Session 1, Monday (Sep 25)
10:00 EDT (4:00 CET):
Peter Bubenik (Invited Talk). Persistence: from Algebra to Analysis
Abstract: Consider the following view of topological data analysis. The first and most difficult step consists of the computation of useful invariants from data. The theoretical foundation for these computations is algebra. The second step is the computation of vector summaries of these invariants, which may then be used in statistics and machine learning. The theoretical foundation for these computations is functional analysis. As the algebraic invariants being computed are increasing in sophistication, we need a better analytic theory for persistence.
This is joint work with Alex Elchesen.
10:30 EDT (4:30 CET):
Jiajie Luo and Gregory Henselman-Petrusek. Algorithmic Interval Decomposition for Persistence Modules of Free Abelian Groups
Abstract: The theory of single-parameter persistence is intimately tied to Gabriel’s Theorem, which states that any functor F from the poset category of finite totally-ordered set I into a category of finitedimensional vector spaces admits an interval decomposition – that is, a decomposition as a direct sum of interval modules.
This theorem typically fails when one loosens the restriction on the target category. In this talk, we introduce a necessary and sufficient condition for the existence of interval decompositions in singleparameter persistence modules valued in the category of finitely-generated free abelian groups, and group homomorphisms. This work complements and informs earlier work characterizing filtered topological spaces whose persistence diagrams are independent of the choice of ground field.
We also provide a polynomial-time algorithm to either (a) compute an interval decomposition of a module of free abelian groups, or (b) certify that no such decomposition exists.
11:00 EDT (5:00 CET):
Isaac Ren. Computing relative Betti diagrams of multipersistence modules using Koszul complexes
Abstract: Relative homological algebra is a fruitful source of numerical invariants for multidimensional persistence modules and modules over arbitrary posets. In this talk, we focus on relative resolutions, which are exact sequences of persistence modules that are projective, in some sense, relative to a given family of “simple” modules. Under certain conditions on this family, these resolutions consist of direct sums of simple modules, whose multiplicities we can then collect in the so-called relative Betti diagrams. We can then compute the these relative Betti diagrams using Koszul complexes, which is simpler than directly computing the full relative resolutions.
This is a joint work with Wojciech Chachólski, Andrea Guidolin, Martina Scolamiero, and Francesca Tombari.