The Legacy of Peter Gabriel – Abstracts
Cleaving diagrams, coverings and the multiplicative bases theorem
Raymundo Bautista
The techniques for giving the structure theorems in the multiplicative bases theorem are based on cleaving diagrams. For this, it is proved that some configurations are forbidden for a quotient of a quiver algebra modulo an admissible ideal, if this algebra is of finite representation type. This is done using cleaving diagrams.
In the talk we consider two examples of the use of cleaving diagrams. We will see in one of the examples its relation with coverings.
Finite representation type is open - then and now
Grzegorz Bobiński
The aim of talk is to present fundamental ideas from a seminal Gabriel's paper "Finite representation type is open". We also plan to discuss how this paper inspired studies of geometric problems in the representation theory of finite dimensional algebras in the last 50 years.
On representation finite and minimal representation infinite algebras
Klaus Bongartz
We discuss the following results for algebras and their modules of finite dimension over an algebraically closed field:
The ray category defined for any distributive algebra in the article on multiplicative bases and its universal covering play a central role in the proofs.
Commutative algebraic groups
Michel Brion
The talk will first present structure results for commutative algebraic groups from the book of Demazure and Gabriel, and related work of Serre and Oort. It will then discuss some recent developments and open questions.
Matrix Problems
Thomas Brüstle
We present matrix problems which are introduced prominently as the opening chapter in the book Representations of Finite-Dimensional Algebras by Gabriel and Roiter. Building on this, we will focus on one-sided matrix problems and present key results by Gabriel and his collaborators concerning the classification into finite, tame, and wild types.
Morita theorem for non-commutative noetherian schemes
Igor Burban
A classical result of Gabriel states that the categories of quasi-coherent sheaves and of two separated noetherian schemes X and Y are equivalent if and only if X and Y are isomorphic. To prove this statement (and in particular to show how the scheme X can be reconstructed from its category of quasi-coherent sheaves), Gabriel developed in his thesis from the year 1962 new concepts of homological algebra, which became nowadays completely classical.
Generalizing Gabriel's approach, Drozd and myself proved a version of a Morita theorem for non-commutative schemes. The key ingredient of the proof uses properties of indecomposable quasi-coherent sheaves in the corresponding categories.
As an application, we obtained a new proof of a conjecture of Caldararu about Morita equivalences of Azumaya algebras on quasi-projective varieties. Another application provides a handleable criterion for two non-commutative projective curves to be Morita equivalent.
TBA
GIT for quivers and free skew fields
Harm Derksen
The free skew field in n variables over a (commutative) field K was developed by Amitsur, Cohn and others. Elements of in a free skew field can be thought of as non-commutative rational functions. Such elements cannot always be written as a quotient of polynomials, but one may need nested inverses to represent them. There are surprising identities, so it is not obvious to see whether rational expressions are equal to each other. There is a polynomial time algorithm to decide this based on work by Garg-Gurvits-Oliveira-Wigderson, Ivanyos-Qiao-Subrahmanyam and work by Makam and the speaker. These algorithms are based on a connection between the notion of sigma-stability in geometric invariant theory for quivers and ranks of matrices over skew fields. I will discuss this connection and, time permitting, discuss some open problems in the theory of free skew fields.
On length categories and their taxonomy
Gustavo Jasso
Length categories are, arguably, the central objects of study in classical representation theory. These are abelian categories in which every object is a finite extension of simple objects. More than half a century ago, Gabriel's 1973 proceedings article "Indecomposable Representations II" introduced the notion of species of a length category to the subject. In this talk, we will discuss old and new aspects surrounding these concepts.
Enhancement for categories and homotopical algebra
Dmitry Kaledin
I am going to describe a framework for working with homotopically enhanced categories loosely based on Grothendieck's idea of a "derivator". The framework is manifestly model-independent, does not use the machinery of model categories, nor simplicial homotopy theory, and is pretty close to the usual categorical intuition and way of thinking. It is also very much motivated by Gabriel's original approach to homotopy theory via fractions, and in fact, it is hard to overestimate how essential are his ideas to the whole story.
