CMSA

The Feynman propagator and self-adjointness
2023-12-05 11:00 - 12:00 (America/New_York) | Virtual
General Relativity Seminar
Speaker: Andras Vasy (Stanford)
Title: The Feynman propagator and self-adjointness
Abstract: In this talk I will discuss the Feynman and anti-Feynman inverses for wave operators on certain Lorentzian manifolds; these are two inverses which from a microlocal analysis perspective are more natural than the standard causal (advanced/retarded) ones. For instance, for the spectral family of the wave operator, these are the natural inverses when the spectral parameter is non-real. Indeed, I will explain that these connect to the self-adjointness of the wave operator, and the positivity properties that follow.

CMSA Q and A Seminar 12/5/2023
2023-12-05 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar
Speakers: Mike Douglas (Harvard CMSA) and Anurag Anshu (Harvard Computer Science)
Topics:
Mike Douglas: What are scaling laws in deep learning?
Anurag Anshu: What’s the difference between classical information theory and quantum information theory (or classical computation vs quantum computation)?

Active structures and flows in living cells
2023-12-07 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Michael Shelley (Flatiron)
Title: Active structures and flows in living cells
Abstract: Flows in the fluidic interior of living cells can serve biological function or act as signatures of how intracellular forces are exerted. I’ll discuss examples of each. One is understanding the emergence of cell-spanning vortical flows in large developing egg cells, while the other arises in studying the nature of force transduction in single cell embryos moving towards their first cell division. Both involve the cytoskeleton, that set of polymers, cross-linkers, and molecular motors that underlie much of the active mechanics within cells, and has led to the development of new coarse-grained active matter models and novel instabilities.

Open Problems in Physics and Novel Solutions from Math
2023-12-08 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Juven Wang
Title: Open Problems in Physics and Novel Solutions from Math

Abstract: (1) How do neutrinos obtain tiny mass and oscillate between e, mu, tau three flavors? What consists of Dark Matter? What causes leptogenesis and baryogenesis?
(2) Why the Parity Symmetry is violated in the weak interaction? How do we quantum mechanically regularize and numerically simulate the Standard Model (chiral fermion/gauge theory) on the lattice in 3+1 spacetime dimensions?
(3) Strong CP problem: Why is the neutron electric dipole moment nearly zero and QCD respects the CP = T symmetry?
(4) Family/Generation problem: Why are there three families/generations of quarks and leptons?
In my talk, I will sketch and propose novel solutions to these challenging open problems in physics based on my own work (available on arXiv, some jointly with my collaborators).
I will show that the eclectic cross-disciplinary fertilization of ideas between the Math-Physics geometry and topology, Quantum Condensed Matter, and High-Energy Physics (theory, pheno, lattice) can help solving these big problems.
The (1) answer relates to Ultra Unification: a Z_{16} class Atiyah-Patodi-Singer eta invariant (Baryon – Lepton)-protected topological superconductor and a symmetry-extended anomalous topological quantum field theory.
The (2) and (3) answers relate to a trivial cobordism class and Symmetric Mass Generation, and disordering and removing the Pecci-Quinn axions.
The (4) answer has to do with the Hirzebruch signature and Atiyah-Singer index theorem, gravitational Chern-Simons theory, and 3 E8 quantum Hall states.

Fermi surface symmetric mass generation and its application in nickelate superconductor
2023-12-08 14:30 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint Quantum Matter in Mathematics and Physics & Topological Quantum Matter Seminar
Speaker: Da-Chuan Lu (UCSD)
Title: Fermi surface symmetric mass generation and its application in nickelate superconductor
Abstract: Symmetric mass generation (SMG) is a novel interaction-driven mechanism that generates fermion mass without breaking symmetry, unlike the standard Anderson-Higgs mechanism. SMG can occur in the fermion system without quantum anomalies. In this talk, I will focus on the SMG for the systems with finite fermion density, i.e., the Fermi surface. I will discuss the Fermi surface anomaly and Fermi surface SMG. Lastly, I will talk about its application in the newly found nickelate superconductors, where the superconductivity emerges without a nearby spontaneous symmetry-breaking phase.

