Combinatorics in Geometry, Physics, and Representation Theory

组合数学是研究离散结构，如排列，组合，子集等。本研究项目研究从代数，几何和理论物理中产生的组合问题。 要研究的课题之一是由许多电阻组成的电阻网络的分类。 在这个问题中，人们研究是否可以通过在特定节点处执行测量来恢复电网络的结构。 另一个要研究的问题是高能粒子散射实验的组合学的理解。 这些计算与排列组合学和其他相关项目有关。 第三个问题是研究高维空间中某些离散的二维曲面，称为膜。研究者的工作是代数组合学。 主题将继续包括：（a）发展理论的洛朗现象代数，最初介绍了联合工作与帕夫洛Pylyavskyy。 这些LP代数通过允许交换多项式有任意多个单项式而不是只有两个单项式来推广簇代数。(b)研究簇代数的代数deRham上同调，它可以应用于计算Verma模的高阶扩张。 (c)多拟阵的一个正类似物的发展，称为多正拟阵。 这些多面体原来有显着的性质，并似乎与某些离散的极小表面称为膜。(d)惠特克函数的抛物型类似物的研究。 目的是研究广义部分旗簇的量子D-模的解。

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.