In a joint work with Frederic Chapoton and Ralf Schiffler, we introduce a category C whose indecomposable objects are the diagonals of a n-polygon. The category C is triangulated and the tilting objects are in correspondence with triangulations of the n-polygon by diagonals (with multiplicity). It is a particular case of the cluster category, introduced by Buan-Marsh-Reiten-Reineke-Todorov, but in this case, the geometrical interpretation of the category provides a connection between tilting theory and Teichmüller spaces theory. We will then give a geometric interpretation of the structure constants of a cluster algebra of type A, via the category C.
There is a set of rational functions which is "dual" in some sense to the set of cluster variables. This set is sometimes called the Y-system. In the case of seeds given by finite Dynkin diagrams, this set is finite because of the periodicity property proved by Fomin and Zelevinsky. Then one can take the sum of values of (a relative of) the Rogers dilogarithm function on this set, and the result appears to be a constant. We will sketch the proof and present some conjectural generalisation for products of two finite Dynkin diagrams.
We use the quantization of Teichmüller space as a Poisson manifold using the graph description due to Penner and Fock to consider the corresponding quantization of the Thurston theory. Since the Thurston's sphere of projectivized foliations of compact support provides a useful compactification for Teichmüller space in the classical case, it is natural to consider corresponding limits of appropriate quantum operators of geodesic laminations. We provide the explicit construction for quantizing the Thurston's boundary in the special case of the once-punctured torus, where there are already substantial analytical and combinatorial challenges. Indeed, an operatorial version of continued fractions is required to prove existence of these limits. It is useful to introduce graph lengths, which are (classical) geodesic lengths in the Strebel uniformization pattern. These lengths simultaneously correspond to the "tropicalization" of the Teichmüller spaces. We show that the L-function that depends on the combination of Poincare and graph lengths is a rational function (in classical case) that can be computed on the corresponding graph.
We will consider several versions of mapping class groups of infinite surfaces modelled on the binary tree. When the genus is zero one obtains extensions of Thompson groups by infinite (spherical) braids and one proves that they are are finitely presented. If the genus is infinite the group is of finite type and its rational cohomology is the stable cohomology of the mapping class groups.
This talk will outline some recent work on Weil-Petersson metrics, following in part a recent survey by Wolpert. I hope to say something about both the finite genus case and about the infinite dimensional situation, where matters become more complicated.
Cluster algebras were defined in 2001 by Fomin and Zelevinsky in order to study the dual canonical basis of a quantum group and total positivity in algebraic groups. Since then they have found application in a variety of different fields. This talk will consider some of the links between cluster algebras and the representation theory of finite dimensional algebras, in particular, with cluster-tilting theory, a generalisation of Auslander-Platzeck-Reiten tilting theory motivated by cluster algebras. This will include joint work with Aslak Buan, Markus Reineke, Idun Reiten and Gordana Todorov.
These introductory (pre-quantum) talks will review and explain the background classical geometry and combinatorics of Teichmüller space and Riemann moduli space. Roughly, the first part will treat the required coordinates and geometry and the second the required combinatorics.
The mysterious periodicity property of the Zamolodchikov Y-system has attracted a lot of attention during the past 12 years. We will review the known cases and some applications of these results to dilogarithm identities, which were obtained in joint work with Edward Frenkel.
Triangulations of surfaces with at least one puncture provide many examples of cluster algebras which are _mutationally_finite_: there are only a finite number of different combinatorial types of clusters. We will survey some of these results and show how, by introducing variables associated to closed loops in addition to arcs, we can hope to prove positivity of these cluster algebras. We will also see how passing to the tropical limit corresponds to looking at laminations of the surface and see how we can start to deal with closed surfaces. A large part of this talk is joint work with Sergey Fomin and Michael Shapiro.
It is possible to construct a morphism from the group of birational transformations of CP² preserving poisson bracket {x,y}=xy to the group of piecewise linear automorphisms of Z² which is known as Thompson group T. This construction naturally gives rise to cluster algebras. In the talk I will expose there apparition.
We will discuss an interplay between Y-systems, introduced and studied by Al.Zamolodchikov in the context of thermodynamic Bethe Ansatz, and cluster algebras. The main observation is that the coefficient dynamics in cluster algebras leads to a natural generalization of Y-systems associated with arbitrary symmetrizable generalized Cartan matrices. In a joint work with S.Fomin, we establish a Laurent phenomenon for such Y-systems (previously known in finite type only) and sharpen the periodicity result from an earlier paper.
[LMS Regional Meeting]
[Teichmüller Theory and Cluster Algebras]
[London Mathematical Society]
The University of Leicester Mathematics