Combinatorics of Cluster Varieties

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Cluster algebras are a fundamental mathematical structure employed to describe highly symmetric mathematical objects. In the last decade, cluster structures have been constructed on spaces throughout representation theory and mathematical physics, most recently in the theory of scattering amplitudes in high-energy particle physics. This project comprises fundamental mathematical research into the combinatorial structure of several crucial cluster algebras, and the geometric spaces related to them. This research project addresses several questions concerning the combinatorial structure of various cluster algebras. Most of these cluster algebras come from natural spaces in algebraic geometry, such as open positroid varieties, which occur in stratifying the Grassmannian and generalized Teichmuller spaces. The first part of the project investigates the coefficients of the Laurent polynomials that occur in changing from one cluster to another in the Grassmannian (and other related spaces). The second part of the project aims to determine the relation between the parametrizations of positroid cells by dimer configurations on tori and the values of Plucker coordinates as charts on those cells. The third part of the project investigates the cohomology of cluster varieties, trying to find presentations for the cohomology ring that are as explicit and combinatorial as the presentations already known for the coordinate rings.