# Members’ Seminar

#### Beginning immediately, until at least April 30, all seminars will take place virtually, through Zoom. Links to connect can be found in the schedule below once they are created.

The 2019-2020 CMSA Members’ Seminar will occur every Friday at 5pm on Zoom. Please email the seminar organizers to obtain a link. The Schedule will be updated below. Previous seminars can be found here.

#### Spring 2020:

Date Speaker Title/Abstract

1/31/2020

Bogdan Stoica

Title: From p-adic to Archimedean Physics: Renormalization Group Flow and
Berkovich Spaces

Abstract: We introduce the p-adic particle-in-a-box as a free particle with periodic boundary conditions in the p-adic spatial domain. We compute its energy spectrum, and show that the spectrum of the Archimedean particle-in-a-box can be recovered from the p-adic spectrum via an Euler product formula. This product formula arises from a flow equation in Berkovich space, which we interpret as a space of theories connected by a kind of renormalization group flow. We propose that Berkovich spaces can be used to relate p-adic and Archimedean quantities generally. Talk based on arXiv:2001.01725.

2/7/2020

Sergiy Verstyuk

Title: Some Shallow Explorations in Deep Learning for Finance

Abstract: I will introduce the existing approaches to understanding (rather than predicting) prices on financial assets. I will then discuss some simple ways of improving upon them using modern machine learning methods.

2/14/2020

Yifan Wang

Title: Modularity in Physics and Mathematics

Abstract: I’ll discuss several incarnations of the modular group SL(2,Z) in quantum field theories and string theories, and how they relate to different areas of mathematics. We’ll see examples where mathematical frameworks lead to nontrivial predictions for physical systems, and how physics methods lead to conjectures that call for new mathematical understanding.

2/21/2020

Du Pei

Title: Quantization: theory and applications

Abstract: How to quantize a classical system to get a quantum system? After briefly surveying the history of this problem, I will explain how to use the topological A-model to better understand quantization. As an application, I will discuss how this approach can shed light on the representation theory of double affine Hecke algebras.

2/28/2020

Cancelled

3/6/2020

G02

Yuewen Chen

Title: Introduction to WENO scheme

Abstract: In this talk, we introduce the analysis and applications of WENO scheme for hyperbolic conservation law and Hamilton-Jacobi equation. WENO scheme is a powerful numerical tool to solve partial differential equationswith shock solutions.

3/13/2020

Michael McBreen

This meeting will be taking place virtually on Zoom.

Title: Modular representations and Lagrangian branes

Abstract: I will give an elementary introduction to representations of Lie algebras in characteristic p, and explain how to study them using symplectic geometry and mirror symmetry.

3/20/2020

Postponed

3/27/2020

Juven Wang

This meeting will be taking place virtually on Zoom.

Title: Quantum Matter Adventure to Fundamental Physics and Mathematics
Abstract: In 1956, T. D. Lee and C. N. Yang questioned the Parity Conservation in Weak Interactions in particle physics. In less than one year, experimentalists confirmed the weak interactions are indeed maximally parity-violating. The parity violation now is a Standard Model physics textbook statement, but it goes without any down-to-earth explanation for long. Why? We will see how the recent physics development in Quantum Matter may guide us to give an adventurous story and possibly a new elementary explanation. We will see how the topology and cobordism in mathematics may come into play of anomalies and non-perturbative interactions in fundamental physics. Perhaps some of you (geometers, string theorists, etc.) can team up with me to understand the “boundary conditions” of the Standard Model and Beyond.

Work-based on and Refs therein: arXiv:1809.11171, 1904.00994, and 1910.14668

p.s. Before the seminar, you may please read the Physics Review Landmarks—Breaking the Mirror https://physics.aps.org/story/v22/st19 so you know parity-violation https://en.wikipedia.org/wiki/Parity_(physics)#Parity_violation.

4/3/2020

4:00pm

Tianqi Wu

This meeting will be taking place virtually on Zoom.

Title: Convergence of Discrete Conformal Change and Computation of Uniformizations

Abstract: The classical uniformization theorem indicates that any closed Riemannian surface is conformally equivalent to a surface of constant curvature 1 or 0 or -1, depending on the genus of the surface. Using a simple notion of discrete conformality for triangulated surfaces, we can introduce the notion of discrete uniformization, and prove the convergence of this discrete uniformization to the classical uniformization. The key ingredient of the proof is an L^\infty estimates for discrete harmonic functions on triangulated surfaces.

4/17/2020

This meeting will be taking place virtually on Zoom.

4/24/2020

This meeting will be taking place virtually on Zoom.

