About the Laboratory

The research project belongs to the intersection of two areas of mathematics: cluster algebras and theory of moduli spaces of flat and holomorphic connections on complex curves. One of the most valuable application of cluster theory describes so-called "shear" coordinates on the Teichmuller space of genus g curves with marked points. This space can be identified with the space of flat PSL_2-connections. Cluster coordinates associated with a triangulation of the curve are Darboux-type coordinates for the Weil-Petersson bracket. Fock and Goncharov generalize this construction to the space of flat connections. The cluster description is convenient for quantization of the moduli spaces. G.Schrader and A.Shapiro applied it successfully to describe positive representations of quantum groups. The generating function for the integrals of characteristic classes of the moduli space of complex curves with marked points satisfies KdV hierarchy. A conceptual proof by Lando and Kazarian of this fact is based on counting Hurwitz numbers, the numbers of ramified covers of the sphere by genus g complex curves. The generating function for Hurwitz numbers satisfies cut-and -join equations and KP-hierarchy. The ELSV formula connects Hurwitz numbers to integrals of characteristic classes over moduli spaces. Lando and Kazarian showed that the KP equation for Hurwitz numbers implies KdV equation for integrals. Zvonkin, Shadrin and Faber proved a generalized version of this theorem for moduli of r-spin structures. Cut-and-join equations are an example of topological recursion: integrals of characteristic classes can be expressed as similar integrals over boundary strata. Similarly, Mirzakhani derived recurrent relation between volumes of strata of moduli of curves with boundaries and connected them to the integrals of characteristic classes. Mirzakhani's recurrent relations are equivalent to a topological reduction for a matrix model and can be generalized to some Hurwitz spaces.

The main goal of the current project is an application of methods used for computation of characteristic classes of the moduli spaces of complex curves to the spaces of flat connections. Modern approach to the computation of integrals of characteristic classes is based on the methods of topological recursion that allows to express integrals recursively. Initial step for this project is a construction of cluster theory for moduli space of curves and its application for computing integrals.

We plan

• to study degeneration of "shear coordinates" and Weil-Petersson forms on boundary strata of Teichmuller space;
• to generalize degeneration formulas to other cluster varieties, e.g. double Bruhat cells;
• using degeneration formulas, to generalize topological recursion methods to boundary strata of cluster varieties;
• to study relation between tropical cluster varieties and tropical curves;
• to find a geometric representation for Belavin-Drinfeld Poisson-Lie brackets and find appropriate space of connections;
• to generalize real Hurwitz numbers defined by Zvonkin and Itenberg for polynomial maps to rational maps.