CIMPA School
Homological Methods, Representation Theory and Cluster Algebras
Mar del Plata
March 7th to 18th, 2016

Mar del Plata

Courses during the first week (given in English)

Advanced homological algebra, by Mariano Suárez-Alvarez (Universidad de Buenos Aires, Argentina)

This course aims at introducing the students to the basic notions of homological algebra, approached first from an elementary viewpoint using derived functors and then using the notion of triangulated category and derived category. We start with a review of functors and categories including several examples and emphasizingthe definitions of kernel and cokernel, product and coproduct, as well as the notions of right and left exact functors. Then we will talk about additive categories, the category of complexes of an additive category and the corresponding homotopy category. This will give rise to the notion of derived category.

Introduction to the representation theory of algebras, by Maria Inés Platzeck (Universidad Nacional del Sur, Argentina)

The objective of the course is to introduce basic notions and results of the representation theory of algebras, with particular emphasis in the functorial techniques introduced by Maurice Auslander. This approach has played a fundamental role in the representation theory of finite dimensional algebras and, more generally, of artin algebras.

Auslander-Reiten theory for algebras of infinite representation type, by Piotr Malicki (Nicholaus Copernicus University)

In the first lecture we shall present some important facts from homological algebra.
In the second lecture we introduce some fundamental concepts such as irreducible morphisms and almost split sequences.
The third lecture will include the following concepts: the Auslander-Reiten translation, the Auslander-Reiten formula and the Auslander-Reiten quiver which is an important combinatorial and homological invariant of the module category of finitely generated modules of an algebra.
In the last lecture we shall present old and new results on the number of terms in the middle of an almost split sequence in the module category of a finite dimensional algebra.
The intention of these lectures is to serve as a source of motivation and information on the main concepts, techniques and results on the Auslander-Reiten theory.
Lecture notes.

Cluster algebras arising from surfaces, by Ralf Schiffler (University of Connecticut, USA)

Cluster algebras are commutative algebras with a special combinatorial structure. A cluster algebra is a subalgebra of a field of rational functions in several variables, which is defined by constructing a specific set of generators in a recursive way.
This course will focus on an important class of cluster algebras, those that are associated to surfaces with boundary and marked points. The generators of these cluster algebra are in bijection with certain curves in the surface and the combinatorial structure of the cluster algebra can be explained in terms of triangulations of the surface.
In the simplest example, a regular polygon with n vertices, the generators of the cluster algebra correspond to the diagonals of the polygon.
The first class will be on general cluster algebras, and the following three classes on cluster algebras from surfaces. I plan to cover the following list of topics. Cluster algebras, definition and results, cluster algebras from surfaces, definition, examples, relation to representation theory, combinatorial expansion formulas, canonical bases, and if time permits upper cluster algebra, skein relations, snake graph calculus.
Lecture notes