CMSA

Modularity and Fibrations in Mirror Symmetry
2024-03-22 12:00 - 13:00 (America/New_York) | 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.

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.

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.

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.

Linearised Second Law for Higher Curvature Gravity and Non-Minimally Coupled Vector Fields
2024-04-02 11:00 - 12:00 | 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).

Derived categories of genus one curves and torsors over abelian varieties
2024-04-04 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Jonathan Rosenberg, University of Maryland

Title: Derived categories of genus one curves and torsors over abelian varieties

Abstract: Studying orientifold string theories on elliptic curves or abelian
varieties motivates studying the derived category of coherent sheaves on
a genus one curve or a torsor over an abelian variety over the reals
(as opposed to the complex numbers).

In joint work with Nirnajan Ramachandran (to appear in MRL), we show that
a genus one curve over a perfect field determines a class in the relative
Brauer group of the Jacobian elliptic curve, and that there is a natural
Mukai-type derived equivalence between the original genus one curve
and the Jacobian twisted by the Brauer class. The proof extends to
torsors over abelian varieties (of any dimension).

Shape morphing with swelling hydrogels and expanding foams
2024-04-04 13:00 - 14:00 | Hybrid – G10
Active Matter Seminar
Speaker: Abby Plummer, Boston University
Title: Shape morphing with swelling hydrogels and expanding foams
Abstract: Materials that increase in size offer intriguing possibilities for shape-morphing applications. Here, we explore two such systems—swelling polyacrylamide hydrogels and expanding polyurethane foams. The hydrogels swell by absorbing water into crosslinked polymer networks. They can therefore be modeled by coupling solvent migration with the deformations of a hyperelastic solid. In contrast, the foams initially behave as liquids with viscosity and volume increasing in time, responding elastically only when close to solidification. We investigate how these expanding materials are sculpted by complex environments with obstacles and trenches.

Discrete geometry and the modular bootstrap
2024-04-05 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Henry Cohn (MIT and Microsoft)
Title: Discrete geometry and the modular bootstrap
Abstract: In this talk, I’ll discuss the remarkable connections between the modular bootstrap and sphere packing or ground state problems discovered by Hartman, Mazáč, and Rastelli in 2019, with a focus on opportunities for further progress.

Phase diagram and confining strings in a minimal model of nematopolar matter
2024-04-05 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Farzan Vafa
Title: Phase diagram and confining strings in a minimal model of nematopolar matter
Abstract: We investigate a minimal model of a nematopolar system. We analytically uncover a phase diagram consisting of a locked phase where the polar order and nematic order are locked, and unlocked phases which could be ordered or disordered. In particular, we develop two complementary perspectives on the locked phase: (i) the nematic order induces polar order, (ii) in the locked phase, all 1/2 integral nematic topological charges are confined. In particular, a polar +1 defect fattens from a point along a string with constant tension and confines a pair of nematic +1/2 defects at its ends.

Current Developments in Mathematics Conference 2024
2024-04-05 14:00 - 2024-04-06 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
CURRENT DEVELOPMENTS IN MATHEMATICS 2024
APRIL 5-6, 2024
HARVARD UNIVERSITY SCIENCE CENTER
LECTURE HALL C
https://www.math.harvard.edu/event/current-developments-in-mathematics-2024/

Speakers:
Daniel Cristofaro-Gardiner – University of Maryland
Samit Dasgupta – Duke University
Jiaoyang Huang – University of Pennsylvania
Daniel Litt – University of Toronto
Lisa Piccirillo – MIT/University of Texas

Download PDF for a detailed schedule of lectures and events.

Friday, April 5

Saturday, April 6

1:30 p.m. – 2:20 p.m. Part 1
2:20 p.m. – 2:30 p.m. Break
2:30 p.m. – 3:20 p.m. Part 2

Jiaoyang Huang
Random Matrix Statistics and Airy Line Ensembles

9:05 a.m. – 9:55 a.m. Part 1
9:55 a.m. – 10:05 a.m. Break
10:05 a.m. – 10:55 a.m. Part 2

Daniel Litt
Motives, mapping class groups, and monodromy

3:20 p.m. – 3:35 p.m.
Break

10:55 a.m. – 11:10 a.m.
Break

3:35 p.m. – 4:25 p.m. Part 1
4:25 p.m. – 4:35 p.m. Break
4:35 p.m. – 5:25 p.m. Part 2

