CMSA

Linearised Second Law for Higher Curvature Gravity and Non-Minimally Coupled Vector Fields
2024-04-02 11:00 - 12:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Zihan Yan, Cambridge University

Title: Linearised Second Law for Higher Curvature Gravity and Non-Minimally Coupled Vector Fields

Abstract: Expanding the work of arXiv:1504.08040, we show that black holes obey a second law for linear perturbations to bifurcate Killing horizons, in any covariant higher curvature gravity coupled to scalar and vector fields. The vector fields do not need to be gauged, and (like the scalars) can have arbitrary non-minimal couplings to the metric. The increasing entropy has a natural expression in covariant phase space language, which makes it manifestly invariant under JKM ambiguities. An explicit entropy formula is given for f(Riemann) gravity coupled to vectors, where at most one derivative acts on each vector. Besides the previously known curvature terms, there are three extra terms involving differentiating the Lagrangian by the symmetric vector derivative (which therefore vanish for gauge fields).

2024 Ding Shum Lecture: Yann LeCun: Objective-Driven AI: Towards AI systems that can learn, remember, reason, and plan
2024-03-28 16:30 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
LECTURE SLIDES (pdf)
On March 28, 2024, the CMSA will host the fifth annual Ding Shum Lecture, given by Yann LeCun.
Time: 4:30–5:30 pm ET
Speaker: Yann Lecun, New York University & META
Location: Harvard Science Center Hall A & via Zoom Webinar
Title: Objective-Driven AI: Towards AI systems that can learn, remember, reason, and plan

Abstract:
How could machines learn as efficiently as humans and animals?
How could machines learn how the world works and acquire common sense?
How could machines learn to reason and plan?
Current AI architectures, such as Auto-Regressive Large Language Models fall short. I will propose a modular cognitive architecture that may constitute a path towards answering these questions. The centerpiece of the architecture is a predictive world model that allows the system to predict the consequences of its actions and to plan a sequence of actions that optimize a set of objectives. The objectives include guardrails that guarantee the system’s controllability and safety. The world model employs a Hierarchical Joint Embedding Predictive Architecture (H-JEPA) trained with self-supervised learning. The JEPA learns abstract representations of the percepts that are simultaneously maximally informative and maximally predictable. The corresponding working paper is available here: https://openreview.net/forum?id=BZ5a1r-kVsf

This event is made possible by the generous funding of Ding Lei and Harry Shum.

Arithmetic Quantum Field Theory Conference
2024-03-25 09:00 - 2024-03-29 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Arithmetic Quantum Field Theory Conference
Dates: March 25-29, 2024
Location: Room G10, Harvard CMSA, 20 Garden Street, Cambridge MA 02138
Arithmetic Quantum Field Theory Conference Youtube Playlist
Organizers:

David Ben-Zvi (University of Texas Austin)
Solomon Friedberg (Boston College)
Natalie Paquette (University of Washington Seattle)
Brian Williams (Boston University)

Scientific Goals: On one hand, there has been tremendous progress in the past decade in our understanding of the algebraic structures underlying quantum field theory as expressed in terms of the geometry and topology of low-dimensional manifolds, both on the level of states (via the formalism of extended, functorial field theory) and on the level of observables (via the formalism of factorization algebras). On the other hand, the arithmetic topology (or “knots and primes”) dictionary provides a sturdy bridge between the topology of 2- and 3-manifolds and the arithmetic of number fields. Thus, one can now port over quantum field theoretic ideas to number theory, as first proposed by Minhyong Kim with his arithmetic counterpart of Chern-Simons theory. Moreover, automorphic objects appear in string theory where they play a role in the study of graviton scattering. Most recently, the work of Kapustin-Witten has been extended towards an understanding of the Langlands program as an arithmetic avatar of electric-magnetic duality in four-dimensional gauge theory to reveal a hidden quantum mechanical nature of the theory of L-functions.
The conference will bring together a wide range of mathematicians and physicists working on adjacent areas to explore the emerging notion of arithmetic quantum field theory as a tool to bring quantum physics to bear on questions of interest for the theory of automorphic forms, representation theory, harmonic analysis and L-functions. Conversely, we will explore potential geometric and physical consequences of arithmetic ideas. Our program will also build on the significant interactions between number theorists and physicists arising from the frequent appearance of modular and automorphic forms in partition functions, scattering amplitudes, and other quantities of interest in quantum field theory and quantum gravity.
Monday, March 25: Connections for Women in Mathematics and Physics
Speakers

Charlotte Chan (U Michigan)
Kim Klinger-Logan (Kansas State)
Sarah Harrison (Northeastern)
Melanie Matchett Wood (Harvard)
Fei Yan (Brookhaven National Lab)

