Geometric representations

首席研究员提出的研究有三个部分。 第一种是使用拓扑学和辛几何的工具来构造表示，特别是Goresky-Kottwitz-MacPherson的理论，并将这些工具扩展到更大的种类。 第二个处理代数和组合的项目，解决问题的几何形状的相关品种。 这个主题通常被称为现代舒伯特演算;具体的项目包括确定旗簇和格拉斯曼簇的上同调环内的乘法公式。 第三个重点是计算和枚举几何性质的品种，如海森伯格品种，出现在领域的表示理论，包括朗兰兹计划。 这些项目将确定基本的几何性质的品种，如纯维数，这是至关重要的表示理论的进步。几何表示理论建立代数结构称为表示从几何对象。 这导致了深刻的见解连接不同的数学学科，包括代数，几何，组合数学和数学物理。至关重要的是，几何表示建立了几何和代数之间的对应表。 一方面，该表通过几何回答了表示论中的问题;另一方面，它使用代数和组合学回答了几何问题。 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提供数学辅导。

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.