2 Expository Talks by Dr. Raphael Bennett-Tennenhaus (Decompositions and classifications, a view towards persistence modules)

Speaker: Dr. Raphael Bennett-Tennenhaus (Bielefeld University, Germany)

Title: Decompositions and classifications with a view towards persistence modules

Abstract: Every point-wise finite-dimensional persistence module admits a decomposition into indecomposables with local endomorphism rings, and any such direct sum is unique up to an isoclass-preserving bijection of summands. This theorem is important in persistent homology since, for example, it ensures the existence and definiteness of the barcode. I will recover this theorem using more general tools, highlighting historical developments along the way.

I will begin by noting the failure of uniqueness when endomorphism rings are not local, or when there are not enough compact objects. Having noted how things go wrong, I will show how to fix them using approximation conditions on the ambient category that are sufficient for uniqueness. For the existence of a decomposition, one can apply finiteness conditions coming from mathematical logic, far milder than requiring point-wise finite-dimensional modules. By combining the approximation and finiteness conditions, I will consider applications, to infinite species, and to derived categories of persistence modules.

Stepping back, I will then survey results where the indecomposables are classified, and consider variations of these decomposition theorems, more in line with the spirit of persistent homology. Time permitted, I will discuss Morita equivalence for functor categories and discuss the notion of corner replacement.

*Talk 1: 14:00-15:00, July 14, 2025

*Location: OIST, Lab 3, Floor C, Room C700

**Talk 2: 11:00-12:00, July 15, 2025

**Location: OIST, Lab 5, Floor D, Room L5D23

Zoom Link: Will be posted closer to the event!

This talk is part of the Thematic Program TDA PARTI: Topological Data Analysis, Persistence And Representation Theory Intertwined.