Below you can find more detailed information about the mini-courses.
Lecturer: Teresa Conde
Duration: 4 hours
Title: On characters and additive functions
Abstract: The notion of dimension or rank provides a measure of the 'size' of algebraic structures. In the context of modules over a ring, the rank of a module is a fundamental invariant that often signifies the maximal number of linearly independent elements. This concept extends naturally to the study of Sylvester rank functions for modules and to characters in the more general context of locally finitely presented categories. It may also be adapted to triangulated categories and other settings. In these lectures, we shall cover some basic results on Sylvester rank functions and explain how functorial methods can be used to derive properties of characters by relating them to more amenable 'measure' functions on abelian categories, called additive functions. We shall also explore the relation between characters, the Ziegler spectrum and endofinite objects, and visit the realm of rank functions on triangulated categories.
Lecturer: William Crawley-Boevey
Duration: 2 hours
Title: Matrix reductions and endofinite modules
Abstract: In the first lecture I would like to explain the idea of matrix reductions, as developed by Roiter, Kleiner, Drozd and others in the 1970s, and I would like to explain some of the different possible formulations - bimodule problems, lift categories, dgas, ditalgebras and bocses. Two theorems I proved in the 1990s relate representation type to endofinite modules: (1) an artin algebra has strongly unbounded representation type if and only if it has a generic module, and (2) a finite dimensional algebra over an algebraically closed field has tame representation type if and only if it has only finitely many generic modules of any given endolength. Both are proved by matrix reductions; in the second lecture I would like to explain some key steps.
Lecturer: Ivo Herzog
Duration: 4 hours
Title: Pure injectives and the Ziegler spectrum
Abstract: The mini course will focus on the Ziegler Spectrum of a ring, exploring the foundations of the theory as well as examples. The techniques will offer a mix of model theoretic and category theoretic concepts that include the free abelian category and its Serre subcategories, positive primitive formulae and types, pure injective and pseudo-finite dimensional representations, and the Grothendieck group. This choice of topics will prepare the audience to understand some outstanding open questions about the Ziegler spectrum of an artin algebra.
Lecturer: Henning Krause
Duration: 2 hours
Title: Finite length in triangulated categories
Abstract: Various notions of finite (cohomological) length in triangulated categories will be discussed; they are closely related to the corresponding notion for abelian categories. For instance, we will consider endofinite objects and discuss several interesting examples. Another notion of finite length involves the central action of a cohomology ring. A useful consequence of finite length is a Krull-Schmidt property.
Lecturer: Jan Št'ovíček
Duration: 4 hours
Title: On (co)silting modules and torsion pairs
Abstract: The purpose of the course is to give an introduction to tilting/silting theory and its connection to model theory of modules. In the first half of the course, we will discuss motivation for (co)tilting objects, relation to torsion pairs and t-structures, and the important result that cosilting objects arising in the nature are pure-injective. In favourable situations, cosilting objects (up to product equivalence) bijectively correspond to maximal rigid subsets in a suitable part of the Ziegler spectrum. In the second half of the course, we will explain combinatorial aspects of large cosilting complexes - the mutation procedure and the relation to the lattice of torsion pairs in module categories of artin algebras. This extends results of Demonet, Iyama, Reading, Reiten and Thomas. Time permitting, we will illustrate the theory on further classes of examples.
Slides of presentation by Diego Alberto Barceló Nieves (click here)
Notes for the second lecture of the minicourse, by Diego Alberto Barceló Nieves (click here)