CMSA

Topics in Deep Learning Theory
2024-10-08 14:30 - 15:45, 2024-10-10 14:30 - 15:45, 2024-10-15 14:30 - 15:45, 2024-10-17 14:30 - 15:45, 2024-10-22 14:30 - 15:45 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topics in Deep Learning Theory
Eli Grigsby

Math and Machine Learning Program Open Discussion/Tea
2024-10-08 16:00 - 17:00, 2024-10-22 16:00 - 17:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Open Discussion/Tea

Skein traces and curve counting
2024-10-08 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Sunghyuk Park, Harvard CMSA
Title: Skein traces and curve counting
Abstract: Skein modules are vector space-valued invariants of 3-manifolds describing the space of line defects modulo skein relations (determined by a choice of a ribbon category). When the 3-manifold is S x I for some surface S, the skein module has a natural algebra structure and is called the skein algebra of S.
In 2010, Bonahon and Wong constructed an algebra embedding (named “quantum trace”) of the sl_2 skein algebra into a quantum cluster variety called the “quantum Teichmuller space” for punctured surfaces, which has applications to the representation theory of skein algebras.
In the first half of this talk, I will give an overview of these concepts and explain how the quantum trace map can be generalized to the 3-dimensional setup.
In the second half, I will discuss how everything above can be generalized to HOMFLYPT skeins and has natural interpretation in terms of counts of holomorphic curves.

Math and Machine Learning Program Discussion
2024-10-09 10:30 - 12:00, 2024-10-11 10:30 - 12:00, 2024-10-14 10:30 - 12:00, 2024-10-16 10:30 - 12:00, 2024-10-18 10:30 - 12:00, 2024-10-21 10:30 - 12:00, 2024-10-23 10:30 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

CMSA Q&A Seminar: Laura DeMarco
2024-10-09 12:00 - 13:00 | 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.

Gravitational collapse to extremal Reissner-Nordström and the third law of black hole thermodynamics
2024-10-15 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Christoph Kehle, MIT
Title: Gravitational collapse to extremal Reissner-Nordström and the third law of black hole thermodynamics
Abstract: In this talk, I will present a proof that extremal Reissner-Nordström black holes can form in finite time in gravitational collapse of charged matter. In particular, this construction provides a definitive disproof of the “third law” of black hole thermodynamics. I will also discuss recent works showing that extremal black holes take on a central role in gravitational collapse, giving rise to a new conjectural picture of “extremal critical collapse.” This is joint work with Ryan Unger (Stanford).

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.

Math Science Lectures in Honor of Raoul Bott: Andrew Neitzke
2024-10-16 16:00 - 17:00, 2024-10-17 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Speaker: Andrew Neitzke, Yale University
Location: Harvard University Science Center Hall D & via Zoom webinar
Dates: October 16 & 17, 2024
Time: 4:00 pm

Wednesday, Oct. 16, 2024
Title: Abelianization in analysis of ODEs
Abstract: I will describe the exact WKB method for asymptotic analysis of families of ODEs in one variable, and its interpretation as a kind of abelianization procedure, which replaces GL(N)-connections over a Riemann surface by GL(1)-connections over an N-fold branched cover. This abelianization procedure connects exact WKB to various subjects in geometry (cluster algebras, moduli of Higgs bundles, enumerative geometry). One application is a conjectural description of Hitchin’s hyperkahler metric on the moduli of Higgs bundles; I will review some recent progress on these conjectures.

Thursday, Oct. 17, 2024
Title: Abelianization in quantum topology
Abstract: I will describe new applications of abelianization to various related subjects: perturbative Chern-Simons invariants, skein algebras, and conformal blocks. The aim is to explain how abelianization gives a unifying perspective on constructions familiar in each of these subjects (e.g. dilogarithmic formulas for Chern-Simons invariants, vertex models for computing quantum invariants of links, and iterated-fusion constructions of conformal blocks for the Virasoro algebra), and also suggests various extensions, which are just beginning to be explored.

Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem.

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?
