Combinatorics in geometry and representation theory

提出的项目研究由表示理论和代数几何引起的组合问题。一个特别的焦点是在环路群或仿射李代数的研究中产生的问题。与合作者将研究仿射李群的旗变的Schubert演算及其与量子上同调的关系。这包括理解舒伯特变换的几何和（co）同调中表示舒伯特变换的多项式的组合学。与合作者一起，PI还计划开发环群的总正性理论，旨在扩展Lusztig在有限维约化群中的工作。将研究与电网络理论的联系，以及与嵌入曲面的图的联系。另一个研究方向是对无限Coxeter群中无限约简词的研究。PI将研究无限约简词上的偏序，并探索随机无限约简词的概率方面。组合学是研究离散结构的学科。这样的结构在科学的不同背景下出现。在提出的工作中出现的这样一种结构是由电阻组成的电网络。一个基本问题是理解网络的电性质和网络的连通性之间的关系。这些问题可以应用于电阻抗断层扫描，这是一种医学成像技术。本文研究的另一种离散结构是具有大量对称性的结构，包括高维空间的规则模式。这些结构的一个基本问题是对它们进行分类并分析它们的大尺度行为。这种结构很重要，例如在编码理论中。

The proposed project studies combinatorial problems arising from representation theory and algebraic geometry. A particular focus is on problems arising from the study of loop groups, or affine Lie algebras. The PI with collaborators will study the Schubert calculus of flag varieties of affine Lie groups, and the relations to quantum cohomology. This includes understanding the geometry of Schubert varieties and the combinatorics of the polynomials representing Schubert varieties in (co)homology. Together with collaborators, the PI also plans to develop a theory of total positivity for loop groups, aiming to extend work of Lusztig for finite-dimensional reductive groups. Connections with the theory of electrical networks, and to graphs embedded in surfaces will be investigated. Another direction of research is the study of infinite reduced words in infinite Coxeter groups. The PI will study partial orders on infinite reduced words, and explore probabilistic aspects of random infinite reduced words.Combinatorics is the study of discrete structures. Such structures come up in varied contexts throughout science. One such structure that arises in the proposed work is an electrical network consisting of resistors. A basic problem is to understand the relationship between the electrical properties of the network and the connectivity properties of the network. These problems have applications to electrical impedance tomography, a technique in medical imaging. Another type of discrete structure studied in the proposed work are structures with a large number of symmetries, including for example tilings of high-dimensional space in a regular pattern. A basic problem with these structures is to classify them and analyze the large scale behavior. Such structures are important for example in coding theory.