This project explores links between classical combinatorics, theory of moduli spaces of holomorphic curves, real algebraic geometry and total positivity, and the newly emerging theory of cluster algebras. In particular,we plan to use the link between decorated Teichmueller spaces and theory of cluster algebras to investigate moduli spaces; to develop a general geometric framework for cluster algebras and to find a sufficiently generic source of geometric examples of formal cluster algebras supplementing those arising from Schubert varieties. In addition, we will apply the cluster algebra approach combined with a geometric Littlewood-Richardson rule to the Shapiro-Shapiro conjecture in the real Schubert calculus. Finally, we plan to generalize ELSV-formula for Hurwitz numbers, namely, to express (double) Hurwitz numbers as integrals of certain characteristic classes over moduli spaces.
Many significant breakthroughs in mathematics are inspired by theoretical physics and achieved through the interaction between different branches of mathematics such as combinatorics, geometry (differential, symplectic and algebraic), the theory of integrable models, and many others. Geometric objects we are interested in can be used to describe parameters of physical systems. In many cases, the cluster algebra formalism, recently discovered by Fomin and Zelevinsky, turns out to be uniquely suited for an investigation of physically important coordinate systems. Extending the scope of the cluster algebra approach will prove useful in topological field theory, 2-D gravity, classical and quantum integrable models and, on a more applicable level, in electrical engineering, in particular, in the design of nonlinear filters.