Lisa Piccirillo
Exotic phenomena in dimension 4

11:10 a.m. – 12 p.m. Part 1
12 p.m. – 1:30 p.m. Lunch
1:30 p.m. – 2:20 p.m. Part 2

Samit Dasgupta
Stark’s conjectures and explicit class field theory

2:20 p.m. – 2:35 p.m.
Break

2:35 p.m. – 3:25 p.m. Part 1
3:25 p.m. – 3:35 p.m. Break
3:35 p.m. – 4:25 p.m. Part 2

Dan Cristofaro-Gardiner
Low-dimensional topology and dynamics

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard).
Sponsored by the National Science Foundation, Harvard University Mathematics, and the Massachusetts Institute of Technology.
Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

Mirror symmetry for fibrations and degenerations of K3 surfaces
2024-04-11 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Alan Thompson (Loughborough University)
Title: Mirror symmetry for fibrations and degenerations of K3 surfaces
Abstract: In 2016, Doran, Harder, and I conjectured a mirror symmetric relationship between Tyurin degenerations and splittings of codimension 1 fibrations on Calabi-Yau manifolds. In this talk I will discuss recent work to make this conjecture rigorous in the case of K3 surfaces. I will give a precise definition of what it means for a Tyurin degeneration of K3’s to be mirror to a splitting of an elliptically fibred K3, and show that this definition enjoys the following compatibilities with existing mirror symmetric theories: 1) The general fibre of the Tyurin degeneration is mirror to the elliptically fibred K3, in the sense of Dolgachev-Nikulin. 2) Components of the Tyurin degeneration and pieces of the splitting satisfy a homological version of the (quasi-) Fano-LG correspondence. 3) Components of the Tyurin degeneration which are weak del Pezzo are mirror to pieces of the splitting that arise as restrictions of the corresponding lattice polarised LG models to discs. This is joint work with Luca Giovenzana.

Global weak solutions of 3+1 dimensional vacuum Einstein equations
2024-04-12 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Puskar Mondal
Title: Global weak solutions of 3+1 dimensional vacuum Einstein equations
Abstract: It is important to understand if the `solutions’ of non-linear evolutionary PDEs persist for all time or become extinct in finite time through the blow-up of invariant entities. Now the question of this global existence or finite time blow up in the PDE settings is well defined if the regularity of the solution is specified. Most physically interesting scenarios demand control of the point-wise behavior of the solution. Unfortunately, most times this level of regularity is notoriously difficult to obtain for non-linear equations. In this talk, I will discuss very low regularity solutions namely distributional (or weak) solutions of vacuum Einsten’s equations in 3+1 dimensions. I prove that on a globally hyperbolic spacetime foliated by closed connected oriented negative Yamabe slices, weak solutions of the Einstein equations exist for all time. The monotonicity of a Coercive Entity called reduced Hamiltonian that controls the minimum regularity required for the weak solution is employed. This is in the same spirit as Leray’s global weak solutions of Navier-Stokes in 3+1 dimensions and the first result in the context of Einstein equations.