M-theory on nodal Calabi-Yau 3-folds and torsion refined GV-invariants
2023-12-11 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Thorsten Schimannek (Utrecht University)

Title: M-theory on nodal Calabi-Yau 3-folds and torsion refined GV-invariants
Abstract: The physics of M-theory and Type IIA strings on a projective nodal CY 3-folds is determined by the geometry of a small resolution, even if the latter is not Kähler. We will demonstrate this explicitly in the context of a family of Calabi-Yau double covers of P^3. Using conifold transitions, we prove that the exceptional curves in any small resolution are torsion while M-theory develops a discrete gauge symmetry.This leads to a torsion refinement of the ordinary Gopakumar-Vafa invariants, that is associated to the singular Calabi-Yau and captures the enumerative geometry of the non-Kähler resolutions. We further argue that twisted circle compactifications of the 5d theory are dual to IIA compactifications on the nodal CY 3-fold with a flat but topologically non-trivial B-field. As a result, the torsion refined invariants are encoded in the topological string partition functions with different choices for the global topology of a flat B-field.
The talk is based on 2108.09311, 2212.08655 (with S. Katz, A. Klemm, and E. Sharpe) and 2307.00047 (with S. Katz).

Homology, higher derived limits, and set theory
2023-12-11 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Justin Moore (Cornell University)
Title: Homology, higher derived limits, and set theory
Abstract: Singular homology has a number of well-known defects when used to study spaces such as the Hawaiian earring and solenoids. It may not reflect the “shape” of the space and can give counterintuitive information about its dimension. One remedy of this is to develop a homology theory based on approximating spaces by polyhedra, computing their homologies, and then taking a limit. This is the approach taken by Steenrod-Sitnikov homology and Lisica and Mardesic’s strong homology. Even within the class of locally compact second countable spaces though, the properties of these homology theories — and the higher derived limits which underly them — are dependent on axioms of set theory beyond ZFC. Recently it was shown that it is consistent with (and therefore independent of) ZFC that strong homology and Steenrod Sitnikov homology coincide in the class of locally compact second countable spaces — and therefore each of these homology theories enjoys the desirable properties of the other. These results also point to how we might develop variants of these homology theories which enjoy their desirable properties, but which are less sensitive to set theory. This is joint work with Nathaniel Bannister, Jeff Bergfalk, and Stevo Todorcevic.

When does a three-dimensions Chern-Simons theory have a time reversal symmetry?
2023-12-14 14:30 - 16:00 | Virtual
Speaker: Roman Geiko (UCLA)
Title: When does a three-dimensions Chern-Simons theory have a time reversal symmetry?
In this talk, I will discuss the time reversal invariance of (spin-) Chern-Simons theory in 3 dimensions at both classical and quantum levels. I will show how to obtain a complete classification of Abelian anyons with the time reversal symmetry expressed in terms of the higher Gauss sums. Then, I will comment on the time reversal symmetry of Chern-Simons in the non-Abelian case.

Exact lattice chiral symmetry in 2d gauge theory
2023-12-15 10:00 - 11:30 | Virtual
Quantum Matter Seminar
Speaker: Aleksey Cherman (UMN)
Title: Exact lattice chiral symmetry in 2d gauge theory
Abstract: Preserving the symmetries of massless fermions is a well-known challenge in lattice field theory. I’ll discuss symmetry-preserving lattice regularizations of 2d QED with one and two flavors of Dirac fermions, as well as the `3450′ chiral gauge theory. The construction leverages bosonization and recently-proposed modifications of Villain-type lattice actions. The internal global symmetries act just as locally on the lattice as they do in the continuum, the anomalies are reproduced at finite lattice spacing, and in each case we’ve found a sign-problem-free dual formulation.

On compactness phenomena
2023-12-15 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Alejandro Poveda
Title: On compactness phenomena
Abstract: In this presentation I shall be preoccupied with compactness phenomena in set theory. Compactness is the phenomenon by which the local properties of a mathematical structure determine its global behavior. This phenomenon is intrinsic to the very architecture of the mathematical universe and manifests to us in a cornucopia of forms. Over the past fifty years, the study of compactness phenomena has been a cornerstone of research in set theory. This talk will present recent discoveries regarding compactness principles, spanning classical themes like the tree property and stationary reflection, while also forging novel connections with other topics, such as Woodin’s HOD Conjecture.

