dimensions, all structures, classification and dynamics

Foundation Seminar
2024-11-04 09:30 - 10:30 | Black Hole Initiative, 20 Garden Street, Cambridge MA, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Foundation Seminar (Joint Seminar with BHI)
Location: BHI

Speaker: Christoph Kehle (MIT)

Title: On the cosmic censorship conjectures
Abstract: I will present the modern formulations of the weak and strong cosmic censorship conjectures and discuss some recent developments in the context of gravitational collapse.

The mathematics of evolution
2024-11-04 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Martin Nowak (Harvard)
Title: The mathematics of evolution
Abstract: All living systems are guided by evolutionary dynamics. Evolution is a search process which occurs in populations of reproducing individuals. The three fundamental forces of evolution are mutation, selection and cooperation. I will present basic ideas in the mathematical description of evolutionary dynamics, including quasi-species theory, evolutionary game theory, and evolutionary graph theory. I will discuss specific problems such as origin of life, emergence of complexity, mechanisms of cooperation, evolution of cancer and how to overcome resistance to targeted therapy.

Introduction to Factorization algebras
2024-11-05 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Dan Freed, Harvard University
Title: Introduction to Factorization algebras

CMSA Q&A Seminar: Michael Douglas
2024-11-06 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Michael Douglas, Harvard CMSA
Topic: What is Argyres-Douglas theory?

Is Behavior Cloning All You Need? Understanding Horizon in Imitation Learning
2024-11-06 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Dylan Foster, Microsoft Research
Title: Is Behavior Cloning All You Need? Understanding Horizon in Imitation Learning
Abstract: Imitation learning (IL) aims to mimic the behavior of an expert in a sequential decision making task by learning from demonstrations, and has been widely applied to robotics, autonomous driving, and autoregressive language generation. The simplest approach to IL, behavior cloning (BC), is thought to incur sample complexity with unfavorable quadratic dependence on the problem horizon, motivating a variety of different online algorithms that attain improved linear horizon dependence under stronger assumptions on the data and the learner’s access to the expert.In this talk, we revisit the apparent gap between offline and online IL from a learning-theoretic perspective, with a focus on general policy classes up to and including deep neural networks. Through a new analysis of behavior cloning with the logarithmic loss, we will show that it is possible to achieve horizon-independent sample complexity in offline IL whenever (i) the range of the cumulative payoffs is controlled, and (ii) an appropriate notion of supervised learning complexity for the policy class is controlled. When specialized to stationary policies, this implies that the gap between offline and online IL is smaller than previously thought. We will then discuss implications of this result and investigate the extent to which it bears out empirically.
Bio: Dylan Foster is a principal researcher at Microsoft Research, New York. Previously, he was a postdoctoral fellow at MIT, and received his PhD in computer science from Cornell University, advised by Karthik Sridharan. His research focuses on problems at the intersection of machine learning, AI, interactive decision making. He has received several awards for his work, including the best paper award at COLT (2019) and best student paper award at COLT (2018, 2019).

Representations of minimal W-algebras: unitarity and modular invariance
2024-11-08 10:00 - 11:30 | Virtual
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Victor Kac (MIT)
Title: Representations of minimal W-algebras: unitarity and modular invariance
Abstract: The minimal W-algebras, obtained by quantum Hamiltonian reduction from affina vertex algebras, form the most interesting class of vertex algebras, which includes all superconformal algebras: Virasoro, Neveu-Scharz, N=2, 3, 4, and big N=4. I will explain a unified classification of their unitary representations, and their character formulas. For N=0, 1, and 2 these vertex algebras are modular invariant (meaning that tr q^L_0-c/24 is a modular function). However for all other minimal W-algebra modular invariance fails, and one needs the “modification” of characters to restore modular invariance. Unfortunately the representation-theoretical or physical meaning of the modification is not known (at least to me).

ADHM spaces and their quantizations
2024-11-08 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Vasily Krylov, CMSA
Title: ADHM spaces and their quantizations
Abstract: In their paper “Construction of Instantons,” Atiyah, Drinfeld, Hitchin, and Manin introduced an algebraic construction of the moduli space of instantons on R^4, now also known as the “ADHM space.” This is a Poisson complex variety; it has been actively studied by both mathematicians and physicists. In this talk, I will review the ADHM construction, present examples, and discuss various geometric and algebraic properties of ADHM spaces. I will also describe natural quantizations of these Poisson varieties. I will explain a joint result with Etingof, Losev, and Simental, providing explicit formulas for the dimensions and characters of all finite-dimensional representations of these quantizations. Time permitting, I will illustrate some predictions of the 3D mirror symmetry in the example of ADHM spaces, following our joint paper with Shlykov.

