Arithmetic Quantum Field Theory Program
2024-02-05 09:00 - 2024-03-29 17:00 (America/New_York) | 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, 2024-02-08 14:30 - 15:30 | 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, 2024-02-09 13:00 - 14:00 | 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, 2024-02-08 16:00 - 17:00, 2024-02-09 14:00 - 15:00, 2024-02-09 15:30 - 16:30 | 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.

Quantum Algorithms to Recognize Phases of Matter and Exactly Solvable 2D Models with Anomalous Entanglement Entropy
2024-02-09 09:30 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Soonwon Choi (MIT)
Title: Quantum Algorithms to Recognize Phases of Matter and Exactly Solvable 2D Models with Anomalous Entanglement Entropy
Abstract: In this talk, I will report my two recent results at the intersection of quantum information and strongly interacting phases of matters.
In the first half of the talk, we describe exact quantum algorithms that recognize a class of 1D gapped phases, namely symmetry protected topological phases or spontaneous symmetry breaking phases protected by abelian internal symmetry. The key idea is to observe the conceptual similarity between renormalization group (RG) flow and error correction, and to implement the latter as unitary circuits emulating the RG flow. Our algorithm guarantees faithful recognition of a target phase with a small number of input quantum state samples.
In the second half, we present a class of 2D Hamiltonians, where the exact ground state wavefunctions can be exactly evaluated and shown to exhibit anomalous entanglement properties. One class of our models exhibit area-law scaling entanglement entropy, but this is mostly due to non-local correlation: one finds that the topological entanglement entropy also scales with the size of subsystem choices. By making simple modifications, we can also devise 2D models with volume-law scaling bipartite entanglement entropy. Our results can be understood as a generation of the 1D Motzkin model to 2D systems.
Based on work done with Ethan Lake and Shankar Balasubramanian
https://arxiv.org/abs/2211.09803
https://arxiv.org/abs/2305.07028

The spectrum of some nonlinear random matrices
2024-02-09 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Benjamin McKenna (Harvard)
Title: The spectrum of some nonlinear random matrices
Abstract: Modern data science often requires one to consider “nonlinear random matrices,” a broad term for random-matrix models whose construction involves a nonlinear function applied entrywise. Such models are typically far from classical random matrix theory, and in principle entrywise nonlinearities can affect the eigenvalues in a complicated way. However, recent years have seen a number of results on nonlinear models whose spectrum is surprisingly simple. We give one such result, emphasizing general random-matrix techniques like free probability and orthogonal polynomials. Joint work with Sofiia Dubova, Yue M. Lu, and Horng-Tzer Yau.

Machine Learning and Scientific Computing: There is plenty of room in the middle
2024-02-12 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Petros Koumoutsakos, Harvard SEAS
Title: Machine Learning and Scientific Computing: There is plenty of room in the middle
Abstract: Over the last thirty years we have experienced more than a billion-fold increase in hardware capabilities and a dizzying pace of acquiring and transmitting massive amounts of data. Scientific Computing and, more lately, Artificial Intelligence (AI) has been key beneficiaries of these advances. In this talk I would outline the need for bridging the decades long advances in Scientific Computing with those of AI. I will use examples from fluid mechanics to argue for forming alloys of AI and simulations for their prediction and control. I will present novel algorithms for learning the Effective Dynamics (LED) of complex systems and a fusion of multi- agent reinforcement learning and scientific computing (SciMARL) for modeling and control of turbulent flows. I will also show our recent work on Optimizing a Discrete Loss (ODIL) that outperforms popular techniques such as PINNs by several orders of magnitude.
I will juxtapose successes and failures and argue that the proper fusion of scientific computing and AI expertise are essential to advance scientific frontiers.

Characteristic Initial Value Problem for the 3D Compressible Euler Equations
2024-02-13 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Sifan Yu, NUS
Title: Characteristic Initial Value Problem for the 3D Compressible Euler Equations
Abstract: We present the first result for the characteristic initial value problem of the compressible Euler equations in three space dimensions without any symmetry assumption. We allow presence of vorticity and consider any equation of state. Compared to the standard Cauchy problem, where initial data can be freely prescribed on a constant-time hypersurface, we formulate the problem by distinguishing between the “free-component” and the “constrained-component” of the initial data. The latter is to be solved by the “free-component” utilizing the properties of the compressible Euler equations on the initial null hypersurfaces. Then, we establish a priori estimates, followed by a local well-posedness and a continuation criterion argument. Moreover, we prove a regularity theory in Sobolev norms. Our analysis critically relies on the vectorfield method due to the nature of the problem. This is a joint work with Jared Speck.