Approaches to the formalization of differential geometry
2024-01-24 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Heather Macbeth, Fordham University
Title: Approaches to the formalization of differential geometry
Abstract: In the last five years, there has been early work on the computer formalization of differential geometry. I will survey the projects I am aware of. I will also describe two projects of my own, as case studies for typical challenges. The first (joint with Floris van Doorn) is an exercise in developing suitable abstractions, the second (joint with Mario Carneiro) is an exercise in developing suitable automation.

Gauging spacetime inversions
2024-01-26 09:30 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Daniel Harlow (MIT)
Title: Gauging spacetime inversions
Abstract: Spacetime inversion symmetries such as parity and time reversal play a central role in physics, but they are usually treated as global symmetries. In quantum gravity there are no global symmetries, so any spacetime inversion symmetries must be gauge symmetries. In particular this includes CRT symmetry (in even dimensions usually combined with a rotation to become CPT), which in quantum field theory is always a symmetry and seems likely to be a symmetry of quantum gravity as well. I’ll discuss what it means to gauge a spacetime inversion symmetry, and explain some of the more unusual consequences of doing this. In particular I’ll argue that the gauging of CRT is automatically implemented by the sum over topologies in the Euclidean gravity path integral, that in a closed universe the Hilbert space of quantum gravity must be a real vector space, and that in Lorentzian signature manifolds which are not time-orientable must be included as valid configurations of the theory.

Anti-Iitaka conjecture in positive characteristic
2024-01-26 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Iacopo Brivio (Harvard)
Title: Anti-Iitaka conjecture in positive characteristic
Abstract: Given a smooth projective variety, its Kodaira dimension kappa(K_X) is an important invariant that measures the rate of growth of m-pluricanonical forms as a function of m. It serves as an higher-dimensional generalization of the genus of a Riemann surface. If f : X –> Y is a fibration with general fiber F, a famous conjecture of Iitaka predicts the inequality kappa(K_X) geq kappa(K_Y) + kappa(K_F). More recently it was shown by Chang that, if the stable base locus of -K_X is vertical, then the inequality kappa(-K_X) leq kappa(-K_Y) + kappa(-K_F) holds. Both Iitaka’s conjecture and Chang’s theorem are known to fail in positive characteristic. In this talk I will explain how one can recover Chang’s theorem for a class of “tame” fibrations in characteristic p > 0. This is based on joint work with M. Benozzo and C.-K. Chang.

CMSA/MATH Bi-Annual Gathering
2024-01-26 16:00 - 18:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On Friday, Jan. 26, 2024 the CMSA will host the CMSA/MATH Bi-Annual Gathering for Harvard CMSA and Math affiliates in the CMSA Common Room at 20 Garden Street, Cambridge MA 02138.

A quasi-local mass in general relativity
2024-01-30 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Aghil Alaee, Clark University
Title: A quasi-local mass in general relativity
Abstract: In this talk, we define a new gauge-independent quasi-local mass and energy with respect to the Minkowski spacetime. In contrast to other quasi-local masses, this new quasi-local mass/energy has a quasi-local proof of positivity. This positivity property is for spacelike surfaces with any topology. Moreover, we show that it has desired asymptotic behaviors at null and spatial infinity of asymptotically flat spacetimes. Rigidity is also established in that vanishing energy implies that the 2-surface arises from an embedding into Minkowski space, and conversely, the mass vanishes for any such surface. This is joint work with M. Khuri and S.T. Yau.

Algebraic billiards and dynamical degrees
2024-02-01 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Max Weinreich (Harvard)
Title: Algebraic billiards and dynamical degrees
Abstract: Billiards is one of the most-studied dynamical systems, modeling the behavior of a point particle bouncing around some space. If the space is a plane region bounded by an algebraic curve, then we may use techniques from algebraic geometry to study its billiards map. We explain how to view billiards as a complex algebraic correspondence, and we prove upper and lower bounds on the dynamical degree, the growth rate of the degrees of the iterates, in terms of the degree of the boundary curve. These degree growth rates are studied in mathematical physics, broadly speaking, as a way to identify integrable (exactly solvable) physical models. In our setting, this theory gives us an upper bound on the entropy, or chaos, of billiards in curves.

