Surfaces, Categorification and Combinatorics of Cluster Algebras – SC3A

Fomin and Zelevinsky invented cluster algebras in early 2000 in order to find a combinatorial approach to the study of Luzstig and Kashiwara's canonical bases in quantum groups, and to total positivity in semisimple groups. The formalism they developped found many applications beyond the scope of their initial goals. A noticeable example is the fact that Fomin and Zelevinsky discovered a phenomenon of simplification of rational fractions, called « Laurent phenomenon ». In the study of rational sequences defined by recurrence relations (such as the Gale-Robinson sequence or the Somos sequences...), the Laurent phenomenon implies that the sequences under consideration take integer values. A second remarkable example is the proof of Zamolodchikov's periodicity conjecture by B. Keller.

Cluster algebras are defined in terms of generators and relations. Contrary to usual presentations, the set of generators and relations is not given a priori. The initial datum is that of an « initial seed » which contains a relatively small subset of the generators (the initial cluster) plus some matrix. That matrix contains all the necessary information in order to construct inductively the whole set of generators, starting from the initial cluster, by means of an operation called « mutation ».

The theory of cluster algebras has had fast developments in many directions: Representation theory of quivers, Poisson geometry, integrable systems, Teichmüller spaces, combinatorial polyhedra, algebraic geometry (stability conditions, Calabi-Yau algebras, DT-invariants), Quantum Field Theory...

In the present project, we focus on some connections between cluster algebras, algebraic and geometric combinatorics, representation theory, triangulated and monoidal categories and integrable systems. Our objectives are to develop new links between these fields, to study some previously known links, and to obtain new applications. Our project gathers three main themes.

A – Riemann surfaces viewed as combinatorial tools: Links with certain triangulated categories (higher generalised cluster categories) and Frobenius categories, or with some combinatorial objects (oriented triangulations and maximal green sequences, multitriangulations, pseudotriangulations).

B – Categorification: Monoidal categorification by means of perverse sheaves on Nakajima's graded quiver varieties. Additive categorification, via the definition of a tilting theory in some triangulated orbit categories associated with the combinatorial objects cited above.

C – Applications to, and of, cluster algebras: Construction of bases, study of friezes, study of the pentagram map... Our team gathers young mathematicians with various mathematical specialities and with different point-of-views, from geometric combinatorics to triangulated categories, representation theory or integrable systems, who all have a common, deep interest in cluster algebras. In relation to this project, several collaborations, as well as a working seminar, have already begun.