Cluster categories: combinatorial constructions via polygons and via strand diagrams
Cluster categories have been introduced in 2005 by Buan-Marsh-Reineke-Reiten-Todorov as quotients of bounded derived categories and by Caldero-Chapoton-Schiffler as categories arising from triangulated polygons. We will describe both approaches, go on to cluster categories from surfaces and then treat Grassmannian categories arising from alternating strand diagrams (Postnikov).
Tilting and Cluster-Tilting Theory
The origins of tilting theory can be traced back to the introduction of Coxeter (or reflection) functors by Bernstein, Gelfand and Ponomarev in 1973, providing a relation between Lie theory (root systems) and representations of quivers. These functors were reformulated by Auslander, Platzeck and Reiten in 1979, and generalized by Brenner and Butler in 1980 who introduced tilting functors. More recently, cluster-tilting theory arose in a similar way in the interpretation of cluster algebra combinatorics in terms of mutation functors on the cluster category introduced by Buan, Marsh, Reineke, Reiten and Todorov in 2004. The lectures present these developments, introduce and compare tilted and cluster-tilted algebras, as well as the posets of tilting and cluster-tilting modules, always having in mind the relation to root systems and cluster algebra combinatorics.
Cluster structures and compatible Poisson brackets
I plan to cover Poisson brackets arising from from networks on surfaces, Poisson-Lie groups and their relations with cluster algebras and give examples of integrable systems that live on Lie groups and algebras.
Modules, projective presentations and bases for cluster algebras.
In this talk, I will review the key features of the categorification of cluster algebras by means of modules over finite-dimensional algebras. I will then discuss how the generic bases of Geiss, Leclerc and Schröer can be obtained from "generic" projective presentations.
Cluster algebras and cluster algebras from surfaces
Cluster algebras are commutative algebras with a combinatorial structure that are related to a variety of research areas in Mathematics and Physics. In these lectures, we will define cluster algebras, explore some of their basic properties and then concentrate on a particular type of cluster algebras which are associated to Riemann surfaces.
Pentagram map, old and new
I shall describe the pentagram map, introduced by R. Schwartz more than 20 years ago. I hope to address the following issues:
- the geometry and combinatorics of the moduli space of projective polygons;
- relation to the Boussinesq equation, a continuous limit of the pentagram map;
- Liouville integrability of the pentagram map;
- higher - and lower! - dimensional generalizations of the pentagram map;
- generalized pentagram maps, weighted directed networks, and cluster algebras.
Time permitting, I shall also talk about configuration theorems of projective geometry related with the pentagram map, and the combinatorics of the integrals of the pentagram map.