Quantum Circuits to local Hamiltonian: role in quantum complexity and new constructions
2024-02-02 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Anurag Anshu (Harvard)
Title: Quantum Circuits to local Hamiltonian: role in quantum complexity and new constructions
Abstract: At the heart of the theory of NP completeness lies a mapping from classical circuits to constraint satisfaction problems (classical local Hamiltonians).
The quantum analogue of this is the remarkable history state construction of Kitaev (building upon Feynman’s work). This talk will provide an introduction to this mapping and its crucial role in bridging quantum computer science and quantum many-body physics research. Then, we will describe a new mapping using tensor networks and quantum fault tolerance (https://arxiv.org/abs/2309.16475). Time permitting, we will discuss the relevance of this mapping to the quantum PCP conjecture.

On complete Calabi-Yau metrics and Monge-Ampere equations
2024-02-02 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Freid Tong (Harvard CMSA)
Title: On complete Calabi-Yau metrics and Monge-Ampere equations
Abstract: Calabi-Yau metrics are central objects in K”ahler geometry and also string theory. The existence of Calabi-Yau metrics on compact manifolds was answered by Yau in his solution of the Calabi conjecture, but the situation in the non-compact setting is much more delicate, and many questions related to the existence and uniqueness of non-compact Calabi-Yau metrics remain unanswered. I will give an introduction to this subject and discuss some ongoing joint work with T. Collins and S.-T. Yau, on a new relationship between complete Calabi-Yau metrics and a new Monge-Ampere equation.

Arithmetic Quantum Field Theory Program
2024-02-05 09:00 - 2024-03-29 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Arithmetic Quantum Field Theory Program
Dates: Feb. 5–Mar. 29, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge MA 02138
Arithmetic Quantum Field Theory Program Youtube Playlist
Organizers:

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

This program features a weekly seminar series, workshops, and a conference.
The object of the program is to develop and disseminate exciting new connections emerging between quantum field theory and algebraic number theory, and in particular between the fundamental invariants of each: partition functions and L-functions.
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 Atiyah-Segal / Baez-Dolan / Lurie formalism of extended, functorial field theory) and on the level of observables (via the Beilinson–Drinfeld / Costello–Gwilliam formalism of factorization algebras). On the other hand, Weil’s Rosetta Stone and the Mazur–Morishita–Kapranov–Reznikov arithmetic topology (the “knots and primes” dictionary) provide 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. Most recently, the work of Ben-Zvi–Sakellaridis–Venkatesh applies 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 program 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, harmonic analysis and L-functions. Conversely, we will explore potential geometric and physical consequences of arithmetic ideas, for example, the Langlands correspondence theory of L-functions for 3-manifolds.

Schedule
The first week of the program will feature several lecture series aimed at a broad local community of mathematicians and physicists, aiming to introduce the main ideas underlying our program and help establish a common reference point.
The program will host a weekly seminar series on Fridays.
The speakers will be selected with the aim of covering a wide panorama of the subjects over the course of the program.
The program will conclude with a week-long Conference on Arithmetic Quantum Field Theory March 25–29, 2024.

AQFT Youtube Playlist
Lecture series
All lectures take place in Room G10, Harvard CMSA, 20 Garden Street Cambridge.
Week 1: Feb. 5–9, 2024
Abstract: In this lecture series we will introduce some of the themes underlying the CMSA program on Arithmetic Quantum Field Theory taking place this winter and the upcoming conference March 25-29, 2024.
Some of the themes we plan to discuss include:
Structures in QFT (like factorization for observables and functorial QFT for states and their relation to geometric / deformation quantization) that are sufficiently algebraic and formal to allow for arithmetic analogs.
The setup of arithmetic topology as a bridge between the background of QFT to that of arithmetic (both “global” and “local”), including the “middle realm” of positive characteristic function fields.
Questions and structures in arithmetic that have been / might be amenable to inspiration from QFT, in particular the theory of L-functions and the Langlands program.
Schedule

Monday, Feb. 5, 2024

11:00 am – 12:00 pm
Minhyong Kim
Arithmetic topology and field theory
Video
(Slides part 1 pdf)