What Algorithms can Transformers Learn? A Study in Length Generalization
2024-02-14 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Preetum Nakkiran, Apple
Title: What Algorithms can Transformers Learn? A Study in Length Generalization
Abstract: Large language models exhibit many surprising “out-of-distribution” generalization abilities, yet also struggle to solve certain simple tasks like decimal addition. To clarify the scope of Transformers’ out-of-distribution generalization, we isolate this behavior in a specific controlled setting: length-generalization on algorithmic tasks. Eg: Can a model trained on 10 digit addition generalize to 50 digit addition? For which tasks do we expect this to work?
Our key tool is the recently-introduced RASP language (Weiss et al 2021), which is a programming language tailor-made for the Transformer’s computational model. We conjecture, informally, that: Transformers tend to length-generalize on a task if there exists a short RASP program that solves the task for all input lengths. This simple conjecture remarkably captures most known instances of length generalization on algorithmic tasks, and can also inform design of effective scratchpads. Finally, on the theoretical side, we give a simple separating example between our conjecture and the “min-degree-interpolator” model of learning from Abbe et al. (2023).
Joint work with Hattie Zhou, Arwen Bradley, Etai Littwin, Noam Razin, Omid Saremi, Josh Susskind, and Samy Bengio. To appear in ICLR 2024.

Quantum Algebra of Chern-Simons Matrix Model and Large N Limit
2024-02-14 16:00 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Sen Hu (Shanghai Institute for Mathematics and Interdisciplinary Study)
Title: Quantum Algebra of Chern-Simons Matrix Model and Large N Limit
Abstract: In this talk we discuss the algebra of quantum observables of the Chern-Simons matrix model which was originally proposed by Susskind and Polychronakos to describe electrons in fractional quantum Hall effects. We establish the commutation relations for its generators and study the large N limit of its representation. We show that the large N limit algebra is isomorphic to the uniform in N algebra studied by Costello, which is conjecturally isomorphic to the deformed double current algebra studied by Guay. Under appropriate scaling limit, we show that the large N limit algebra degenerates to a Lie algebra which admits a surjective map to the affine Lie algebra of u(p). This leads to a complete proof of the large N emergence of the u(p) current algebra as proposed by Dorey, Tong and Turner. This also suggests a rigorous derivation of edge excitation of a fractional quantum Hall droplet. This is a joint work with Si Li, Dongheng Ye and Yehao Zhou (arXiv: 2308.14046).

Programmable Simulations of Molecules and Materials with present-day Reconfigurable Quantum Processors
2024-02-16 10:30 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Susanne Yelin (Harvard)
Title: Programmable Simulations of Molecules and Materials with present-day Reconfigurable Quantum Processors
Abstract: Simulations of quantum chemistry and quantum materials are believed to be among the most important potential applications of quantum information processors, but realizing practical quantum advantage for such problems is challenging. We introduce a simulation framework for strongly correlated quantum systems that can be represented by model spin Hamiltonians. Our approach leverages reconfigurable qubit architectures to programmably simulate real-time dynamics and introduces an algorithm for extracting chemically relevant spectral properties via classical co-processing of quantum measurement results. We develop a digital-analog simulation toolbox for efficient Hamiltonian time evolution utilizing digital Floquet engineering and hardware-optimized multi-qubit operations to accurately realize complex spin-spin interactions, and as an example present an implementation proposal based on Rydberg atom arrays. Then, we show how detailed spectral and other relevant chemical information can be extracted from these dynamics through snapshot measurements and single-ancilla control, enabling the evaluation of excitation energies and finite-temperature susceptibilities from a single-dataset. To illustrate the approach, we show how this method can be used to compute key properties of a polynuclear transition-metal catalyst and 2D magnetic materials.

Symmetries and algebraicity in the flux landscape
2024-02-16 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Damian van de Heisteeg (Harvard CMSA)
Title: Symmetries and algebraicity in the flux landscape
Abstract: In this talk I consider potentials coming from fluxes in string theory. The minima of these potentials trace out special loci in the moduli space of Calabi-Yau manifolds. I discuss the structure that underlies these minima from a Hodge-theoretic point of view.