Tuesday, March 26–Friday, March 29: Arithmetic Quantum Field Theory
Speakers

Anne-Marie Aubert (IMJ-PRG)
Roman Bezrukavnikov (MIT)
Sasha Braverman (Toronto / Perimeter)
Alejandra Castro (Cambridge)
YoungJu Choie (POSTECH)
Pavel Etingof (MIT)
Davide Gaiotto (Perimeter)
Axel Kleinschmidt (Max Planck Institute for Gravitational Physics)
Kobi Kremnitzer (Oxford)
Spencer Leslie (Boston College)
David Nadler (Berkeley)
Bảo Châu Ngô (U Chicago)
George Pappas (Michigan State)
Sam Raskin (Yale)
Peng Shan (Tsinghua)
Zhiwei Yun (MIT)

Conference Schedule
Arithmetic Quantum Field Theory Conference
March 25–29, 2024
Download Program (pdf)

Monday, March 25, 2024 – Women in Math and Physics

8:30 – 9:00 am

Breakfast

9:00 – 10:00 am

Melanie Matchett Wood (Harvard)
Title: Statistics of Number fields, function fields, and 3-manifolds
Abstract: Motivated by conjectures of Cohen, Lenstra, and Martinet on the distribution of class groups of number fields, we describe the analogous questions of understanding distributions of class groups and fundamental groups of curves over finite fields, and the distribution of fundamental groups of 3-manifolds. We describe results on these distributions in the cases of curves over finite fields and 3-manifolds, joint with Liu, Zureick-Brown, and Sawin, and discuss how ideas have passed back and forth between the number field, curves over finite fields, and 3-manifold settings.

10:00 – 10:20 am

Coffee break

10:20 – 11:20 am

Charlotte Chan (U Michigan)
Title: Generic character sheaves on parahoric subgroups
Abstract: Lusztig’s theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give an overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the “generic” part of this desired theory. In the simplest nontrivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne–Lusztig induction. This talk is intended to describe in broad strokes what we know about these generic character sheaves, especially within the context of the Langlands program.

11:30 – 12:30 pm

Kim Klinger–Logan (Kansas State)
Title: Connections between special values of L-functions and scattering amplitudes
Abstract: In this talk we will attempt make a connection between zeros and special values of L-functions and scattering amplitudes. The connection is best seen through solutions to differential equations of the form $(Delta-lambda)f = S$ on $X=SL(2,Z)SL(2,R)/SO(2,R)$ for $Delta=y^2(partial_x^2+partial_y^2)$ and $H^{-infty}(X)cup M$ where $M$ is the space of moderate growth functions. Recently, Bombieri and Garrett (following work of Hass, Hejhal, and Colin de Verdiere) laid out the possibly connection with eigenvalue solutions to equations of this form with zeros of L-functions. On the other hand, physicists such as Green, Russo, Vanhove found that eigenfunction solutions to equations of this form give coefficients of the 4-graviton scattering amplitude. We will elaborate on these connections and discuss some recent work on finding solutions for such equations. This work is in collaboration with Ksenia Fedosova, Stephen D. Miller, Danylo Radchenko and Don Zagier.
Slides (pdf)

12:30 – 2:15 pm

Lunch

2:15 – 3:15 pm

Fei Yan (Brookhaven National Lab)
Title: Topological defects on the lattice
Abstract: Topological defects, endowed with a rich mathematical structure, play important roles in condensed matter physics, high energy theory and quantum information science. Realization of such defects on the lattice not only has interesting theoretical consequences, but also opens the pathway to quantum simulation of physical systems. In this talk, I will discuss lattice realizations of topological defects in simple (1+1)-d systems, taking the transverse field Ising model and the three-state Potts model as examples. Time permitting, I will also briefly comment on topological defects in non-equilibrium systems, such as periodically-driven Floquet systems.

3:15 – 3:30 pm

Coffee break

3:30 – 4:30 pm

Sarah Harrison (Northeastern)
Title: Liouville Theory and Weil-Petersson Geometry
Abstract: Two-dimensional conformal field theory is a powerful tool to understand the geometry of surfaces. Liouville conformal field theory in the classical (large central charge) limit encodes the geometry of the moduli space of Riemann surfaces. I describe an efficient algorithm to compute the Weil–Petersson metric to arbitrary accuracy using Zamolodchikov’s recursion relation for conformal blocks, focusing on examples of a sphere with four punctures and generalizations to other one-complex-dimensional moduli spaces. Comparison with analytic results for volumes and geodesic lengths finds excellent agreement. In the case of M_{0,4}, I discuss numerical results for eigenvalues of the Weil-Petersson Laplacian and connections with random matrix theory. Based on work with K. Coleville, A. Maloney, K. Namjou, and T. Numasawa.
Slides (pdf)

Tuesday, March 26, 2024

9:00 – 9:30 am

Breakfast

9:30 – 10:30 am

Roman Bezrukavnikov (MIT)
Title: From affine Hecke category to invariant distributions
Abstract: By a result of Ben-Zvi, Nadler and Preygel the cocenter of the affine Hecke category can be identified with coherent sheaves on the appropriate stack of commuting pairs in the Langlands dual group. In a joint work (in progress) with Ciubotaru, Kazhdan and Varshavsky we recover the space of unipotent invariant distributions on the p-adic group from that category and develop applications to harmonic analysis, including endoscopic property of unipotent L-packets. Time permitting, I will explain how a part of this result can be recovered from a geometric realization of Lusztig’s asymptotic affine Hecke algebra J (joint with Karpov and Krylov), and present a conjecture generalizing the story to other depth zero representations; another special case of this generalization appears in a joint work with Varshavsky.