Abelian categories from Mac Lane to Gabriel. A historical approach
Ralf Krömer
In my talk, I will overview the historical development leading to Gabriel's 1961 Ph.D. thesis on abelian categories (published 1962 in the bulletin of the SMF). The conceptual development starts with Mac Lane's 1950 work on duality for groups. The Cartan-Eilenberg approach to homological algebra led to a first step towards generalization of the apparatus of derived functors by David Buchsbaum. But the most important step was made when the focus switched to sheaves with Serre's FAC and, most notably, Grothendieck's 1957 Tohoku paper. We will also investigate the role of foundational issues in the theory up to Gabriel's work, and take a short outlook on later developments, as discussed in recent historical work by Marquis.
A categorification of combinatorial Auslander–Reiten quivers
Ricardo Felipe Rosada Canesin
Combinatorial Auslander–Reiten quivers, introduced by Oh and Suh, serve as an important tool in the representation theory of quantum affine algebras. These quivers generalize the AR quiver associated with a Dynkin quiver Q, but are defined purely in terms of the Coxeter combinatorics of the Weyl group of Q. This naturally raises the question of whether these combinatorial objects admit a representation-theoretic interpretation analogous to that of classical AR quivers. In this talk, we provide such a categorification using the derived category of the derived preprojective algebra of the same Dynkin type. We construct two categories that generalize the category of Q-representations and its derived category, and discuss some of their properties.
Presentable infinity-categories and factorization homology
Federica Pasqualone
Factorization homology provides an elegant mathematical formalism for various homology theories, and, in particular, it's a very effective tool in mathematical physics to deal with observables of a TQFT.
This talk will provide the audience with an introduction to factorization homology, starting from prefactorization algebras, and illustrate the topos-theoretical foundations of the theory: operads, presentability, and posites.
No prerequisite knowledge of physics is required.
Tensor triangular geometry for finite group schemes
Julia Pevtsova
Associated to a finite group scheme in some appropriate universe one can form the stable category of its finitely generated representations, stab G, which has the structure of a tensor triangulated category. Then one can study stab G using tensor triangular (tt-) geometry, and, in particular, try to compute its spectrum. I'll give a brief introduction to tt-geometry in this representation theoretic setting and try to describe the current landscape: what is known and what are the open questions.
Rediscovering coverings
Pierre-Guy Plamondon
We will review covering techniques in representation theory, using as a starting point the seminal paper of Bongartz and Gabriel.
Gabriel-Zisman homology and homotopy colimits for diagrams in chain complexes
Birgit Richter
In Appendix II.3 of "Calculus of Fractions and homotopy theory" Gabriel and Zisman define the notion of the homology of a small category with coefficients in a functor and relate it to the derived functors for colimits. This was heavily used by Quillen in his work on algebraic K-theory. Later, many authors referred back to Quillen and not to the original source. The Gabriel-Zisman approach was applied and generalized in recent work of Gálvez-Carrillo, Neumann and Tonks and plays an important role in the notion of homotopy colimis for diagrams of chain complexes. I'll sketch the development and present some sample applications.
Locally presentable categories
Jan Šťovíček
The notion of locally presentable category is technical, but extrmely useful when one needs to work with categories that are not small. The theory from the 1971 monograph by Gabriel and Ulmer (sometimes through the lens of more modern expositions) among others allows one to safely stay away from set-theoretic paradoxes or provides useful adjoint functor theorems. The aim of the talk is to present the main ideas as well as more recent developments where the insight of Gabriel and Ulmer played a key role.
Grothendieck categories, injectives, and reconstruction
Greg Stevenson
The article Des Catégories Abéliennes, based on Gabriel's thesis, contains a wealth of beautiful mathematics which has been profoundly influential. I'll attempt to give a survey of its contents and then focus on the thread starting with locally noetherian abelian categories and ending with his reconstruction theorem for noetherian schemes.
Transfer for finite group schemes
Peter Symonds
The transfer is an invaluable tool in the modular representation theory of finite groups and in the study of their homological algebra. We define a version for finite group schemes and discuss which properties carry over from the finite group case and which don't. Joint with Kostas Karagiannis.
Multiplicative bases for infinite-dimensional algebras
Jan Trlifaj
Multiplicative bases and their generalizations relate the complexity of multiplication in an algebra to its linear structure. One of the highlights of classic representation theory--due to Bautista, Gabriel, Roiter, and Salmeron--says that each finite-dimensional algebra of finite representation type over an algebraically closed field possesses a normed multiplicative basis. We prove a dual result, the existence of conormed multiplicative bases, for all commutative semiartinian von Neumann regular algebras of countable type (joint work with K.Fukova).