Talk Title: Quasiperiodicity, sliding window embeddings and the Künneth theorems
Event: Topology, Geometry, and Data Analysis Seminar
Where: Department of Mathematics, The Ohio State University
Date: October 1, 2019
Abstract: Classically, sliding window embeddings were used in the study of dynamical systems to reconstruct the topology of underlying attractors from generic observation functions. In 2015, Perea and Harer developed a technique for recurrence detection in time series data using sliding window embeddings of L^2 periodic functions and persistent homology---which is an algebraic and computational tool used to quantify multiscale features of shapes. We define a quasiperiodic function as a superposition of periodic functions with non-commensurate harmonics. It turns out that sliding window embeddings of quasiperiodic functions are dense in high dimensional tori. The study of persistence of Rips filtration on these embeddings motivated our work on persistent Künneth theorems---that is, results relating persistent homology of two filtered spaces to persistent homology of their products (we define two such products). In this talk, I will present some theoretical results and demonstrate that in certain cases, a Künneth theorem helps recover quasiperiodicity better.