1:30 – 2:30 pm
Brian Williams
Algebraic quantum field theory
Video
(Lecture Notes)

2:30 – 3:30 pm
David Ben-Zvi
The Langlands program via arithmetic QFT
Video

Wednesday, Feb. 7, 2024

11:00 am – 12:00 pm
Minhyong Kim
Arithmetic topology and field theory
Video
(Slides part 2 pdf)

2:30 – 3:30 pm
Brian Williams
Algebraic quantum field theory
Video
(Lecture Notes)

Thursday, Feb.8, 2024

2:30 – 3:30 pm
Minhyong Kim
Arithmetic topology and field theory
Video
(Slides part 3 pdf)

4:00 – 5:00 pm
David Ben-Zvi
The Langlands program via arithmetic QFT
Video

Friday, Feb. 9, 2024

1:00 – 2:00 pm
Brian Williams
Algebraic quantum field theory
Video
(Lecture Notes)

2:00 – 3:00 pm
David Ben-Zvi
The Langlands program via arithmetic QFT 1
Video

3:30 – 4:30 pm
David Ben-Zvi
The Langlands program via arithmetic QFT 2
Video

Monday, Feb. 26, 2024

1:00 – 2:00 pm
Omer Offen (Brandeis)
Period integrals of automorphic forms and the residue method
Video

Tuesday, Feb. 27, 2024

2:00 – 3:00 pm
Wei Zhang (MIT)
Shtuka special cycles and their generating series
Video

Friday, March 1, 2024

11:00 am – 12:00 pm
Chen Wan (Rutgers Newark)
Some examples of the relative Langlands duality
Video

2:00 – 3:00 pm
Peng Shan (Tsinghua)
Skein algebras and quantized Coulomb branches
Video

Thursday, March 7, 2024

1:30 – 2:30 pm
An Huang (Brandeis)
Tate’s thesis and p-adic strings
Video

3:00 – 4:00 pm
John Francis (Northwestern)
Integrating braided categories over 3-manifolds
Video

Friday, March 8, 2024

1:00 – 2:00 pm
Dihua Jiang (U Minnesota)
Shalika Periods: Functoriality and Arithmetic
Video

Friday, March 15, 2024

11:45 – 1:00 pm
Baiying Liu (Purdue)
Recent progress on certain problems related to local Arthur packets of classical groups
Video

2:15 – 3:30 pm
Tasho Kaletha (Michigan)
Covers of reductive groups and functoriality
Video

Monday, March 18, 2024

1:00 – 3:00 pm
Xinwen Zhu (Stanford)
The tame categorical local Langlands correspondence
Video

4:30 – 5:30 pm
Natalie Paquette (U Washington)
Koszul duality & twisted holography for asymptotically flat spacetimes

Wednesday, March 20, 2024

11:00 – 12:15 pm
Stephen D. Miller (Rutgers)
What 4-graviton scattering amplitudes had to say about the unitary dual

Friday, March 22, 2024

1:45 – 3:00 pm
Jayce Getz (Duke)
The Poisson summation conjecture and the fiber bundle method
Video

Program Visitors

Mina Aganagic, University of California, Berkeley
Anne-Marie Aubert, Institut de Mathématiques de Jussieu-Paris Rive Gauche, March 15-29
Clark Barwick, University of Edinburgh, February 19-March 15
Alexander Braverman, Perimeter Institute
Alejandra Castro, Cambridge University, March 25-29
YoungJu Choie, Pohang University of Science and Technology, February 12-16; March 22-28
John Francis, Northwestern University, March 1-14
David Gaiotto, Perimeter Institute, March 25-29
Jayce Getz, Duke University, March 18-22
Ezra Getzler, Northwestern University, March 11-22
Sam Gunningham, Montana State University, February 9-12
Sarah Harrison, Northeastern University
Dihua Jiang, University of Minnesota, February 29-March 9
Tasho Kaletha, University of Michigan, March 12-20
Minhyong Kim, University of Edinburgh, February 1-29
Axel Kleinschmidt, Max Planck Institute for Gravitational Physics, Potsdam, March 18-28
Kim Klinger-Logan, Kansas State University, March 25-29
Kobi Kremnitzer, Oxford University, March 25-29

