[Topology.beta] CTAAAG and colloquium.

[Rot, T.O. (Thomas)]

This is happening soon!

From: "Rot, T.O. (Thomas)" <t.o.rot at vu.nl>
Date: Sunday, 23 February 2025 at 20:42
To: "_list_topology.beta" <topology.beta at listserver.vu.nl>
Subject: CTAAAG and colloquium.

Dear all,

This week on Tueday 11:00 we have Ran Levi in the CTA^3G. Ran gifts us a second talk in the colloquium on Wednesday at 16:00.

Hope to see you at both events!

Thomas


CTAAAG: Foundations of Differential Calculus for modules over small categories

Let 𝑘 be a field and let C be a small category. A 𝑘-linear representation of C, or a 𝑘C-module, is a functor from C to the category of finite dimensional vector spaces over 𝑘. A motivating example for this work is the concept of a tame generalised persistence module, which can be reduced to the case where C is a finite poset. Unsurprisingly, it turns out that when the category C is more general than a linear order, then its representation type is generally infinite and in most cases wild. Hence the task of understanding such representations in terms of their indecomposable factors becomes difficult at best, and impossible in general. In a joint project with Jacek Brodzki and Henri Rihiimaki we proposed a new set of ideas designed to enable studying modules locally. Specifically, inspired by work in discrete calculus on graphs, we set the foundations for a calculus type analysis of 𝑘C-modules, under some restrictions on the category C. In this talk I will review the basics of the theory and describe some more recent advances.


Colloquium: Title: p-local group theory: Groups from a homotopy theoretic point of view

Abstract: A standard link between groups and topological spaces is the classifying space construction due to Milnor, designed originally for the classification of principal G-bundles over an arbitrary space X, where G is a topological group. If G is discrete, then its classifying space BG is a space whose only non vanishing homotopy group is its fundamental group which is G itself. For finite groups G, its cohomology with coefficients in a p-local abelian group is given by the Cartan-Eilenberg stable elements theorem, which shows that it depends only on the Sylow p-subgroup of G and the conjugacy relations of subgroups within it. This motivates the application of the Bousfield-Kan p-completion functor to BG, which is a process that does not change the mod-p cohomology of BG, but in general endows the resulting space with arbitrarily many non-vanishing homotopy groups. Thus p-completion turns classifying spaces into a family of quite mysterious objects on one hand, and at the same time related to topics such as stable homotopy groups and algebraic K-theory. The project I will report on in this talk has its roots in the late 1990s in collaboration with Bob Oliver and Carles Broto. A p-local group is not a group but rather an algebraic object that encodes the p-local algebraic and homotopical properties that make a p-completed classifying space. The theory enables a better understanding of the p-local homotopy theory of finite, compact Lie and other types of groups, and also includes many examples of exotic objects, namely such that do not arise from a group. Central concepts in p-local group theory are also strongly related to ideas in group theory and representation theory. In the talk I will attempt to introduce the background and cover some of the main achievements of p-local group theory and remaining challenges.


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