Frontier of Formal Theorem Proving with Large Language Models: Insights from the DeepSeek-Prover Series
2024-11-13 10:00 - 23:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Huajian Xin, DeepSeek
Title: Frontier of Formal Theorem Proving with Large Language Models: Insights from the DeepSeek-Prover Series
Abstract: Recent advances in large language models have markedly influenced mathematical reasoning and automated theorem proving within artificial intelligence. Yet, despite their success in natural language tasks, these models face notable obstacles in formal theorem proving environments such as Lean and Isabelle, where exacting derivations must adhere to strict formal specifications. Even state-of-the-art models encounter difficulty generating accurate and complex formal proofs, revealing the unique blend of mathematical rigor required in this domain. In the DeepSeek-Prover series (V1 and V1.5), we have explored specialized methodologies aimed at addressing these challenges. This talk will delve into three foundational areas: the synthesis of training data through autoformalization, reinforcement learning that utilizes feedback from proof assistants, and test-time optimization using Monte Carlo tree search. I will also provide insights into current model capabilities, persistent challenges, and the future potential of large language models in automated theorem proving.

Quantum Criticality in Black Hole Scattering
2024-11-15 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Uri Kol
Title: Quantum Criticality in Black Hole Scattering
Abstract: Perturbation theory around rotating black holes captures a few important effects in the physics of gravitational waves emitted from binary mergers. Despite a long and rich history, developing a qualitative understanding of the system remains a challenging problem. In this talk I will describe an emergent critical phenomena arising in black hole perturbation theory, which is reminiscent of the structure found in quantum many-body systems. A critical point is identified at zero temperature, giving rise to a wide “quantum” critical region at finite temperatures that is dominated by critical fluctuations. In the critical region, the physics is exclusively described by a set of critical exponents, therefore leading to robust predictions.

Freedman CMSA Seminar
2024-11-15 14:30 - 17:30 | Virtual
Freedman CMSA Seminar
*Note: via Zoom only*

2:00-3:30 pm ET
Speaker: Michael Freedman, Harvard CMSA
Title: Some questions and theorems about closed 3 manifolds embedded in S^4
Abstract: Much is unknown about smooth embeddings of 3-manifolds in S^4; the Schoenflies problem (Is there only one smoothly embedded 3-sphere in S^4 up to isotopy?) is the best-known example. There has long been a hope that 3-manifold reasoning applied to level-sets will be helpful. I’ll mention some successes and failures of this method and revisit a classical theorem of Hantzsche in this light. (Hantzsche: If a 3-manifold embeds in S^4 its linking form is hyperbolic.)

3:30-4:00 pm ET
Break/Discussion

4:00-5:30 pm ET
Speaker: Slava Krushkal, University of Virginia
Title: A higher order torsion linking form for 3-manifolds
Abstract: This talk is based on a joint work with Mike Freedman defining a triple linking form for rational homology spheres, assuming that the classical torsion linking pairing of three classes pairwise vanishes. I will discuss its vanishing for 3-manifolds in S^4, and its relation to the Matsumoto triple intersection form on 4-manifolds.

Emergent Non-Invertible Symmetries —The Adjoint QCD Example
2024-11-18 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Shani Nadir Meynet (Uppsala)
Title: Emergent Non-Invertible Symmetries — The Adjoint QCD Example
Abstract: After reviewing some general properties of generalized symmetries and the renormalization group (RG) flow for quantum field theories (QFT), I’ll describe how the recently discovered non-invertible symmetries can be used to study theories at strong coupling. I’ll illustrate these facts using (3+1)-dimensional adjoint QCD with two flavors as an example. This theory can be obtained by mass deforming a pure N=2 super Yang-Mills theory. Relying on supersymmetric results, dynamical abelianization and monopole condensation, we are able to get to the description of an infrared (IR) phase as an abelian theory flowing to a CP1 sigma model. In this scenario, the IR phase has an emergent non-invertible symmetry, which is matched with the non-invertible symmetry of the IR CP1 phase. This result illustrates how an emergent non-invertible symmetry can be used to provide a bridge connecting gauge theories at strong coupling and their IR via dynamical abelianization.
