mini-workshop2015

Aims: This Mini-Workshop consists of four introductory lectures (basic level) for students of the Master program in Mathematics. The main aim is to present some techniques in Category Theory which are of interest in Algebra, Algebraic Geometry, Computer Science and Logic.

Math-Lab: Gli studenti della Laurea Magistrale sono invitati e, partecipando ad almeno due delle quattro lezioni, potranno avere una firma per Math-Lab. E' richiesta l'iscrizione, vedere link qui sotto.

Registration: Although there is no registration fee, we ask those interested in partecipating to the Mini-Workshop to register. In the form one can also register to the social dinner that will be on the day of the conference at 8 pm and it will take place at the Osteria "Ghiottus Taberna" in via Urbano Rattazzi 2/h at the price of 25€ per person. (Click here to register)

PROGRAM:

• 9:15 - 10:45 Laiachi El Kaoutit “Introduction to commutative Tannakian Theory.“ (aula C)

Abstract

We first present a brief account on symmetric monoidal categories, tensorial-functors and commutative Hopf algebras. Next we recall in details the definition of Tannakian category with fibre functor in the category of finite dimensional vector spaces. The re-construction theorem will be also formulated in detailed way. We conclude by showing how a Tannakian category is, up to a symmetric monoidal equivalence, recovered as the category of dualizable comodules on the reconstructed Hopf algebra. (Slides)

• 11:00 - 12:30 Felice Cardone “Glimpses of categories for computation“ (aula C)

Abstract

Among the many applications of category theory to computer science, we illustrate the main categorical approaches to the solution of the fixed-point equations needed for the semantical treatment of recursion in programming. These include, in particular, initial algebras and their duals, final coalgebras; the iterative algebraic theories of Elgot, and traced monoidal categories. We develop these notions starting from motivating examples, in an attempt to make the presentation self-contained and (hopefully) useful to an audience with interests in algebra, logic or computer science.

• Lunch break
• 14:00 - 15:00 Federica Galluzzi “Brauer groups from fields to categories” (aula 4)

Abstract

In this talk we first present the Brauer group over a field k which was classically studied to classify division algebras over k.

Then we move to topological spaces and schemes where the Brauer group classifies Azumaya algebras up to Morita equivalence.

Finally we discuss some possible generalizations to categories.

• 15:15 - 16:45 Isar Goyvaerts “Variations on a theme of categories of representations of finite groups” (aula 4)

Abstract

We start this talk by recalling certain properties of categories of finite-dimensional representations of finite groups. We then consider variations on this theme, exploring some extra pleasant structures these categories exhibit. The question whether considering such extra structures allows us to determine a finite group (up to isomorphism) is addressed. Along the way we will come across notions that return in the talks by Felice Cardone and Laiachi El Kaoutit.

This talk takes on some linear algebra and some basic definitions from representation theory of finite groups. It is also useful to be familiar with the definitions of category, functor and natural transformation.

This talk is suitable for Master students and vegetarians.

Organizers: Alessandro Ardizzoni, Cristiana Bertolin, Felice Cardone

Poster: pdf