10:30 – 11:00am

Coffee break

11:00 – 12:00 pm

Sasha Braverman (Toronto/Perimeter)
Title: Hecke operators for algebraic curves over local non-archimedian fields: a survey of some recent results
Abstract: The main goal of this talk is to discuss Hecke operators and Hecke eigen-functions for the moduli space of G-bundles on a smooth projective algebraic curve X defined over a local non-archimedian field K (possibly with level structures at finitely many points). The plan is to discuss the following subjects: 1) Definition of Hecke operators and the space on which they act 2) Relation to “classical story” – i.e. eigen-functions of Hecke operators for curves over a finite field. 3) Detailed discussion of the examples when X has genus zero and we consider bundles with trivialization at two points. In this case we’ll discuss the relation to classical representation theory of p-adic groups and two representation theory of Cherednik algebras. Based on joint works with P. Etingof, D.Kazhdan, and A. Polishchuk.

12:00 – 12:15 pm

Group photo.

12:15 – 1:30 pm

Lunch

1:30 – 2:30 pm

Peng Shan (Tsinghua)
Title: Modularity for W-algebras, affine Springer fibres and associated variety
Abstract: I will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibres. I will also explain how to match the modular group action on the characters of representations with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. If time permits, I will discuss some extension of these results to non-admissible levels and some conjectures about their associated varieties. This is based on joint work with Dan Xie, Wenbin Yan, and Qixian Zhao.

2:30 – 3:00 pm

Coffee break

3:00 – 4:00 pm

Bảo Châu Ngô (U Chicago)
Title: On the nonabelian Fourier kernel and the Lafforgue transform
Abstract: In the case of SL2, we present an analytic formula for the nonabelian Fourier kernel responsible for the functional equation of automorphic L-functions. We use the Gelfand-Graev formula for Langlands’ stable transfer factor and a linear map between the Bernstein center and the cocenter that we call the Lafforgue transform. This is a joint work with Zhilin Luo.

Wednesday, March 27, 2024

9:00 – 9:30 am

Breakfast

9:30 – 10:30 am

YoungJu Choie (POSTECH)
Title: Schubert Eisenstein series and Poisson summation for Schubert varieties
Abstract: Schubert Eisenstein series by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety has been studied. In the case of GL3 over Q it was proved that these Schubert Eisenstein series have meromorphic continuations in all parameters and conjectured the same is true in general. We revisit the conjecture and relate it to the program of Braverman, Kazhdan, Lafforgue, Ngˆo, and Sakellaridis aimed at establishing generalizations of the Poisson summation formula. This is a joint work with Jayce Getz.
Slides (pdf)

10:30 – 11:00 am

Coffee break

11:00 – 12:00 pm

Axel Kleinschmidt (MPI)
Title: Automorphic representations in string amplitudes
Abstract: I will review how automorphic representations arise in the low-energy expansion of string scattering amplitudes, highlighting the connection found by Green/Miller/Vanhove between wavefront sets and BPS conditions. To study the wavefront sets I will present reduction principles for the calculation of Fourier coefficients. String theory also predicts new types of automorphic objects that are characterised by lacking finiteness under the center of the universal enveloping algebra.
Slides (pdf)

12:00 – 1:30 pm

Lunch

1:30 – 2:30 pm

Pavel Etingof (MIT)
Title: Analytic Langlands correspondence over C and R
Abstract: I will review the analytic component of the geometric Langlands correspondence, developed recently in my joint work with E. Frenkel and D. Kazhdan (based on previous works by other authors), with a special focus on archimedian local fields, especially R. This is based on our work with E. Frenkel and D. Kazhdan and insights shared by D. Gaiotto and E. Witten.
Slides (pdf)

2:30 – 3:00 pm

Coffee break

3:00 – 4:00 pm

Davide Gaiotto (Perimeter)
Title: Unexpected Unitarity
Abstract: Much of the mathematical content of Supersymmetric Quantum Field Theories can be extracted through “twisted theories”: simplified QFTs which are topological (or holomorphic) in a derived sense and often amenable of a rigorous mathematical treatment. The twisting procedure destroys or obfuscates certain properties of the underlying SQFTs, including unitarity. I will discuss a variety of situations where some form of unitarity can be restored, endowing the twisted theories with unexpected structures. This includes the recently developed Analytic Langlands program, an analytic version of Symplectic Duality, an A-model description of quantization (as opposed to deformation quantization) and other constructions of Hodge-theoretic or twistorial flavour.

