Quiver Representations in Topological Data Analysis II

Magnus Botnan (Vrije Universiteit Amsterdam)

11-Nov-2020, 09:10-10:00 (3 years ago)

Abstract: The goal of these three lectures is to highlight the role of quiver representations in the field of topological data analysis (TDA). Emphasis will be put on the interplay between the pure and applied. Familiarity with simplicial (co-)homology will be assumed.

Lecture 1: Persistent homology in a single parameter

Persistent homology is a central topic in the burgeoning field of topological data analysis. The key idea is to study topological spaces constructed from data and infer the ‘‘shape’’ of the data from topological invariants. The term ‘’persistent’’ refers to the fact that the construction of these spaces usually depends on one or more parameters, and in order to obtain information about the data in a stable and robust way, it is crucial to consider how the family of resulting invariants relate across scales. This naturally leads to a representation of a totally ordered set.

In this first lecture I will motivative persistent homology in a single parameter, introduce the necessary terminology, and state foundational results.

Lecture 2: Multiparameter persistent homology part 1

Multiparameter persistent homology is a vibrant subfield of topological data analysis which has attracted much attention in recent years. It has become evident that the transition from a single to multiple parameters comes with significant computational and mathematical challenges. At the level of representation theory, this can be understood by the fact that one is studying representations of a partially ordered set of wild representation type.

In this lecture we shall identify settings for which the theory in the first lecture generalizes to more general posets. Of particular interest is level-set zigzag persistent homology.

Lecture 3: Multiparameter persistent homology part 2

In this lecture we will consider models for constructing representations of posets for which most of the theory developed in the first lecture does not generalize in a reasonable way. However, we shall see that we still can extract useful invariants for the purpose of data analysis. Our primary motivation will come from clustering (in the data-scientific sense).

Mathematics

Audience: researchers in the discipline


ICRA 2020

Series comments: The Workshop and International Conference on Representations of Algebras (ICRA) will take place online between 9th November and 25th November 2020.

Visit our website to register and for further information: www.icra2020.info

Deadline for submitting research snapshots: November 1st, 2020

Organizers: Lidia Angeleri Hügel, Aslak Bakke Buan, Gustavo Jasso*, Henning Krause, Rosanna Laking, Øyvind Solberg
*contact for this listing

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