Seminars and Colloquia by Series

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

Please Note: 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.

Using Morse homology to understand persistence modules II

Series
Geometry Topology Working Seminar
Time
Friday, December 2, 2022 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Daniel IrvineGeorgia Tech

Morse theory and Morse homology together give a method for understanding how the topology of a smooth manifold changes with respect to a filtration of the manifold given by sub-level sets. The Morse homology of a smooth manifold can be expressed using an algebraic object called a persistence module. A persistence module is a module graded by real numbers, and in this setup the grading on the module corresponds to the aforementioned filtration on the smooth manifold.

This is the second of a series of talks that aims to explain the relationship between Morse homology and persistence modules. In this second talk, I will define persistence modules, explain how to compute Morse homology using persistence modules, and explain how the Künneth theorem and the cup product work with persistence modules. The material from the first part of this series will be assumed.

Sparse Quadratic Optimization via Polynomial roots

Series
ACO Student Seminar
Time
Friday, December 2, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kevin ShuGeorgia Tech Math

We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.

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.
 

 

Quantum mechanics and diffusion on metric graphs

Series
Math Physics Seminar
Time
Thursday, December 1, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Evans HarrellSchool of Mathematics, Georgia Tech

Quantum mechanics and diffusion on a network, in the sense of a metric graph, are locally one-dimensional, but the way the graph is connected can add multidimensional features and some strange phenomena.  Quantum graphs have been an active area of research since the 1990s.  I’ll review the subject and share some ideas about analyzing Schrödinger and heat equations on metric graphs, through the associated eigenvalue problem and the heat kernel.

This talk is based on a 2022 article with David Borthwick and Kenny Jones, and on work in progress with David Borthwick, Anna Maltsev, and Haozhe Yu. 

Everything Alexander in the context of mapping class group

Series
Geometry Topology Student Seminar
Time
Wednesday, November 30, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jaden WangGeorgia Tech

Disks are nice for many reasons. In this casual talk, I will try to convince you that it's even nicer than you think by presenting the Alexander's lemma. Just like in algebraic topology, we are going to rely on disks heavily to understand mapping class groups of surfaces. The particular method is called the Alexander's method. Twice the Alexander, twice the fun! No background in mapping class group is required.

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)

Intersection number and intersection points of closed geodesics on hyperbolic surfaces by Tina Torkaman

Series
Geometry Topology Seminar
Time
Monday, November 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Tina TorkamanHarvard University

 In this talk, I will talk about the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I will discuss the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.

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.

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. 

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