Thursday, March 28, 2024

8:30 – 9:00 am

Breakfast

9:00 – 10:00 am

Spencer Leslie (Boston College)
Title: Relative Langlands and endoscopy
Abstract: Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural question of distinguishing different forms of a given spherical variety in arithmetic settings, giving a solution for symmetric varieties. It turns out that the answer is intimately connected with the construction of the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I will explain the source of these questions in the theory of endoscopy for symmetric varieties, with application to the (pre)-stabilization of relative trace formulas.

10:00 – 10:30 am

Coffee break

10:30 – 11:30 am

Anne-Marie Aubert (IMJ–PRG)
Title: The Local Langlands correspondence: from extended quotients to affine Hecke algebras
Abstract: We will introduce the notion of extended quotient, illustrate it on examples, and show how it can be used to construct the local Langlands correspondence in the nonarchimedean case. Next, we will connect extended quotients, that are attached to the Bernstein decomposition of the category of smooth representations of p-adic groups, and their Langlands duals, to representations of affine Hecke algebras in order to get a description of the LLC as a correspondence between the categories of modules of two collections of such algebras.
Slides (pdf)

11:45 – 12:45 pm

Kobi Kremnitzer (Oxford)
Title: Functional analysis over the integers, L-functions and global Hodge theory
Abstract: In this talk I will explain how using bornological methods one can develop functional analysis over the integers unifying Archimedean and non-Archimedean analysis. I will give examples of algebras of functions and distributions defined over the integers which base change to the usual algebras over the reals and p-adics. Using these it is possible to write some version of L-functions over the integers. I will then introduce an analytic stack over the integers for which the category of quasi-coherent sheaves gives global Hodge structures. I will relate the integral L-functions to trivialisations of line bundles on this stack. I will also explain how to define a cohomology theory for schemes valued in global Hodge structures (possibly related to q-deRham) and speculate on the relation between the determinant of cohomology and L-functions. This is work in progress joint with Federico Bambozzi and Jack Kelly.

12:45 – 2:00 pm

Lunch

2:00 – 3:00 pm

David Nadler (Berkeley)
Title: Going to the boundary
Abstract: I’ll describe several situations where degenerating a marked smooth curve to a marked nodal curve leads to interesting structures on automorphic moduli spaces. In particular, I’ll discuss its implications for the cocenter of the affine Hecke category, real-symmetric duality in relative Langlands, and some other conjectural situations. The talk will borrow from joint work with various authors including D. Ben-Zvi, T.-H. Chen, P. Li, and Z. Yun.

3:00 – 3:30 pm

Coffee break

Friday, March 29, 2024

9:00 – 9:30 am

Breakfast

9:30 – 10:30 am

George Pappas (Michigan State)
Title: Finite and p-adic Chern-Simons type invariants
Abstract: We will define arithmetic invariants of Galois covers and of ‘etale local systems which are inspired by the classical constructions of Dijkgraaf-Witten and Chern-Simons. We will discuss various conjectures and recent results about these invariants.

10:30 – 11:00 am

Coffee break

11:00 – 12:00 pm

Sam Raskin (Yale)
Title: The geometric Langlands conjecture
Abstract: I will describe the main ideas that go into the proof of the (unramified, global) geometric Langlands conjecture. All of this work is joint with Gaitsgory and some parts are joint with Arinkin, Beraldo, Chen, Faergeman, Lin, and Rozenblyum.

12:00 – 1:30 pm

Lunch

1:30 – 2:30 pm

Alejandra Castro (Cambridge)
Title: The light we can see: Extracting black holes from weak Jacobi forms
Abstract: Modular forms play a pivotal role in the counting of black hole microstates. The underlying modular symmetry of counting formulae was key in the precise match between the Bekenstein-Hawking entropy of supersymmetric black holes and Cardy’s formula for the asymptotic growth of states. The goal of this talk is to revisit the connection between modular forms and black hole entropy, and tie it with other consistency conditions of AdS/CFT. We will focus our attention on weak Jacobi forms.
I will quantify how constraints on polar states affect the asymptotic growth of non-polar states in weak Jacobi forms. The constraints I’ll consider are sparseness conditions on the Fourier coefficients of these forms, which are necessary to interpret them as gravitational path integrals. In short, the constraints will leave an imprint on the subleading corrections to the asymptotic growth of heavy states. With this we will revisit the UV/IR connection that relates black hole microstate counting to modular forms. In particular, I’ll provide a microscopic interpretation of the logarithmic corrections to the entropy of supersymmetric black holes and tie it to consistency conditions in AdS_3/CFT_2.

2:30 – 3:00 pm

Coffee break

3:00 – 4:00 pm

Zhiwei Yun (MIT)
Title: Theta correspondence and relative Langlands
Abstract: A reductive dual pair (such as a symplectic group and an orthogonal group) acting on the tensor product of their standard representations is an example of hyperspherical varieties, and is the geometric avatar for theta correspondence. I will explain two geometric results/constructions motivated by the theta correspondence over finite fields. The first one describes how principal series representations behave under theta correspondence using Springer correspondence (joint with Jiajun Ma, Congling Qiu and Jialiang Zou). The second one is a definition of character sheaves in the setup of theta correspondence (joint with Shamgar Gurevich). I will speculate how the first result fits into relative Langlands duality.

