This project explores links between classical combinatorics, modern theory of Teichmueller spaces, real algebraic geometry and total positivity, and the rapidly developing theory of cluster algebras. In particlular, PIs utilize the link between decorated Teichmueller spaces and the algebra of geodesics on the one hand and the theory of cluster algebras on the other hand to investigate the structure of the dual canonical basis of a cluster algebra. Furthermore, they use cluster algebra point of view to study directed nets on surfaces and describe compatible Poisson-Lie structures for nets and solutions of corresponding inverse problems and to study associated integrable hierarchies. The latter will be applied to investigate new relations between double Hurwitz numbers of coverings of the sphere by higher genus curves and, on the other hand, to analyze a new two- and multi-matrix models and associated biorthogonal polynomials and apply them to problems of enumeration of bicolored embedded graphs.
Space discretization using networks on surfaces is an important in the theory of random processes, in mathematical physics, including 2D gravity, the theory of electric potential and especially theory of electrical networks and in other fields. Combinatorial properties of surface networks capture crucial features of complex mathematical and physical structures. Recently it was observed that surface networks also exhibit many features that are typical for cluster algebra structures. Introduced only a few years ago by Fomin and Zelevinsky, the cluster algebra formalism is proved to be widely applicable in investigation of algebraic and geometric objects with symmetries often associated with important physical systems. Interplay between the two concepts will be instrumental in the study of combinatorial quantities and geometric phenomena of physical relevance and classical and quantum exactly solvable models.
