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: geometric motivations, metric stability, and representation theory of posets (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 lecture on Wednesday). Lectures will be between 9:30-12:00, with a 30-minute break between the two lectures.

Location: OIST campus, Center Building, Room C209

Zoom Link: Will be posted closer to the event!

※ 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※

On Wednesday (July 2), at 14:00, there will be an Expository Talk by Prof. Hideto Asashiba on the related topics. Further information can be found here.

• 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: geometric motivations, metric stability, and representation theory of posets
  Abstract: The theory of persistence [1] 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? Initially motivated by geometric data science, persistence has since found applications in several areas of geometry and analysis [2].
  
  The first part of this minicourse will be geared towards beginners, with motivation and an overview of one-parameter persistence, including the bottleneck stability theorem, its fundamental result. The second part will focus on multiparameter persistence [3]. Specifically, motivations from geometric data science and analysis, and recent work connecting two-dimensional Morse theory with the representation theory of the bigraded polynomial ring in two variables, shedding new light on these classical objects [4]. I will assume some familiarity with rings, modules, categories, functors, topological spaces, and homology.
  
  [1] Persistence Theory: From Quiver Representations to Data Analysis. Steve Oudot
  [2] Topological Persistence in Geometry and Analysis. Leonid Polterovich, Daniel Rosen, Karina Samvelyan, Jun Zhang
  [3] An Introduction to Multiparameter Persistence. Magnus Bakke Botnan, Michael Lesnick
  [4] Counts and end-curves in two-parameter persistence. Thomas Brüstle, Steve Oudot, Luis Scoccola, Hugh Thomas