COURSE 2: Algebraic structures in Topological Data Analysis: Persistent homology
Lecturers: Thomas Brüstle | Université de Sherbrooke and Bishop’s University

Course Description:

Topological data analysis (TDA) uses topology to identify relevant geometric features of data, such as clusters and loops. This is accomplished by first modelling the data by a family of topological spaces indexed over a poset, and then identifying the topological features that persist over several indices. The field of algebraic topology provides tools to TDA via the language of homology, and persistent homology is the branch of TDA that makes use of methods from representation theory in order to study these representations of posets, referred to as persistence modules.

In this course we will present the basic theory of pointwise finite persistence modules with coefficients in a field. This includes classification results for one-parameter persistence modules where the poset is totally ordered, as well as description of invariants of persistence modules in the multiparameter setting. We will also discuss some notions that are useful in the applications, such as the stability of an invariant, which, roughly speaking means that two persistence modules that come from similar data should have close values of their corresponding invariants.

We will also show how the theory of persistence modules helps in practice in the study of the shape of data.

Course Materials:

Preliminary Readings:
+ Steve Oudot: Persistence Theory: From Quiver representations to Data Analysis, [ Ссылка ]... - Chapter 1

+ Magnus Bakke Botnan and Michael Lesnick: An Introduction to Multiparameter Persistence - [ Ссылка ]

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