Program on Mathematical Aspects of Scattering Amplitudes
2024-04-15 09:00 - 2024-05-24 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Aspects of Scattering Amplitudes Program
Dates: April 15 – May 24, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge, MA 02138
This program will bring together and foster interaction between theoretical physicists and mathematicians working on various topics connected to recent developments in our understanding of scattering amplitudes in quantum field theory. The field has advanced considerably since 2019 when the CMSA hosted the program “Spacetime and Quantum Mechanics, Total Positivity and Motives.” Recent developments have primed this area for further significant advances, which will be facilitated by bringing together many of the relevant experts for intensive discussion and collaboration.
The program will feature a weekly seminar series.
Tuesday, April 16, 2024
4:15 pm
Sabrina Pasterski, Perimeter Institute
Title: Radiation in Holography
[Physics Talk]
Wednesday, April 17, 2024
4:30 pm – Cruft 309
Ana-Maria Raclariu, King’s College London
Thursday, April 18, 2024
4:15 pm
Lionel Mason, University of Oxford
Title: Hidden symmetries of SD Poincare Einstein metrics in split signature
[Physics Talk]
Tuesday, April 23, 2024
4:30 pm – Jefferson 256
Juan Maldacena, Institute for Advanced Study
Thursday, April 25, 2024
4:15 pm
Tomasz Taylor, Northeastern University
Title: Progress in Yang-Mills-Liouville Theory
[CMSA Colloquium]
Monday, April 29, 2024
4:30 – 5:30 pm
Lance Dixon, Stanford
Title: The DNA of Particle Scattering
Tuesday, April 30, 2024
9:00 am- Jefferson 453
Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 1
1:00 pm – Jefferson 453
Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 2
4:00 pm – Jefferson 453
Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 3
Wednesday, May 1, 2024
11:00 am – Science Center 507
Jaroslav Trnka, UC Davis
Title: Loops of loops expansion in the Amplituhedron
3:00 pm
Yu-tin Huang, National Taiwan University
Title: Loop in trees: Chambers in amplitudes and correlation functions
4:00 pm
Livia Ferro, University of Hertfordshire
Title: Scattering Amplitudes from Null-cone Geometry
5:00 pm
Stephan Stieberger, Max Planck Institute
Title: One-loop Double Copy Relation in String Theory and Twisted (Co)homology
Thursday, May 2, 2024
11:00 am – Science Center 310
Daniil Rudenko, Chicago
Title: Introduction to Cluster Polylogarithms
Friday, May 3, 2024
11:00 am
Andrew McLeod, Edinburgh
Title: Genealogical Constraints on Feynman Integrals
Tuesday, May 7, 2024
3:00 pm
Jacob Bourjaily, Penn State
Title: The Algebraic and Transcendental Structure of Perturbative QFT
Wednesday, May 8, 2024
3:00 pm
Ruth Britto, Trinity
Title: Cuts and Symbols
Tuesday, May 14, 2024
3:00 pm
James Drummond, University of Southampton
Title: Multiple light-like Wilson loops in N=4 super Yang-Mills theory
Wednesday, May 15, 2024
3:00 pm
Matteo Parisi, Harvard CMSA
Title: The amplituhedron and cluster algebras
Tuesday, May 21, 2024
11:00 am
Michael Borinsky, ETH Zurich
Title: On the Euler characteristic of the commutative graph complex and the top-weight cohomology of the moduli space of curves
Wednesday, May 22, 2024
11:00 am
Chaim Even-Zohar, Technion
Title: Amplituhedron tiles and twistor polynomials

Organizers:

Nima Arkani-Hamed (Institute for Advanced Study)
Marcus Spradlin (Brown University)
Andrew Strominger (Harvard University)
Anastasia Volovich (Brown University)
Lauren Williams (Harvard University)

Participants:

Michael Borinsky, ETH Zurich
Jacob Bourjaily, Pennsylvania State University
Ruth Britto, Trinity College
Lance Dixon, Stanford Linear Accelerator Center
James Drummond, University of Southampton
Chaim Even-Zohar, Technion
Livia Ferro, University of Hertfordshire
Carolina Figueiredo, Princeton University
Hadleigh Frost, Oxford University
Bruno Gimenez, University of Southampton
Omer Gurdogan, University of Southampton
Xuhua He, Chinese University of Hong Kong
Paul Heslop, Durham University
Yu-Tin Huang, National Taiwan University
Dani Kaufman, University of Copenhagen
Jianrong Li, University of Vienna
Tomasz Lukowski, University of Hertfordshire
Yelena Mandelshtam, University of California, Berkeley
Lionel Mason, University of Oxford
Andrew McLeod, University of Edinburgh
Natalie Paquette, University of Washington
Matteo Parisi, Harvard University
Sabrina Pasterski, Perimeter Institute
Dmitri Pavlov, Max Planck Institute for Mathematics in the Sciences, Leipzig
Lizzie Pratt, University of California, Berkeley
Claudia Rella, University of Geneva
Daniil Rudenko, University of Chicago
Giulio Salvatori, Max Planck Institute for Physics
Melissa Sherman-Bennett, Massachusetts Institute of Technology
Jonah Stalknecht, University of Hertfordshire
Stephan Stieberger, Max Planck Institute
Tomasz Taylor, Northeastern University
Ran Tessler, Weizmann Institute of Science
Hugh Thomas, Université du Québec à Montréal
Jaroslav Trnka, University of California, Davis
Cristian Vergu, Pennsylvania State University

New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity
2024-04-16 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Xiaoyi Liu, UCSB
Title: New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity
Abstract: We consider four-dimensional Euclidean gravity in a finite cavity. We point out that there exists a one-parameter family of boundary conditions, parameterized by a real constant, where a suitably Weyl-rescaled boundary metric is fixed, and all give a well-posed elliptic system, as opposed to the Dirichlet boundary condition. Focussing on static Euclidean solutions, we derive a thermodynamic first law. Restricting to a spherical spatial boundary, the infillings are flat space or the Schwarzschild solution and have similar thermodynamics to the Dirichlet case. We study the stability behavior of several geometries under these boundary conditions in both Euclidean and Lorentzian signatures and find two puzzles.

