Geometric representations

首席研究员提出的研究有三个部分。第一种是使用拓扑和辛几何的工具，特别是Goresky-Kottwitz-MacPherson的理论来构造表示，并将这些工具扩展到更大的变种类。第二部分处理代数和组合项目，解决相关品种的几何问题。这门学科通常被称为现代舒伯特微积分；具体项目包括确定旗种和格拉斯曼种的上同调环内的乘法公式。第三部分侧重于品种的计算和枚举几何性质，如黑森伯格品种，出现在包括朗兰兹程序在内的表示理论领域。这些项目将确定品种的基本几何特性，如纯维度，这对表征理论的进步至关重要。几何表示理论建立称为几何对象表示的代数结构。这导致了深刻的见解连接不同的数学学科，包括代数，几何，组合和数学物理。至关重要的是，几何表示建立了几何和代数之间对应关系的表。一方面，表通过几何学回答表征理论中的问题；另一方面，它用代数和组合来回答几何问题。该提案包括促进各级学生发展的项目，并为妇女和其他代表性不足的少数民族在数学方面提供指导。

The principal investigator's proposed research has three parts. The first constructs representations using tools from topology and symplectic geometry, especially the theory of Goresky-Kottwitz-MacPherson, and extends those tools to larger classes of varieties. The second tackles algebraic and combinatorial projects that solve problems in the geometry of the associated varieties. This subject is often called modern Schubert calculus; specific projects include determining multiplication formulas inside the cohomology ring of flag varieties and Grassmannians. The third focuses on computational and enumerative geometric properties of varieties, such as Hessenberg varieties, that arise in areas of representation theory including the Langlands program. The projects will identify fundamental geometric properties of the varieties, like pure-dimensionality, that are critical to advances in representation theory.Geometric representation theory builds algebraic structures called representations from geometric objects. This leads to deep insights connecting different mathematical disciplines, including algebra, geometry, combinatorics, and mathematical physics. Crucially, a geometric representation establishes a table of correspondences between geometry and algebra. On one hand, the table answers questions in representation theory via geometry; on the other, it answers questions in geometry using algebra and combinatorics. The proposal includes projects to foster student development at all levels and to provide mentoring for women and other underrepresented minorities in mathematics.