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to cstillman@math.harvard.eduno later than Sunday, February 25, 2024.
Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.

This event will be co-funded by the National Science Foundation.
The conference is part of the Arithmetic Quantum Field Theory Program, Feb 4-March 29, 2024.

AQFT Lecture Series
2024-03-22 13:45 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Jayce Getz (Duke University)
Title: The Poisson summation conjecture and the fiber bundle method
Abstract: The Poisson summation conjecture of Braverman-Kazhdan, L. Lafforgue, Ngo, and Sakellaridis predicts that spherical varieties over a global field admit Schwartz spaces, Fourier transforms, and a generalized Poisson summation formula. In this talk I will state a rough form of the conjecture and explain how to deduce new cases of it from known cases using the fiber bundle method. Time permitting I will sketch an approach to proving the functional equation of triple product L-functions using this method. The approach is joint work with Pam Gu, Chun-Hsien Hsu, and Spencer Leslie.

Modularity and Fibrations in Mirror Symmetry
2024-03-22 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Chuck Doran (Harvard CMSA)
Title: Modularity and Fibrations in Mirror Symmetry
Abstract: We will introduce appearances of modularity in the study both of families of Calabi-Yau threefolds and of their enumerative invariants. An important role is played by the structure of fibrations and the DHT fibration-degeneration mirror correspondence, which clarifies how these notions of modularity are (and are not) related. This is joint work with Boris Pioline and Thorsten Schimannek.

Non-Invertible Peccei-Quinn Symmetry and the Massless Quark Solution to the Strong CP Problem
2024-03-22 09:00 - 10:30 | Virtual
Quantum Matter in Mathematics and Physics Seminar
Speaker: Sungwoo Hong, Korea Advanced Institute of Science and Technology (KAIST)
Title: Non-Invertible Peccei-Quinn Symmetry and the Massless Quark Solution to the Strong CP Problem
Abstract: In this talk, I will discuss a new solution to the strong CP problem, one of the most important and challenging problems, of the Standard Model (SM) based on the generalized global symmetry. To this end, I will first show that there exist non-invertible chiral symmetries acting on quark fields once the SM is extended with gauged quark flavor symmetry. Interestingly enough, such symmetries, which we named them as “non-invertible Peccei-Quinn symmetries”, exist only because the SM has the same number of generations as colors, yet another feature that deserves a fundamental explanation. Then I discuss how these new generalized symmetries set the down Yukawa to be exactly zero and lead to a massless quark solution to the strong CP problem. Completion of the solution requires (i) generation of down quark Yukawas to be consistent with both observed quark mass spectrum and mixings and (ii) generation of O(1) CP violating phase called CKM phase, crucially without destabilizing our solution to the strong CP problem. I will discuss that these rather non-trivial tasks can be achieved by a UV completion in an SU(9) quark color-flavor unification.

Decoding The Origins of Fluidity in Multicellular Systems
2024-03-21 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar
Speaker: Max Bi (Northeastern University)
Title: Decoding The Origins of Fluidity in Multicellular Systems
Abstract: Organisms continually adapt to mechanical forces at the cellular and tissue levels, a process crucial for sustaining vital life functions. In pivotal physiological processes, such as cancer progression and embryonic development, tissues are often poised near solid-like and fluid-like states. My talk will delve into three critical aspects of this phenomenon: (1) utilizing computational models that draw parallels with soft matter physics, we examine shear-induced rigidity and the origins of fluidity in epithelial tissues; (2) exploring the intricate relationship between external mechanical stresses and internal cellular dynamics, unraveling a range of rheological behaviors, such as shear thinning and thickening, which are key for understanding rheological responses in varying physical contexts; and (3) investigating how cellular processes like division and apoptosis influence tissue states, with a specific focus on the emergence of hexatic phases, an intermediate state exhibiting properties of both solids and liquids.

AQFT Lecture Series
2024-03-21 11:00 - 12:15 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Ezra Getzler (Northwestern)
Title: Flat connections on derived stacks and the Gauss-Manin connection in derived algebraic geometry

The KSBA moduli space of log Calabi-Yau surfaces
2024-03-21 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Pierrick Bousseau, University of Georgia
Title: The KSBA moduli space of log Calabi-Yau surfaces
Abstract: The KSBA moduli space, introduced by Kollár–Shepherd-Barron, and Alexeev, is a natural generalization of “the moduli space of stable curves” to higher dimensions. It parametrizes stable pairs (X,B), where X is a projective algebraic variety satisfying certain conditions and B is a divisor such that K_X+B is ample. This moduli space is described concretely only in a handful of situations: for instance, if X is a toric variety and B=D+epsilon C, where D is the toric boundary divisor and C is an ample divisor, it is shown by Alexeev that the KSBA moduli space is a toric variety. Generally, for a log Calabi-Yau variety (X,D) consisting of a projective variety X and an anticanonical divisor D, with B=D+epsilon C where C is an ample divisor, it was conjectured by Hacking–Keel–Yu that the KSBA moduli space is still toric (up to passing to a finite cover). In joint work with Alexeev and Argüz, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log smooth deformation theory and mirror symmetry.

