NEWS in the Department of Mathematics
Rachel Neville - PhD preliminary examination
Date: Tuesday, December 13, 2016
Place: Engineering E-205
Time: 9:00 a.m.
Title: Topological Techniques for Characterization of Patterns in Differential Equations
Advisor: Dr. Patrick Shipman
Committee: Dr. Chris Peterson, Dr. Clayton Shonkwiler, Dr. Henry Adams, Dr. Amber Krummel
Abstract: Persistent homology has proven to be a useful tool for characterizing spatio-temporally complex pattern by capitalizing on the geometric and topological structure of data arising from differential equations. In this dissertation proposal, we discuss various methods for characterizing real and simulated complex data by leveraging the topological structure. First, we will discuss a pattern arising in the persistence diagram of a class of one-dimensional discrete dynamical systems--even in chaotic parameter regimes. This pattern reflects the underlying dynamics of the system. Next, we will discuss using topological characteristics as a way to learn parameters driving pattern-forming systems. We make use of persistence images to represent persistence diagrams so that machine learning techniques may be leveraged in this endeavor. This technique is modified and applied to several different systems including the anisotropic Kuramoto-Sivashinsky equation displaying chaotic bubbling and nanodot formations that emerge when a binary compound is bombarded with ions. Last, persistence can provide a way to gauge order and measure roughness in surfaces. We create a persistence-based measures of order of nearly hexagonal lattices, and roughness for snowpack surfaces.