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Introduction to persistent homology

Abstract

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Next lectures
26.06.2024, 13:00 (E2 10 (Leon-Lichtenstein))
03.07.2024, 13:00 (E2 10 (Leon-Lichtenstein))
05.07.2024, 11:00 (E2 10 (Leon-Lichtenstein))
In this mini-course, we introduce persistent homology of a continuous function on a topological space. We define two descriptors, called the persistence Betti numbers functions and persistence diagrams and study the relations between them. In particular, we show that they are equivalent topological descriptions of the function and that one can be recovered from the other. We conclude by showing that persistent homology is a stable descriptor of a continuous function with respect to the uniform norm.

The lectures will cover the following topics:

  • May 29, 13-14:30. Persistent homology group, persistent Betti numbers function (PBNF), monotonicity of PBNF
  • June 12, 13-14:30. Discontinuities of PBNF, multiplicity
  • June 19, 13-14:30. Persistence diagram (PD), cornerpoints of PD
  • June 26, 13-14:30. Discontinuities of PD, local finiteness
  • July 3, 13-14:30. Representation theorem
  • July 5, 11-12:30. Bottleneck distance, stability theorem

Keywords
Persistence diagram, persistence Betti numbers function, bottleneck distance

Prerequisites
Basic notions of linear algebra and calculus are enough. Knowing the definition of homology may also help.

Audience
Anyone who is interested in Topological Data Analysis

Language
English

lecture
01.04.24 31.07.24

Regular lectures Summer semester 2024

MPI for Mathematics in the Sciences / University of Leipzig see the lecture detail pages

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail

Upcoming Lectures in this Semester