Combinatorics in Geometry, Physics, and Representation Theory

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Combinatorics is the study of discrete structures such as permutations, combinations, subsets, and so on. This research project studies combinatorial problems that arise from algebra, geometry, and theoretical physics. One of the topics to be studied is the classification of electrical resistor networks consisting of finitely many resistors. In this problem, one studies whether the structure of an electrical network can be recovered by performing measurements at particular nodes. Another problem to be studied is the understanding of the combinatorics of high-energy particle scattering experiments. These computations have been related to the combinatorics of permutations and other related projects. A third problem is the study of certain discrete two-dimensional surfaces in high dimensional space, called membranes.The investigator's work is in algebraic combinatorics. Topics to be pursued include: (a) Development of the theory of Laurent phenomenon algebras, originally introduced in joint work with Pavlo Pylyavskyy. These LP algebras generalize cluster algebras by allowing exchange polynomials to have arbitrarily many monomials, rather than just two. (b) The study of the algebraic deRham cohomology of cluster algebras, which may have applications to the calculation of higher extensions of Verma modules. (c) A development of a positive analogue of polymatroids, called polypositroids. These polytopes turn out to have remarkable properties, and appear to be related to certain discrete minimal surfaces called membranes. (d) The study of a parabolic analogue of Whittaker functions. The aim is to study solutions to the quantum D-module for a generalized partial flag variety.