Solving olympiad geometry without human demonstrations
2024-03-20 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Trieu H. Trinh, Google Deepmind and NYU Dept. of Computer Science
Title: Solving olympiad geometry without human demonstrations
Abstract: Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

AQFT Lecture Series
2024-03-20 11:00 - 12:15 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Stephen D. Miller (Rutgers University)
Title: What 4-graviton scattering amplitudes had to say about the unitary dual
Abstract: I’ll give an update on the problem of describing all unitary representations of a Lie group, including joint work with Michael Green and Pierre Vanhove that used intuition from string theory to show the unitarity of the “next to minimal” representation of E8, and more recent work with Joe Hundley and Jeff Adams, Marc van Leeuwen, and David Vogan.

CMSA/Tsinghua Math-Science Literature Lecture: Cameron Gordon
2024-03-20 09:00 - 10:30 | Hybrid
CMSA/Tsinghua Math-Science Literature Lecture
Prof. Cameron Gordon presented a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.

Date: Wednesday, March 20, 2024
Time: 9:00–10:30 am ET
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA and via Zoom Webinar
Title: The Unknotting Number of a Knot
Abstract: One of the oldest and most natural knot invariants is the unknotting number, which is the minimum number of times a knot must be allowed to pass through itself in order to unknot it. Although this invariant was discussed by Tait almost 150 years ago, it is still poorly understood. For instance it is not known if it is algorithmically computable, and indeed there is an 8-crossing knot whose unknotting number is unknown. Nevertheless, the many developments in knot theory since Tait have led to some understanding of unknotting number, for example through its connection with 4-dimensional topology. We will give a historical account of this progress, and discuss some of the questions that are still open.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

CMSA COVID-19 Policies

CMSA Q&A Seminar 3/19/2024
2024-03-19 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar

Speaker: Arthur Jaffe, Harvard University

Question: What is mathematical picture language?

Scattering Rigidity Problem
2024-03-19 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Jin Jia, Hunan University
Title: Scattering Rigidity Problem
Abstract: If the asymptotic behavior of a solution to a nonlinear equation is the same as that of a solution to its linearized equation, it is called a scattering solution. Scattering phenomena are widely observed near steady-state solutions of various mathematical physics equations, such as water wave equations, Einstein equations, MHD equations, and Vlasov-Poisson equations. This report takes these equations as examples to provide the definition of scattering operator and its relation to radiation fields. It explains the concepts of scattering rigidity problem and scattering completeness problem, and introduces the latest discoveries of the speaker in the general solution approach to scattering rigidity problem.

Koszul duality & twisted holography for asymptotically flat spacetimes
2024-03-18 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Natalie Paquette, University of Washington
Title: Koszul duality & twisted holography for asymptotically flat spacetimes
Abstract: Koszul duality has been understood in recent years to characterize order-type defects in twists of supersymmetric field theories. This notion has been generalized, from a physical point of view, by studying couplings between D-branes and closed string theories in the topological string. Computing the D-brane backreaction, and studying the resulting open/closed string duality, is the purview of the twisted holography program. Twisted holography seeks to study supersymmetric sectors of the AdS/CFT correspondence using these methods, and leverage the appropriate generalization of Koszul duality to elucidate the bulk/boundary map. When applying these methods to a topological string configuration on twistor space, one can construct an instance of twisted holography in which a 2d chiral algebra, supported on the “celestial sphere”, is dual to a 4d theory in an asymptotically flat spacetime. This is the first such top-down example of holography in a 4d asymptotically flat spacetime. This talk describes joint work done, variously, with Kevin Costello, Brian Williams, and Atul Sharma.

AQFT Lecture Series
2024-03-18 13:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Xinwen Zhu (Stanford)
Title: The tame categorical local Langlands correspondence.
Abstract: I will discuss a conjectural categorical form of the local Langlands correspondence for p-adic groups and establish the tame part of such correspondence (currently for unramified groups with connected center, and for Q_l-coefficients). I will also explain how to extract a (n enhanced) discrete Langlands parameter for depth zero supercuspidal representations from the categorical equivalence.

AQFT Lecture Series
2024-03-15 14:15 - 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Tasho Kaletha (University of Michigan)
Title: Covers of reductive groups and functoriality
Abstract: When studying problems arising from Langlands’ functoriality principle, one often encounters groups that are extensions of complex reductive groups by Galois groups, but that do not necessarily satisfy all properties to be L-groups of reductive groups. We will show, in the case of a local base field F, that such group can be understood as L-groups of covers of reductive groups. This generalizes to the case of arbitrary local fields work of Adams–Vogan for real groups. These covers, for a fixed connected reductive group G, can be understood either as arising from a certain “universal” cover of the topological group G(F) by a certain “fundamental” group tildepi_1(G). We will present two concrete applications of this, one that gives a characterization of the local Langlands correspondence for supercuspidal L-parameters when p is sufficiently large, and one to the construction of transfer factors in the theory of endoscopy.

