This Week's Seminars and Colloquia

Identifiability of overcomplete independent component analysis

Series
Algebra Seminar
Time
Monday, April 8, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ada WangHarvard University

There will be a pre-seminar in Skiles 005 at 11 am.

Independent component analysis (ICA) is a classical data analysis method to study mixtures of independent sources. An ICA model is said to be identifiable if the mixing can be recovered uniquely. Identifiability is known to hold if and only if at most one of the sources is Gaussian, provided the number of sources is at most the number of observations. In this talk, I will discuss our work to generalize the identifiability of ICA to the overcomplete setting, where the number of sources can exceed the number of observations.The underlying problem is algebraic and the proof studies linear spaces of rank one symmetric matrices. Based on joint work with Anna Seigal https://arxiv.org/abs/2401.14709

Diffusion Models: Theory and Applications (in PDEs)

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 8, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Yulong LuUniversity of Minnesota, Twin Cities

Diffusion models, particularly score-based generative models (SGMs), have emerged as powerful tools in diverse machine learning applications, spanning from computer vision to modern language processing. In the first part of this talk, we delve into the generalization theory of SGMs, exploring their capacity for learning high-dimensional distributions. Our analysis show that SGMs achieve a dimension-free generation error bound when applied to a class of sub-Gaussian distributions characterized by certain low-complexity structures.  In the second part of the talk, we consider the application of diffusion models in solving partial differential equations (PDEs). Specifically, we present the development of a physics-guided diffusion model designed for reconstructing high-fidelity solutions from their low-fidelity counterparts. This application showcases the adaptability of diffusion models and their potential to scientific computation.  

Self-similar singular solutions in gas dynamics

Series
PDE Seminar
Time
Tuesday, April 9, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Juhi JangUniversity of Southern California

In this talk, we will discuss mathematical construction of self-similar solutions exhibiting implosion arising in gas dynamics and gaseous stars, with focus on self-similar converging-diverging shock wave solutions to the non-isentropic Euler equations and imploding solutions to the Euler-Poisson equations describing gravitational collapse. The talk is based on joint works with Guo, Hadzic, Liu and Schrecker. 

On tight $(k, \ell)$-stable graphs

Series
Graph Theory Seminar
Time
Tuesday, April 9, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zixia SongUniversity of Central Florida

For integers $k>\ell\ge0$, a graph $G$ is $(k,\ell)$-stable if  $\alpha(G-S)\geq \alpha(G)-\ell$ for every    
$S\subseteq V(G)$ with $|S|=k$. A recent result of Dong and Wu [SIAM J.
Discrete Math. 36 (2022) 229--240] shows that every $(k,\ell)$-stable 
graph $G$  satisfies $\alpha(G) \le  \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell$.  A $(k,\ell)$-stable graph $G$   is   tight if $\alpha(G) = \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell$; and  $q$-tight for some integer $q\ge0$ if $\alpha(G) = \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell-q$.
In this talk, we first prove  that for all $k\geq 24$, the only tight $(k, 0)$-stable graphs are $K_{k+1}$ and  $K_{k+2}$, answering a question of Dong and Luo [arXiv: 2401.16639]. We then prove that  for all nonnegative integers $k, \ell, q$ with $k\geq 3\ell+3$, every $q$-tight $(k,\ell)$-stable graph has at most  $k-3\ell-3+2^{3(\ell+2q+4)^2}$ vertices, answering a question of Dong and Luo in the negative.   \\  

This is joint work with Xiaonan Liu and Zhiyu Wang. 

The intertwined derivative Schrödinger system of Calogero--Moser--Sutherland type

Series
Math Physics Seminar
Time
Wednesday, April 10, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ruoci SunSchool of Mathematics, Georgia Tech

Available via zoom at: https://gatech.zoom.us/j/98258240051

This presentation is dedicated to extending both defocusing and focusing Calogero–Moser–Sutherland derivative nonlinear Schrödinger equations (CMSdNLS), which are introduced in Abanov–Bettelheim–Wiegmann [arXiv:0810.5327], Gérard-Lenzmann [arXiv:2208.04105] and R. Badreddine [arXiv:2303.01087, arXiv:2307.01592], to a system of two matrix-valued variables. This new system is an integrable extension and perturbation of the original CMSdNLS equations. Thanks to the conjugation acting method, I can establish the explicit expression for general solutions on the torus and on the real line in my work [hal-04227081].

