Random Topology and its Applications
Abstract: The study of random topology focuses on describing high-level qualitative properties of random spaces. This field has been rapidly developed in the past decade, and is driven both by deep theoretical questions and state-of-the-art data analysis applications. Topological Data Analysis (TDA) broadly refers to the use of concepts from mathematical topology to analyze data and networks. In the past decade a variety of powerful topological tools has been introduced, and were proven useful for applications in various fields (e.g. shape analysis, signal processing, neuroscience, and genomic research). The theory developed in random topology aims to provide a solid foundation for the statistical analysis of these methods, which to date is at a very preliminary stage.
This talk will be divided into three parts. Firstly, I will provide an introduction and motivation to random topology and TDA. Next, we will discuss the theory of random geometric complexes. In particular, we will consider their Betti numbers (counting “cycles” in different dimensions), and phase transitions related to these. Finally, we will present a few examples of combining the theory of random complexes with statistical problems related to TDA. In particular, we are interested in analyzing ``topological noise” that appears in such problems, for the purposes of filtering and hypothesis testing.