Department Seminar by Takashi Owada

Please Note: This seminar will be given online.

Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex.

Topological Data Analysis (TDA) is a growing research area that broadly refers to the analysis of high-dimensional datasets, the main goal of which is to extract robust topological information from datasets. Among many fields in TDA, this talk deals with the problems related to the time evolution of topological structure. More specifically, we shall consider the multi-parameter simplicial complex model as a higher-dimensional generalization of the Erdos-Renyi graph.  Topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since the randomness there is usually governed by a single parameter. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex and, more importantly, of its dynamic analogue that we introduce here. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. The dynamic versions of the clique complex and the Linial-Meshulum complex are special cases of our setup. Our key result concerns the regime where face-counts of a particular dimension dominate. We show that the Betti numbers (i.e., basic quantifiers of topological complexity) corresponding to this dimension, and the Euler characteristic satisfy functional strong law of large numbers and functional central limit theorems. Surprisingly, in the latter result, the limiting Gaussian process depends only upon the dynamics in the smallest non-trivial dimension. This is joint work with Gennady Samorodnitsky (Cornell) and Gugan Thoppe (Indian Institute of Science).