Asymptotic decay for defocusing semilinear wave equations on Schwarzschild spacetimes
2024-02-20 09:30 - 10:30 | Virtual
General Relativity Seminar
Speaker: He Mei, Shenzhen University
Title: Asymptotic decay for defocusing semilinear wave equations on Schwarzschild spacetimes
Abstract: In this talk, I will present a work on the long time dynamics of solutions to the defocusing semilinear wave equations on the Schwarzschild black hole spacetimes. For sufficiently smooth and localized initial data, we show that the solution decays in the domain of outer communication. The proof relies on a vector field method of Dafermos-Rodnianski together with Strichartz estimates for linear waves by Marzuola-Metcalfe-Tataru-Tohaneanu.

CMSA Q&A Seminar 2/20/2024
2024-02-20 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar

Speaker: Solomon Friedberg, Boston College

Question: What is the Langlands program?

Math Science Lectures in Honor of Raoul Bott: Maggie Miller: Fibered ribbon knots vs. major 4D conjectures
2024-02-20 16:00 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Fibered ribbon knots vs. major 4D conjectures
Location: Harvard University Science Center Hall A & via Zoom webinar
Dates: Feb 20 & 22, 2024
Time: 4:00-5:30 pm
Maggie Miller is an assistant professor in the mathematics department at the University of Texas at Austin and a Clay Research Fellow.
This is the fourth annual Math Science Lecture Series held in Honor of Raoul Bott.
Talk topic: Fibered ribbon knots vs. major 4D conjectures

Feb. 20, 2024
Title: Fibered ribbon knots and the Poincaré conjecture
Abstract: A knot is “fibered” if its complement in S^3 is the total space of a bundle over the circle, and ribbon if it bounds a smooth disk into B^4 with no local maxima with respect to radial height. A theorem of Casson-Gordon from 1983 implies that if a fibered ribbon knot does not bound any fibered disk in B^4, then the smooth 4D Poincaré conjecture is false. I’ll show that unfortunately (?) many ribbon disks bounded by fibered knots are fibered, giving some criteria for extending fibrations and discuss how one might search for non-fibered examples.

Feb. 22, 2024
Title: Fibered knots and the slice-ribbon conjecture
Abstract: The slice-ribbon conjecture (Fox, 1962) posits that if a knot bounds any smooth disk into B^4, it also bounds a ribbon disk. The previously discussed work of Casson-Gordon yields an obstruction to many fibered knots being ribbon, yielding many interesting potential counterexamples to this conjecture — if any happy to bound a non-ribbon disk. In 2022, Dai-Kong-Mallick-Park-Stoffregen showed that unfortunately( ?) many of these knots don’t bound a smooth disk into B^4 and thus can’t disprove the conjecture. I’ll show a simple alternate proof that a certain interesting knot (the (2,1)-cable of the figure eight) isn’t slice and discuss remaining open questions. This talk is joint with Paolo Aceto, Nickolas Castro, JungHwan Park, and Andras Stipsicz.

Talk Chair: Cliff Taubes (Harvard Mathematics)
Moderator: Freid Tong (Harvard CMSA)

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

Computers and mathematics in partial differential equations: New developments and challenges
2024-02-21 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Javier Gomez Serrano, Brown University
Title: Computers and mathematics in partial differential equations: new developments and challenges
Abstract: In this talk I will address the interaction between traditional and more modern mathematics and how computers have helped over the last decade providing rigorous (computer-assisted) proofs in the context of partial differential equations. I will also describe new exciting future directions in the field. No background is assumed.

Geometric origins of values of the Riemann Zeta functions at positive integers
2024-02-22 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Yan Zhou, Northeastern
Title: Geometric origins of values of the Riemann Zeta functions at positive integers
Abstract: Given a Fano manifold, Iritani proposed that the asymptotic behavior of solutions to the quantum differential equation of the Fano should be given by the so-called ‘Gamma class’ in its cohomology ring. Later, Abouzaid-Ganatra-Iritani-Sheridan reformulated the ‘Gamma conjecture’ for Calabi-Yau manifolds via the tropical SYZ mirror symmetry and proposed that values of the Riemann Zeta function at positive integers have geometric origins in the tropical periods and singularities of the SYZ geometry. In this talk, we will first review the content of the Gamma conjecture. Then, we will discuss the first step of generalizing AGIS’ approach to Gamma conjecture for the Gross-Siebert mirror families of a Fano manifold in dimension 2 cases, based on joint work with Bohan Fang and Junxiao Wang.
