Mini-courses and Speakers

In this mini-course, we delve into key aspects of curves over finite fields, drawing from a curated selection of topics. In particular, we discuss bounds for the number of rational points, their improvements, and applications. Throughout the five lectures, several open problems will be discussed.

A classical result by Auslander states that there is a correspondence between Artin algebras of finite representation type and algebras whose global dimension at most 2 and dominant dimension at least 2. At the end of the 2000's Iyama realized and proved that this result constitutes a special case of a more general theorem stating a correspondence of "d-representation finite" algebras and algebras whose global dimension at most d+1 and dominant dimension at least d+1. The main insight of Iyama was to realize that, instead of looking at the whole of the module category, a generalization of Auslander's result required considering a maximal orthogonal rigid subcategory, also known as a "d-cluster tilting" subcategory, which behaves much like an abelian category where the shortest non-split exact sequences have d+2 terms. In other words, in a d-cluster tilting subcategory the bifunctor Ext^i(-,-) is zero for every i between 1 and d-1. In this course we will go over the basics on representation theory of Artin algebras, we will give the definition of d-cluster tilting subcategories and we will see how the classical notions adapt to the higher setting. The topics covered in this course include (higher) homological algebra, (higher) Auslander-Reiten theory and (higher) torsion classes.

The course will give an elementary account of $p$-adic methods in de Rham cohomology of algebraic hypersurfaces with explicit examples and applications in number theory and combinatorics. Lectures are based on a series of our joint papers with Frits Beukers entitled Dwork crystals. These methods also have applications in mathematical physics and arithmetic geometry.

Gradings (also known as gradations) have played an important role in algebra for a long time: for example, the grading of the algebra of polynomials by total degree or by multidegree, the grading of a complex semisimple Lie algebra by its root lattice, the natural grading of a crossed product. Since the 1990's, there has been an increasing interest in classifying gradings by arbitrary groups on various algebras and studying the properties of the resulting graded algebras (for example, their ``contractions'', graded polynomial identities, graded representations).