Baiying Liu, Purdue University, March 13-16

Steven Miller, Rutgers University

Greg Moore, Rutgers University, February 5-9
David Nadler, University of California, Berkeley, March 17-30
Bảo Châu Ngô, University of Chicago, March 25-29
George Pappas, Michigan State University, March 25-29
Daniel Persson, Chalmers Institute of Technology, March 25-29
Sam Raskin, Yale University, March 26-29
Yiannis Sakellaridis, Johns Hopkins University, March 18-22
Peng Shan, Tsinghua University, February 12-April 14
Akshay Venkatesh, Institute for Advanced Study
Roberto Volpato, University of Padova, February 4-10
Chen Wan, Rutgers University, February 29-March 9
Fei Yan, Brookhaven National Laboratory, March 18-29
Xinwen Zhu, Stanford University

AQFT Lecture Series | Minhyong Kim
2024-02-05 11:00 - 12:00, 2024-02-07 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Arithmetic Quantum Field Theory Program Lecture Series
Speaker: Minhyong Kim, University of Edinburgh
Topic: Arithmetic topology and field theory
Abstract: The setup of arithmetic topology as a bridge between the background of QFT to that of arithmetic (both “global” and “local”), including the “middle realm” of positive characteristic function fields.
Slides (pdf)

AQFT Lecture Series | Brian Williams
2024-02-05 13:30 - 14:30, 2024-02-07 14:30 - 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Arithmetic Quantum Field Theory Program Lecture Series
Speaker: Brian Williams, Boston University
Topic: Algebraic quantum field theory
Abstract: Questions and structures in arithmetic that have been / might be amenable to inspiration from QFT, in particular the theory of L-functions and the Langlands program.

AQFT Lecture Series | David Ben-Zvi
2024-02-05 14:30 - 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Arithmetic Quantum Field Theory Program Lecture Series
Speaker: David Ben-Zvi
Topic: The Langlands program via arithmetic QFT
Abstract: Structures in QFT (like factorization for observables and functorial QFT for states and their relation to geometric / deformation quantization) that are sufficiently algebraic and formal to allow for arithmetic analogs.

Noncompact n-dimensional Einstein spaces as attractors for the Einstein flow
2024-02-06 10:00 - 11:00 | Virtual
General Relativity Seminar
Speaker: Jinhua Wang, Xiamen University
Title: Noncompact n-dimensional Einstein spaces as attractors for the Einstein flow
Abstract: We prove that along with the Einstein flow, any small perturbations of an $n$($ngeq4$)-dimensional, non-compact negative Einstein space with some “non-positive Weyl tensor” lead to a unique and global solution, and the solution will be attracted to a noncompact Einstein space that is close to the background one. The $n=3$ case has been addressed by Wang-Yuan, while in dimension $ngeq 4$, as we know, negative Einstein metrics in general have non-trivial moduli spaces. This fact is reflected on the structure of Einstein equations, which further indicates no decay for the spatial Weyl tensor. Furthermore, it is suggested in the proof that the mechanic preventing the metric from flowing back to the original Einstein metric lies in the non-decaying character of spatial Weyl tensor. In contrary to the compact case considered in Andersson-Moncrief, our proof is independent of the theory of infinitesimal Einstein deformations. Instead, we take advantage of the inherent geometric structures of Einstein equations and develop an approach of energy estimates for a hyperbolic system of Maxwell type.
References – arXiv: 2209.15244, 2309.15152, 2311.00868

CMSA Q&A Seminar 2/6/2024
2024-02-06 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar
Speaker: Greg Moore, Rutgers University
Question: What is supersymmetry?

