08:45 - 09:30 |
Woojin Kim: The Generalized Rank Invariant: Möbius invertibility, Discriminating Power, Computation, and Connection to Other Invariants ↓ Unlike one-parameter persistent homology, the absence of a canonical method for quantifying ‘persistence’ in multiparameter persistent homology remains a hurdle in its application. One of the best-known quantifications of persistence for multiparameter persistent homology, or more broadly persistence modules over arbitrary posets, is the rank invariant. Recently, the rank invariant has evolved into the generalized rank invariant by naturally extending the domain of the rank invariant to the collection of all connected subposets of the domain poset. This extension enables us to measure ’persistence’ across a broader range of regions in the indexing poset compared to the rank invariant. Additionally, restricting the generalized rank invariant can enhance computational efficiency, albeit with a potential trade-off in discriminating power. This talk overviews various aspects of the generalized rank invariant: Möbius invertibility, discriminating power, computation, and its relation to other invariants of multiparameter persistence modules. (TCPL 201) |
09:45 - 10:30 |
Dolors Herbera: An approach to relative homological algebra for persistence modules ↓ The aim of this talk is to present some notions of relative homological algebra that are proving to be useful in the developing of the theory of persistence modules. We will follow closely Sections 3, 4 and 5 of the nice paper [BBH},
which in turn follows the track of the theory of relative homological algebra developed by Auslander and Soldberg for artin algebras in [AS], and that was extended to more general settings in [DRSS].
Let A be an abelian category, and fix a class of objects X. Let FX denote the class of short exact sequences in A that remain exact when applying the covariant functor HomA(X,−) for any X∈X. Dually, let FX denote the class of exact sequences that remain exact when applying the contravariant functor HomA(−,X) for any X∈X.
Both FX and FX define exact estructures over A, so one can make relative homological algebra with respect to both of them. Basic problems, in this setting, are to determine the relative projective objects and the relative injective objects, whether such classes of relative projectives/injectives are resolving/corresolving, whether there are minimal resolutions/corresolutions, do we have relative homological invariants? can we compute relative global dimensions?.
The answer to such questions, in general, is difficult and we will outline solutions in settings that, according to [BBH], are of interest for persistence theory.
References:
[AS] M. Auslander and Ø.\ Solberg, Relative homology and representation theory I: relative homology and homologically finite subcategories. Comm. Alg. 21 (1993), no. 9, 2995-3031.
[BBH] BENJAMIN BLANCHETTE, THOMAS BRUSTLE, AND ERIC J. HANSON. Exact Structures for Persistence Modules. arXiv:2308.01790 (2023).
[DRSS] P. Dräxler, I. Reiten, S. O. Smalø , and Ø. Solberg, with an appendix by B. Keller, Exact categories and vector space categories. Trans. Amer. Math. Soc. 351 (1999), no. 2, 647-682. (TCPL 201) |
11:00 - 11:45 |
Eric Hanson: Homological invariants of persistence modules ↓ A common approach to studying multiparameter persistence modules is to introduce some "invariant" to determine the similarity between two given modules. In this mostly expository talk, we discuss recent research which utilizes techniques from (relative) homological algebra to interpret classical examples of invariants and define new invariants. The Hilbert function/dimension vector, barcode, and (generalizations of) the rank invariant serve as our main examples. If time permits, we will also discuss the relationship between homological invariants and poset embeddings. Portions of this talk are based on joint works with Claire Amiot, Benjamin Blanchette, and Thomas Brüstle. (TCPL 201) |