The compactness of multilinear Calderón-Zygmund operators

Series
Analysis Seminar
Time
Wednesday, April 10, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anastasios FragkosWashington University St Louis

We prove a wavelet T(1) theorem for compactness of multilinear Calderón -Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator. This talk is based on joint work with Walton Green and Brett Wick.   

Minimal surfaces in negatively curved manifolds

Series
School of Mathematics Colloquium
Time
Thursday, April 11, 2024 - 11:00 for
Location
Skiles 005
Speaker
Andre NevesUniversity of Chicago

The asymptotic behavior of closed  geodesic on negatively curved spaces occupies a central place in Riemannian geometry. Minimal surfaces are higher dimensional analogies of geodesics and I will talk about some recent developments regarding the growth rate of minimal surfaces in negatively curved manifolds.

From Ehrhard to Generalized Bobkov inequality, and more

Series
Stochastics Seminar
Time
Thursday, April 11, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Galyna LivshytsGeorgia Tech

We discuss a general scheme that allows to realize certain geometric functional inequalities as statements about convexity of some functionals, and, inspired by the work of Bobkov and Ledoux, we obtain various interesting inequalities as their realizations. For example, we draw a link between Ehrhard’s inequality and an interesting extension of Bobkov’s inequality, and several new and more general inequalities are discussed as well. In this talk we discuss a joint project with Barthe, Cordero-Erausquin and Ivanisvili, and also mention briefly some results from a joint project with Cordero-Erausquin and Rotem.

Numerical Methods for Optimal Transport Problems

Series
Dissertation Defense
Time
Friday, April 12, 2024 - 13:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 268
Speaker
Daniyar OmarovSchool of Mathematics, Georgia Tech

I will present numerical methods for solving the optimal transport (OT) problems in three settings. Firstly, I will discuss discrete OT problems from the perspective of linear programming and assignment problems. Additionally, I will provide a solution for a discrete problem with an obstacle in the domain.

Next, I will consider and compare several different numerical methods to solve the classic continuous OT problem with the squared Euclidean cost function. I will compare two numerical methods used for the fluid dynamics formulation with a direct discretization of the Monge-Ampère PDE. Furthermore, I will introduce a new class of problems called separable, for which very accurate methods can be devised. 

Lastly, I propose a novel implementation of Newton's method for solving semi-discrete OT problems for cost functions that are a positive combination of $p$-norms, $1

Riemannian geometry and contact topology IV

Series
Geometry Topology Working Seminar
Time
Friday, April 12, 2024 - 14:00 for 2 hours
Location
Skiles 006
Speaker
John EtnyreGeorgia Tech

This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series). 

Enumeration of interval graphs and d-representable complexes (Amzi Jeffs, CMU)

Series
Combinatorics Seminar
Time
Friday, April 12, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Speaker
Amzi JeffsCarnegie Mellon University

How many different ways can we arrange n convex sets in R^d? One answer is provided by counting the number of d-representable complexes on vertex set [n]. We show that there are exp(Theta(n^d log n))-many such complexes, and provide bounds on the constants involved. As a consequence, we show that d-representable complexes comprise a vanishingly small fraction of the class of d-collapsible complexes. In the case d = 1 our results are more precise, and improve the previous best estimate for the number of interval graphs.

Eremenko’s Conjecture and Wandering Lakes of Wada

Series
CDSNS Colloquium
Time
Friday, April 12, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
James WatermanStonybrook University

In 1989, Eremenko investigated the set of points that escape to infinity under iteration of a transcendental entire function, the so-called escaping set. He proved that every component of the closure of the escaping set is unbounded and conjectured that all the components of the escaping set are unbounded. Much of the recent work on the iteration of entire functions is involved in investigating properties of the escaping set, motivated by Eremenko's conjecture. We will begin by introducing many of the basic dynamical properties of iterates of an analytic function, and finally discuss constructing a transcendental entire function with a point connected component of the escaping set, providing a counterexample to Eremenko's conjecture. This is joint work with David Martí-Pete and Lasse Rempe.