Fri, 26 Nov 2021

15:00 - 16:00
Virtual

Morse inequalities for the Koszul complex of multi-persistence

Claudia Landi
(University of Modena and Reggio Emilia)
Abstract

In this talk, I'll present inequalities bounding the number of critical cells in a filtered cell complex on the one hand, and the entries of the Betti tables of the multi-parameter persistence modules of such filtrations on the other hand. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for the number of critical cells. Furthermore, we prove a sharp upper bound for the minimal number of critical cells, expressed again in terms of the entries of Betti tables. This is joint work with Andrea Guidolin (KTH, Stockholm). The full paper is posted online as arxiv:2108.11427.

Fri, 09 Oct 2020

15:00 - 16:00
Virtual

Invariants for tame parametrised chain complexes

Barbara Giunti
(University of Modena and Reggio Emilia)
Abstract

Persistence theory provides useful tools to extract information from real-world data sets, and profits of techniques from different mathematical disciplines, such as Morse theory and quiver representation. In this seminar, I am going to present a new approach for studying persistence theory using model categories. I will briefly introduce model categories and then describe how to define a model structure on the category of the tame parametrised chain complexes, which are chain complexes that evolve in time. Using this model structure, we can define new invariants for tame parametrised chain complexes, which are in perfect accordance with the standard barcode when restricting to persistence modules. I will illustrate with some examples why such an approach can be useful in topological data analysis and what new insight on standard persistence can give us. 

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