14:15: Thomas Brüstle (Université de Sherbrooke)
Homological approximations in persistence theory
Multiparameter persistence modules appear in topological data analysis when dealing with noisy data. They are defined over a wild algebra and therefore they do not admit a complete discrete invariant. One thus tries in persistence theory to “approximate” such a module by a more manageable class of modules. Using that approach we define a class of invariants for persistence modules based on ideas from homological algebra. This is a report on joint work with Claire Amiot, Benjamin Blanchette and Eric Hanson. No prior knowledge of topological data analysis is required.
15:15 Amit Shah (Aarhus Universitet)
Characterising Jordan-Hölder extriangulated categories via Grothendieck monoids
Abstract: The notions of composition series and length are well-behaved in the context of abelian categories. And, in addition, each abelian category satisfies the so-called Jordan-Hölder property/theorem. Unfortunately, these ideas are poorly behaved for triangulated categories. However, with the introduction of extriangulated categories, it is interesting to see what sense we can make of these concepts for extriangulated categories.
I'll present a result that characterises Jordan-Hölder, length extriangulated categories using the Grothendieck monoid of an extriangulated category. This is motivated by the exact category setting as considered by Enomoto, in which it becomes apparent that Grothendieck monoid is more appropriate to look at than the Grothendieck group. I'll present some examples coming from stratifying systems. In fact, developing stratifying systems for extriangulated categories was the original motivation of our article. This is joint work with Thomas Brüstle, Souheila Hassoun and Aran Tattar.
16:15 Henrik Holm (Københavns Universitet)
Compact and perfect objects in triangulated categories of quiver representations.
A complex of modules over a ring, A, can be viewed as an A-module valued representation of a certain quiver with relations, Q. The vertices of Q are the integers, there is an arrow q -> q-1 for each integer q, and the relations are that consecutive arrows compose to zero. Hence the classic derived category of A can be viewed as a triangulated category of representations of Q. It is well-known that the derived category is compactly generated and that the compact objects are precisely the so-called perfect complexes, i.e. complexes that are isomorphic to a bounded complex of finitely generated projective A-modules.
While complexes do play a prominent role, other types of quiver representations are important as well. It turns out that if Q is a suitably nice quiver with relations, then there exists a triangulated category whose objects are the A-module valued representations of Q. This category is called "the Q-shaped derived category" of the ring A.
We will prove that the Q-shaped derived category is always a compactly generated triangulated category. We will also extend the notion of perfect complexes (in the classic derived category) to perfect objects in the Q-shaped derived category. It turns out that, up to direct summands, perfect and compact objects in the Q-shaped derived category are the same.
The talk is based on a joint paper (arXiv:2208.13282) with Peter Jørgensen from Aarhus University.