AQFT Lecture Series
2024-03-15 11:45 - 12:45 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Baiying Liu (Purdue)
Title: Recent progress on certain problems related to local Arthur packets of classical groups
Abstract: In this talk, I will introduce recent progress on certain problems related to local Arthur packets of classical groups. First, I will introduce a joint work with Freydoon Shahidi towards Jiang’s conjecture on the wave front sets of representations in local Arthur packets of classical groups, which is a natural generalization of Shahidi’s conjecture, confirming the relation between the structure of wave front sets and the local Arthur parameters. Then, I will introduce a joint work with Alexander Hazeltine and Chi-Heng Lo on the intersection problem of local Arthur packets for symplectic and split odd special orthogonal groups, with applications to the Enhanced Shahidi’s conjecture, the closure relation conjecture, and the conjectures of Clozel on unramified representations and on unramified components of automorphic representations. This intersection problem also has been worked out independently at the same time by Hiraku Atobe. In a recent joint work with Alexander Hazeltine, Chi-Heng Lo, and Freydoon Shahidi, we made an upper bound conjecture on wavefront sets of admissible representations of connected reductive groups. In the last part of my talk, I will introduce our recent progress towards this conjecture.

Monopoles, scattering, generalized symmetries
2024-03-15 10:00 - 11:30 | Virtual
Quantum Matter in Mathematics and Physics Seminar
Speaker: Diego Delmastro (Simons Center for Geometry and Physics)
Title: Monopoles, scattering, generalized symmetries

Abstract: Gauge theory is a very mature subject by now. Surprisingly, some of its symmetries have only been properly understood in the last couple of years. Specifically, such theories typically have a very rich set of symmetries, involving modern notions such as higher-form symmetries, higher-group symmetries, and categorical symmetries. A proper understanding of these generalized symmetries is not really crucial if we are interested in scattering processes involvingfundamental particles only, but it does become quite essential if we want to add monopoles into the mix. In this talk I will review where these generalized symmetries come from, and how they impose constraints on scattering amplitudes of elementary particles against heavy monopoles. This will allow us to resolve a decades-old puzzle concerning such processes, where incoming electrons appear to become fractional particles after the scattering event.

Cluster state as a noninvertible SPT phase
2024-03-13 10:00 - 12:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Shu-Heng Shao (SUNY Stony Brook)
Title: Cluster state as a noninvertible SPT phase

Abstract: We will point out a non-invertible symmetry of the ordinary 1+1d Z2xZ2 cluster Hamiltonian. Therefore, the cluster state can be viewed as a non-invertible SPT phase under this fusion category symmetry, which is Rep(D8). We furthermore find two new stabilizer states that are distinct from the cluster state as Rep(D8) SPT phases, but identical as Z2xZ2 SPT phases. Finally, we discuss the degenerate edge modes at the interfaces between these three Rep(D8) SPT phases.

AQFT Lecture Series
2024-03-08 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Dihua Jiang (U Minnesota)
Title: Shalika Periods: Functoriality and Arithmetic
Abstract: Shalika periods of automorphic forms were first used by H. Jacquet and J. Shalika (1990) in their construction of global zeta integrals for exterior square L-functions of GL(2n). They were also used by S. Friedberg and H. Jacquet (1993) in their construction of global zeta integrals for the standard L-functions with connections to the linear periods. A. Ash and D. Ginzburg (1994) used them in their study of p-adic L-functions of GL(2n). In this talk, we will first review the implication of the Shalika periods to the Langlands functoriality and the connections with automorphic descents and theta correspondence. Then we turn to my recent work joint with B. Sun and F. Tian on an application of Shalika periods to algebraicity of critical L-values of cuspidal automorphic of GL(2n) of symplectic type.

Symmetry in quantum field theory
2024-03-08 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Dan Freed (Harvard Mathematics and CMSA)
Title: Symmetry in quantum field theory
Abstract: In joint work with Greg Moore and Constantin Teleman we show how ideas and techniques in topological field theory apply to the study of symmetry in quantum field theory. I will discuss how this came about, beginning with some discussion of symmetry in mathematics more generally, and give some examples.

