Persistent Homology for Random Fields and Complexes

TL;DR

This paper reviews recent advances in applied algebraic topology, focusing on persistent homology and barcodes, and explores their applications in manifold learning, random complexes, and the topology of random fields' excursion sets.

Contribution

It provides a comprehensive review of recent developments in persistent homology and their applications to random fields and complexes, highlighting new insights and methods.

Findings

Connections between persistent homology and manifold learning.

Analysis of algebraic structures of random simplicial complexes.

Insights into the topology of excursion sets of random fields.

Abstract

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the algebraic topology of the excursion sets of random fields.