Mini-courses on Topological Data Analysis (by Dr. Emerson Escolar & Dr. Luis Scoccola)

These two introductory mini-courses are offered by experts in Topological Data Analysis, as part of the thematic program TDA PARTI: Topological Data Analysis, Persistence And Representation Theory Intertwined (TP25TD).

• Invariants for persistence (by Dr. Emerson Escolar)
• Introduction to persistence and bottleneck stability (by Dr. Luis Scoccola)

Each mini-course consists of four lectures. Each one of the two lecturers will give a 1-hour talk on Monday, Tuesday, Thursday and Friday (there will be no talk on Wednesday). Lectures will be between 9:30-12:00, with a 30-minute break between the two lectures.

***Further information on the mini-courses and the schedule will be posted on the main website of the thematic program TDA PARTI (TP25TD): https://www.oist.jp/visiting-program/tp25td

※ Please note that these lecture series will be streamed via Zoom and may be recorded. Upon the consent of the lecturer, the mini-course may be uploaded on the OIST website※

• Miniourse by Dr. Emerson Escolar.
  Title: Invariants for Persistence
  Abstract: Persistent homology is one of the main tools of topological data analysis (TDA) and has been applied to many problems in various scientific fields. In its theoretical foundation, central is the concept of persistence modules, i.e., representations of the poset Rn (or perhaps more conveniently the n-dimensional commutative grid viewed as a finite discretization) in the category of vector spaces. This introductory minicourse focuses on the “structure” of persistence modules. In the first part, I review the well-understood case of n = 1. In particular, I discuss the persistence barcode from multiple points of view (including the fact that it is obtained from an indecomposable decomposition of a persistence module into intervals). It is known that the n-dimensional (with n > 1) commutative grid of large enough size is of wild representation type, and so classifying their indecomposables is virtually impossible. Hence many algebraic invariants have been developed in TDA. In the second (and main) part of this minicourse, I review some of the invariants that have been proposed, including (generalized) rank invariants and compressed multiplicities, their M¨obius inversions, and invariants coming from (relative) homological algebra. Furthermore, identifying the relationships among invariants, or ordering them by their discriminating power, is a fundamental question. Thus, in the final part of this minicourse I discuss some very recent work about comparing invariants using their discriminating powers. Some familiarity with the following will be assumed: basic (linear) algebra and basic definitions from category theory.
   
• Miniourse by Dr. Luis Scoccola​.
  Title: Introduction to persistence and bottleneck stability
  Abstract: Persistent  theory was born from the observation that the critical values of a Morse function admit a canonical pairing, which induces a direct sum decomposition of the sublevel set homology of the function into interval poset representations. What makes persistence theory distinct from Morse theory - besides the fact that it is usually framed using the language of representation theory - is its focus on perturbation-stability, providing answers to questions such as: How can the critical values of a Morse function change when the function is perturbed? The initial motivation for the study of perturbation-stability came from problems in geometric data science, but has since found applications in symplectic geometry and functional analysis. I will give an overview of the representation theoretic and combinatorial aspects of persistence theory, including motivation and applications. I will assume that participants are familiar with the concepts of ring, module, category, and functor.