Seminars and Colloquia Schedule

Lattices on shuffle words

Series
Algebra Seminar
Time
Monday, November 28, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Thomas McConvilleKennesaw State University

The shuffle lattice is a partial order on words determined by two common types of genetic mutation: insertion and deletion. Curtis Greene discovered many remarkable enumerative properties of this lattice that are inexplicably connected to Jacobi polynomials. In this talk, I will introduce an alternate poset called the bubble lattice. This poset is obtained from the shuffle lattice by including transpositions. Using the structural relationship between bubbling and shuffling, we provide insight into Greene’s enumerative results. This talk is based on joint work with Henri Mülle. 

A Nonlocal Gradient for High-Dimensional Black-Box Optimization in Scientific Applications

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Guannan ZhangOak Ridge National Laboratory (ORNL)

In this talk, we consider the problem of minimizing multi-modal loss functions with many local optima. Since the local gradient points to the direction of the steepest slope in an infinitesimal neighborhood, an optimizer guided by the local gradient is often trapped in a local minimum. To address this issue, we develop a novel nonlocal gradient to skip small local minima by capturing major structures of the loss's landscape in black-box optimization. The nonlocal gradient is defined by a directional Gaussian smoothing (DGS) approach. The key idea is to conducts 1D long-range exploration with a large smoothing radius along orthogonal directions, each of which defines a nonlocal directional derivative as a 1D integral. Such long-range exploration enables the nonlocal gradient to skip small local minima. We use the Gauss-Hermite quadrature rule to approximate the d 1D integrals to obtain an accurate estimator. We also provide theoretical analysis on the convergence of the method on nonconvex landscape. In this work, we investigate the scenario where the objective function is composed of a convex function, perturbed by a highly oscillating, deterministic noise. We provide a convergence theory under which the iterates converge to a tightened neighborhood of the solution, whose size is characterized by the noise frequency. We complement our theoretical analysis with numerical experiments to illustrate the performance of this approach.

Anticoncentration in Ramsey graphs and a proof of the Erdos-McKay conjecture

Series
Graph Theory Seminar
Time
Tuesday, November 29, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mehtaab SawhneyMassachusetts Institute of Technology

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size Clog n (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge-statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of "random-like’’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables.

The proof proceeds via an "additive structure’’ dichotomy on the degree sequence, and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics, and low-rank approximation. One of the consequences of our result is the resolution of an old conjecture of Erdos and McKay, for which he offered one of his notorious monetary prizes.
(Joint work with Matthew Kwan, Ashwin Sah and Lisa Sauermann)

Solving ODE eigenvalue problems with rigorous computation

Series
Time
Friday, December 2, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Blake BarkerBrigham Young University

https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

Abstract: ODE eigenvalue problems often arise in the study of stability of traveling waves, in showing the second variation of a functional is positive definite, and in many other applications. For many eigenvalue problems, it is not possible to obtain an explicit eigen pair. Thus, one uses numerical methods to approximate the solution. By rigorously bounding all errors in the computation, including computer rounding errors via use of an interval arithmetic package, one may obtain a computer assisted proof that the true solution lies in a small neighborhood of an approximation. This allows one to prove stability of traveling waves, for example. In this talk, we discuss recent work regarding computer assisted proof of stability of waves, and discuss other areas of application, such as in identifying most probable paths of escape in stochastic systems.
 

 

High-Girth Steiner Triple Systems

Series
Combinatorics Seminar
Time
Friday, December 2, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Ashwin SahMassachusetts Institute of Technology

We prove a 1973 conjecture due to Erdős on the existence of Steiner triple systems with arbitrarily high girth. Our proof builds on the method of iterative absorption for the existence of designs by Glock, Kü​hn, Lo, and Osthus while incorporating a "high girth triangle removal process". In particular, we develop techniques to handle triangle-decompositions of polynomially sparse graphs, construct efficient high girth absorbers for Steiner triple systems, and introduce a moments technique to understand the probability our random process includes certain configurations of triples.

(Joint with Matthew Kwan, Mehtaab Sawhney, and Michael Simkin) ​

Evolutionary de Rham-Hodge method and its applications in SARS-CoV-2 studying

Series
Mathematical Biology Seminar
Time
Friday, December 2, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jiahui ChenMichigan State University -- Department of Mathematics

The classroom version of this event will be held in Skiles 005. Everyone on campus at Georgia Tech is highly encouraged to attend this version. The virtual version will be administered through Zoom (https://gatech.zoom.us/j/99514218896).

This talk will discuss an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exact topological invariants. To demonstrate the utility of the proposed method, the application is considered for the predictions of binding free energy (BFE) changes of protein-protein interactions (PPIs) induced by mutations with machine learning modeling. It has a great application in studying the SARS-CoV-2 virus' infectivity, antibody resistance, and vaccine breakthrough, which will be presented in this talk.