Flavor hierarchy from smooth confinement & Towards a complete classification of 6d supergravities
2024-02-06 16:00 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Yuta Hamada (KEK, Tsukuba)
Title: Flavor hierarchy from smooth confinement & Towards a complete classification of 6d supergravities
Abstract: The talk consists of two independent parts. In the first part, I will talk about a new model to explain the Standard Model flavor hierarchy. Our model is based on explicit smooth confinement. The smallness of the first- and second-family fermion masses is explained by the exponential hierarchy via dimensional transmutation. In the second part, I will talk about a classification of 6D supergravities. We make progress towards a complete classification of 6D supergravities with minimal supersymmetry and non-abelian gauge group.
References – arXiv: 2209.15244, 2309.15152, 2311.00868

CMSA/Tsinghua Math-Science Literature Lecture: Amie Wilkinson
2024-02-07 09:00 - 10:30 | Virtual
CMSA/Tsinghua Math-Science Literature Lecture
Prof. Amie Wilkinson gave a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.
Date: Wednesday, February 7, 2024
Time: 9:00–10:30 am ET
Title: Stretching and shrinking: 85 years of the Hopf argument for ergodicity
Abstract: The early 20th century witnessed an explosion of activity, much of it centered at Harvard, on rigorizing the property of ergodicity first proposed by Boltzmann in his 1898 Ergodic Hypothesis for ideal gases. Earlier, in the 1880’s, Henri Poincaré and Felix Klein had also initiated a study of discrete groups of hyperbolic isometries. The geodesics in hyperbolic manifolds were discovered to carry a rich structure, first investigated from a topological perspective by Emil Artin and Marston Morse. The time was ripe to investigate geodesics in hyperbolic manifolds from an ergodic theoretic (i.e., statistical) perspective, and indeed Gustav Hedlund proved in 1934 that the geodesic flow for closed hyperbolic surfaces is ergodic.

In 1939, Eberhard Hopf published a proof of the ergodicity of geodesic flows for negatively curved surfaces containing a novel method, now known as the Hopf argument. The Hopf argument, a “soft” argument for ergodicity of systems with some hyperbolicity (the “stretching and shrinking” in the title) has since seen wide application in geometry, representation theory and dynamics. I will discuss three results relying on the Hopf argument:

Theorem (E. Hopf, 1939, D. Anosov, 1967): In a closed manifold of negative sectional curvatures, almost every geodesic is directionally equidistributed.

Theorem (G. Mostow, 1968) Let M and N be closed hyperbolic manifolds of dimension at least 3, and let f:M->N be a homotopy equivalence. Then f is homotopic to a unique isometry.

Theorem (R. Mañé, 1983, A. Avila- S. Crovisier- A.W., 2022) The C^1 generic symplectomorphism of a closed symplectic manifold with positive entropy is ergodic.

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.

Large language models, mathematical discovery, and search in the space of strategies: an anecdote
2024-02-07 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Jordan Ellenberg (UW Madison)
Title: Large language models, mathematical discovery, and search in the space of strategies: an anecdote
Abstract: I spent a portion of 2023 working with a team at DeepMind on the “cap set problem” – how large can a subset of (Z/3Z)^n be which contains no three terms which sum to zero? (I will explain, for those not familiar with this problem, something about the role it plays in combinatorics, its history, and why number theorists care about it a lot.) By now, there are many examples of machine learning mechanisms being used to help generate interesting mathematical knowledge, and especially interesting examples. This project used a novel protocol; instead of searching directly for large cap sets, we used LLMs trained on code to search the space of short programs for those which, when executed, output large cap sets. One advantage is that a program is much more human-readable than a large collection of vectors over Z/3Z, bringing us closer to the not-very-well-defined-but-important goal of “interpretable machine learning.” I’ll talk about what succeeded in this project (more than I expected!) what didn’t, and what role I can imagine this approach to the math-ML interface playing in near-future mathematical practice.
The paper: https://www.nature.com/articles/s41586-023-06924-6

On (semi)stable reduction and KSBA moduli in positive characteristic
2024-02-08 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Iacopo Brivio (Harvard CMSA)
Title: On (semi)stable reduction and KSBA moduli in positive characteristic
Abstract: The moduli space M_g of genus g stable curves is perhaps the most studied of all algebraic varieties. Its higher-dimensional generalization is the moduli functor M_{n,v} of n-dimension stable varieties of volume v. It was proven only recently, and thanks to the joint effort of many over many years, that such functors are represented by projective algebraic spaces when working over the complex numbers. In this talk I will give some examples showing that the same moduli functors in positive characteristic are not even proper and, more in general, that the MMP fails to be functorial even in very nice families. In the second part I am going to explore some possible generalizations of the notion of stable variety that could be used as a replacement in positive characteristic.