A model for studying the sign problem on quantum computers: Z(3) gauge theory with three fermion flavors in 1+1 dimensions
2024-03-08 09:00 - 10:00 | Virtual
Quantum Matter in Mathematics and Physics Seminar
Speaker: Semeon Valgushev (Iowa State University)
Title: A model for studying the sign problem on quantum computers: Z(3) gauge theory with three fermion flavors in 1+1 dimensions
Abstract: Many properties of quantum field theories can be understood with the help of classical numerical methods. Yet there are several outstanding problems — most remarkably the behaviour of nuclear matter at finite density and out-of-equilibrium real-time dynamics — that require the use of quantum algorithms. We present a simple yet physically rich toy model of QCD/nuclear matter at finite density, termed “QZD”, where Z stands for Z(3) gauge theory coupled to Nf = 3 fermion flavors, suitable for simulation on near-term quantum devices. We study it using tensor networks and find that in weak coupling the spectrum is that of the expected mesons and baryons, although the corrections in weak coupling are nontrivial. In strong coupling, besides the usual baryon, the singlet meson is a baryon anti-baryon state. For two special values of the coupling constant, the lightest baryon is degenerate with the lightest octet meson, and the lightest singlet meson, respectively.

AQFT Lecture Series
2024-03-07 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: John Francis (Northwestern)
Title: Integrating braided categories over 3-manifolds
Abstract: I’ll describe a form of factorization homology, which gives an invariant of framed 3-manifolds given as input a rigid braided-monoidal category with duals. More generally, the construction gives an invariant of framed n-manifolds from an E_{n-1}-monoidal category with duals. This construction relies on a form of the tangle hypothesis, that a dualizable object in an E_{n-1}-monoidal category uniquely determines a functor from the category of tangles in n-space. This is joint work with David Ayala.

AQFT Lecture Series
2024-03-07 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: An Huang (Brandeis)
Title: Tate’s thesis and p-adic strings
Abstract: I shall explain the relation between a family of deformations of genus zero p-adic string worldsheet action and Tate’s thesis, which in particular, gives rise to an attempt of physically deriving quadratic reciprocity. I shall then propose a genus one p-adic string worldsheet action, where the key is an explicit definition of a p-adic Laplacian operator on a Tate curve, and the computation of its Green’s function. I shall also mention a very recent attempt of developing a Lorentzian version of the p-adic worldsheet action.

Geometric construction of toric NCRs
2024-03-07 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Jesse Huang, University of Alberta
Title: Geometric construction of toric NCRs
Abstract: The Rouquier dimension of a toric variety is recently shown to be achieved by the Frobenius pushforward of O via coherent-constructible correspondence. From the perspective of noncommutative geometry, this result leads to a geometric construction of toric NCR of the invariant ring of the Cox ring with respect to a multi-grading which also gives the information about its global dimension. From the perspective of mirror symmetry, the same construction provides a universal “wall skeleton” capturing VGIT wall-crossings, which contains a window for each chamber as a full subcategory. From the perspective of commutative algebra, the same construction indicates the existence of virtual resolutions of the multigraded diagonal bimodule, which agrees with a recent result of Hanlon-Hicks-Larzarev constructing one such resolution explicitly. In this talk, I will survey these perspectives. The talk is based on joint works with P. Zhou, joint works with D. Favero, and work in progress with D. Favero.

LILO: Learning Interpretable Libraries by Compressing and Documenting Code
2024-03-06 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Gabe Grand, MIT CSAIL and Dept. of EE&CS
Title: LILO: Learning Interpretable Libraries by Compressing and Documenting Code
Abstract: While large language models (LLMs) now excel at code generation, a key aspect of software development is the art of refactoring: consolidating code into libraries of reusable and readable programs. In this paper, we introduce LILO, a neurosymbolic framework that iteratively synthesizes, compresses, and documents code to build libraries tailored to particular problem domains. LILO combines LLM-guided program synthesis with recent algorithmic advances in automated refactoring from Stitch: a symbolic compression system that efficiently identifies optimal lambda abstractions across large code corpora. To make these abstractions interpretable, we introduce an auto-documentation (AutoDoc) procedure that infers natural language names and docstrings based on contextual examples of usage. In addition to improving human readability, we find that AutoDoc boosts performance by helping LILO’s synthesizer to interpret and deploy learned abstractions. We evaluate LILO on three inductive program synthesis benchmarks for string editing, scene reasoning, and graphics composition. Compared to existing neural and symbolic methods – including the state-of-the-art library learning algorithm DreamCoder – LILO solves more complex tasks and learns richer libraries that are grounded in linguistic knowledge.

CMSA Q&A Seminar 3/5/2024
2024-03-05 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar

Speaker: Yannai Gonczarowski, Harvard University

Question: What do people mean when they say ‘the intersection between theoretical computer science and economic theory’?

CMSA Q&A Seminar 3/5/2024
2024-03-05 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar

Speaker: Cumrun Vafa

Question: What is swampland?

High order WENO finite difference scheme for Einstein-Yang-Mills equations
2024-03-05 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Yuewen Chen, Tsinghua University
Title: High order WENO finite difference scheme for Einstein-Yang-Mills equations
Abstract: In this talk, we will show the convergence analysis of the first-order finite difference scheme for static spherically symmetric $SU(2)$ Einstein-Yang-Mills (EYM) equations. We also construct a new WENO scheme for EYM.