CMSA Q&A Seminar 4/16/2024
2024-04-16 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar

Speaker: Cengiz Pehlevan, Harvard

Question: What is feature learning?

Mathematical Aspects of Scattering Amplitudes Lecture
2024-04-16 16:15 - 17:15 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Aspects of Scattering Amplitudes Lecture
Speaker: Sabrina Pasterski, Perimeter Institute
Title: Radiation in Holography

Geometric local systems on very general curves
2024-04-18 10:15 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Aaron Landesman, MIT
Title: Geometric local systems on very general curves
Abstract: What is the smallest genus h of a non-isotrivial curve over the generic genus g curve? In joint work with Daniel Litt, we show h is more than $sqrt{g}$ by proving amore general result about variations of Hodge structure on sufficiently general curves. As a consequence, we show that local systems on a sufficiently general curve of geometric origin are not Zariski dense in the character variety parameterizing such local systems. This gives counterexamples to conjectures of Esnault-Kerz and Budur-Wang.

Fusion Rule Measurement in a Topological Qubit
2024-04-19 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Chetan Nayak, Microsoft and UCSB
Title: Fusion Rule Measurement in a Topological Qubit

Member Seminar
2024-04-19 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Sunghyuk Park, Harvard CMSA
Title: 3D quantum trace map
Abstract: I will speak about my recent work (joint with Sam Panitch) constructing the 3d quantum trace map, a homomorphism from the Kauffman bracket skein module of an ideally triangulated 3-manifold to its (square root) quantum gluing module, thereby giving a precise relationship between the two quantizations of the character variety of ideally triangulated 3-manifolds. Our construction is based on the study of stated skein modules and their behavior under splitting, especially into face suspensions.

Quantum Energy Inequalities
2024-04-23 10:00 - 11:00 | Virtual
General Relativity Seminar
Speaker: Chris Fewster, York University
Title: Quantum Energy Inequalities
Abstract: Many theorems of mathematical relativity, including singularity and positive mass theorems, include the classical energy conditions among their hypotheses. However, matter described by quantum field theory can violate the classical energy conditions and indeed there is no lower bound to the energy density at a given point as the quantum state is varied. Do the classical theorems apply to quantised matter?
In this talk I discuss Quantum Energy Inequalities, lower bounds on averages of the energy density along timelike curves or over spacetime regions, that have been proved in QFT and are the remnants of the classical energy conditions after quantisation. I will also discuss some of their consequences.

CMSA Q&A Seminar 4/23/2024
2024-04-23 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar

Speaker: Melanie Weber, Harvard

Question: What is the Ricci curvature of a graph?

The logarithmic double ramification locus
2024-04-25 10:30 - 11:30 | Virtual
Algebraic Geometry in String Theory Seminar
Speaker: Alessandro Chiodo, IMJ-Paris Rive Gauche (Jussieu)
Title: The logarithmic double ramification locus
Abstract: Given a family of smooth curves C -> S with a line bundle L on C, it is natural to study the locus of points x in S where L_x is trivial on C_x. When the family is stable, the definition can be extended, not directly on the base scheme S, but more naturally on a (logarithmic) blow-up S’ of S. The problem is in many ways analogue to the problem of defining a Néron model on the moduli space of stable curves (instead of a DVR). Over the past years, David Holmes and his collaborators pioneered a new approach on a logarithmic modification of the entire moduli space of curves. In this talk, we determine this logarithmic double ramification cycle and several variants and alternative presentations of it (work in collaboration with David Holmes).

Mathematical Aspects of Scattering Amplitudes Lecture
2024-04-25 16:15 - 17:15 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Aspects of Scattering Amplitudes Lecture
Speaker: Tomasz Taylor, Northeastern University
Title: Progress in Yang-Mills-Liouville Theory

What Observables are Safe to Calculate?
2024-04-26 10:30 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Jesse Thaler, MIT
Title: What Observables are Safe to Calculate?
Abstract: In collider physics, perturbative quantum field theory is the workhorse framework for computing theoretical predictions to compare to experimental measurements. An observable is called “safe” if its cross section can be predicted order-by-order in perturbation theory with controlled non-perturbative corrections. In this talk, I show that naive definitions of “safety” are inadequate to determine which observable are perturbatively calculable. I then argue for a more refined definition of safety based on principles from optimal transport theory.