5/1/2020

5/8/2020

5/15/2020

G02

#### Fall 2019:

Date Speaker Title/Abstract
9/6/2019 Spiro Karigiannis Title: Constructions of compact torsion-free $G_2$-manifolds

Abstract: Compact torsion-free $G_2$-manifolds are 7-dimensional analogues of Calabi-Yau threefolds, being compact Ricci-flat Riemannian manifolds with reduced holonomy that are important ingredients in theories of physics. All known constructions use an abstract existence theorem of Dominic Joyce to perturb “almost” solutions of a quasilinear elliptic PDE to honest solutions, and construct the “almost” solutions via glueing methods. I will first summarize some basic facts about $G_2$-manifolds and Joyce’s existence theorem, and then briefly mention the previous constructions by Joyce (1994), Kovalev (2003), and Corti-Haskins-Nordstrom-Pacini (2014). Then I will focus on a new construction (joint work of myself and Joyce, to appear in JDG) that is significantly more involved for several reasons, which I will elucidate. In particular one key step in our construction involves solving a linear first order elliptic PDE on a noncompact 4-manifold with prescribed asymptotics at infinity. (arXiv: 1707.09325)

9/13/2019 Wei Gu Title: Sigma models and mirror symmetry

Abstract: In this talk, I will roughly review why physicists be interested in Calabi-Yau manfiolds, and will introduce some tools we used to probe Calabi-Yaus and other spaces like Fanos which we called sigma models.  I will also briefly mention how physicists using sigma models to study mirror symmetry. This is not a technical talks, rather, I will just focus on the pictures of the connections between math and physics from sigma models.

9/20/2019 Ryohei Kobayashi Title: Fermionic phases of matter on unoriented spacetime

Abstract: We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin± case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin− invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the Z8 classification of (1+1)d topological superconductors. We also compute the indicator formula of Z16 valued time-reversal anomaly for (2+1)d pin+ TQFT based on our construction.

9/27/2019 Yun Shi  Title: On motivic Donaldson-Thomas theory on local P2

Abstract: Donaldson-Thomas (DT) theory is an enumerative theory which counts ideal sheaves of curves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory is a categorification of the DT theory. This categofication contains more refined information of the moduli space, just like the topological space or cohomology contains more information than an Euler characteristic. In this talk, I will give a brief introduction to motivic DT theory. I will also discuss some results on this theory for moduli spaces of sheaves on the local projective plane.

10/4/2019 Yoosik Kim  Title: Towards SYZ mirror symmetry of flag varieties.

Abstract: SYZ mirror symmetry has provided a geometric way of understanding mirror symmetry via T-duality. In this talk, I will discuss how to obtain SYZ mirrors of partial flag varieties using Floer theory.

10/11/2019 Rongxiao Mi Title: On the change of Gromov-Witten theory under extremal transitions.

Abstract: Extremal transitions are a topological surgery that conjecturally connects the moduli space of Calabi-Yau 3-folds (often known as “Reid’s Fantasy”). Through extremal transitions, we may be able to build new mirror pairs from old ones, provided we understand how mirror symmetry is preserved. In this talk, I will outline a conjectural framework that relates the genus zero Gromov-Witten theory under an extremal transition. I will explain how it works for a large family of extremal transitions among toric hypersurfaces.

10/18/2019 No Seminar
10/25/2019 Ruth J Williams  Title: Reflected Diffusions and (Bio)Chemical Reaction Networks

Abstract: Continuous-time Markov chain models are often used to describe them stochastic dynamics of networks of reacting chemical species, especially in the growing field of systems biology. Discrete-event stochastic simulation of these models rapidly becomes computationally intensive. Consequently, more tractable diffusion approximations are commonly used in numerical computation, even for modest-sized networks. However, existing approximations (e.g., linear noise and Langevin), do not respect the constraint that chemical concentrations are never negative.

In this talk, we propose an approximation for such Markov chains, via reflected diffusion processes, that respects the fact that concentrations of chemical species are non-negative. This fixes a difficulty with Langevin approximations that they are frequently only valid until the boundary of the positive orthant is reached. Our approximation has the added advantage that it can be written down immediately from the chemical reactions. Some numerical examples illustrate the advantages of our approximation over direct simulation of the Markov chain or use of the linear noise approximation.

11/1/2019 No Seminar
11/8/2019 Zhengping Gui  Title:  Deformation quantization and Algebraic index theorem

Abstract: Deformation quantization is one approach to encapsulating the algebraic aspects of observables in a quantum mechanical system. By constructing a trace map on the algebra of quantum observables, correlation functions are defined. Using this paradigm, an algebraic analogue of the Atiyah-Singer index theorem was established by Fedosov and jointly by Nest and Tsygan.

In this talk, I will discuss how to use topological quantum mechanics to prove the algebraic index theorem when quantum algebra is twisted by vector bundles.

11/15/2019 Ryan Thorngren   Title: Introduction to Bulk-Boundary Correspondences in Condensed Matter Physics

Abstract: A hallmark of topological phases are featureless insulators which are metallic at their edges. The bulk-boundary correspondence relates the ground state entanglement of the bulk with the anomalous properties of the boundary. I will give a gentle introduction to these ideas in a couple of simple models relevant to graphene and superconducting nanowires, respectively. If time permits, I will describe some recent work extending these ideas to bulk “phases” (actually critical points) described by conformal field theory.

11/22/2019 Cancelled
11/29/2019 Cancelled
12/6/2019 Sergiy Verstyuk