# Graduate Student Seminar-Persistent Homology, Spectral Sequences, and Topological Data Analysis

Date and Time: Wednesday, April 26th 4-5 p.m.

Place: Rome 206

Speaker: Yingfeng Hu, GWU

**Title:** Persistent Homology, Spectral Sequences, and Topological Data Analysis

**Abstract:** Persistent homology and spectral sequences are powerful mathematical tools for analyzing topological features of data. In this presentation, we will introduce the concept of spectral sequences and persistent homology and demonstrate how they can be constructed from a filtered chain complex. We will also show how to obtain a filtered chain complex from a data cloud and discuss how spectral sequences and persistent homology are connected. The presentation will begin with an overview of the basic concepts of homology and cohomology. We will then introduce the filtration of a chain complex, which is the first step towards constructing a spectral sequence. Next, we will show how to define persistent homology, which captures the evolution of topological features across different scales of the data. We will demonstrate how to construct spectral sequences and persistent homology from a filtered chain complex, and how to use these tools to analyze data. We will also discuss how spectral sequences and persistent homology are related, and how they can be used in tandem to gain a more complete understanding of the topological structure of data. This presentation is aimed at a general mathematical audience with a basic understanding of algebraic topology. No prior knowledge of spectral sequences or persistent homology is required, as we will provide all necessary background material. By the end of the presentation, attendees will have a solid understanding of how to use spectral sequences and persistent homology to analyze data and extract topological information.