Member Seminar
2024-04-26 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Matteo Parisi, Harvard CMSA

Workshop on Global Categorical Symmetries
2024-04-29 09:00 - 2024-05-03 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
The CMSA will be hosting a Workshop on Global Categorical Symmetries from April 29–May 3, 2024.
Participation in the workshop is by invitation.
The workshop will hold three Symmetry Colloquia open to the community on Thursday, May 2, 2024.
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Time: 2:00 – 2:50 pm
Speaker: Clay Còrdova, University of Chicago
Title: Particle-Soliton Degeneracies from Spontaneously Broken Non-Invertible Symmetry
Abstract: We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Time: 3:00 – 3:50 pm
Speaker: Thomas Dumitrescu, UCLA
Title: Symmetries, Invertible Field Theories, and Gauge Theory Phases
Abstract: I will start with a brief overview of gauge theory phases in 3+1 dimensions through the lens of higher symmetries — in particular the realization of 1-form symmetries acting on loop order parameters. I will then review recent progress in refining this characterization using invertible field theories, or equivalently symmetry protected topological phases (SPTs). This refinement leads to new results in gauge theories with fundamental matter, such as quantum chromodynamics (QCD), which do not possess 1-form symmetries. I will explain why these theories must sometimes undergo a phase transition between their confining and Higgs regimes, despite the fact that classic results and standard lore say they should be continuously connected.

Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Time: 4:30 – 5:20 pm
Speaker: Theo Johnson-Freyd, Dalhousie University and Perimeter Institute
Title: The Universal Target Category
Abstract: Hilbert’s Nullstellensatz says that the complex numbers C satisfy a universal property among all R-algebras: every not-too-large nonzero commutative R-algebra maps to C. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category over R maps to the category sVec of super vector spaces. In other words, sVec (and not Vec!) is “algebraically closed”. These statements help explain why quantum field theory requires imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence C, sVec, …. This is joint work in progress with David Reutter, and builds on closely-related work by GCS collaborators Freed, Scheimbauer, and Teleman and Schlank et al.

Organizers:
Dan Freed (Harvard CMSA & Math)
Constantin Teleman (UC Berkeley)
This event is co-sponsored by the Simons Foundation.

The DNA of Particle Scattering
2024-04-29 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Lance Dixon (SLAC, Stanford University)
Title: The DNA of Particle Scattering
Abstract: At the Large Hadron Collider, the copious scattering of quarks and gluons in quantum chromodynamics (QCD) produces Higgs bosons and many backgrounds to searches for new physics. At short distances, scattering in QCD can be evaluated in perturbation theory and leads to highly intricate, multivariate mathematical functions such as generalized polylogarithms. To gain further insight, one can study a cousin of QCD called planar N=4 super-Yang-Mills theory. Some processes in this theory can be computed to eighth order in perturbation theory, versus second or third order in QCD. The computation and analysis of these results rely on a Hopf algebra coaction on polylogarithms. Its maximal iteration is called the ‘symbol’, which serves as a ‘genetic code’ for amplitudes. The symbol is a linear combination of words, sequences of letters analogous to sequences of DNA base pairs. Understanding the alphabet, and then reading the code, exposes the physics and mathematics of quantum scattering, including bizarre new symmetries. For example, the two scattering amplitudes that are known to the highest orders in perturbation theory (8 loops) are related to each other by an ‘antipodal duality’, which involves reading the code backwards as well as forwards. A third scattering amplitude, which contains the other two as limits, has an antipodal self-duality which ‘explains’ the other duality. However, we still don’t know ‘who ordered’ this property, or what it really means.

Mathematical Aspects of Scattering Amplitudes Lecture
2024-04-30 09:00 - 10:00 | Jefferson 453, 17 Oxford St, Cambridge, MA 02138, MA
Mathematical Aspects of Scattering Amplitudes Lecture
Speaker: Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 1
(Note change of venue to Jefferson 453)

CMSA Q&A Seminar 4/30/2024
2024-04-30 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Lakshminarayanan Mahadevan

Question: What is morphogenesis? (Morphogenesis: geometry and biology)

Mathematical Aspects of Scattering Amplitudes Lecture
2024-04-30 13:00 - 14:00 | Jefferson 453, 17 Oxford St, Cambridge, MA 02138, MA
Mathematical Aspects of Scattering Amplitudes Lecture
Speaker: Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 2
(Note change of venue to Jefferson 453)
