Geometric representations

首席调查者提出的研究包括三个部分。第一种是使用拓扑学和辛几何的工具，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别。第二个解决代数和组合项目，解决相关簇的几何问题。这门学科通常被称为现代舒伯特微积分；具体的项目包括确定旗型变种和格拉斯曼尼亚变种的上同调环内的乘法公式。第三部分集中在表示理论领域中出现的簇的计算和计数几何性质，例如Hessenberg簇，包括朗兰兹计划。这些项目将确定变量的基本几何性质，如纯维性，这对表示理论的进步至关重要。几何表示理论从几何对象建立称为表示的代数结构。这导致了将不同的数学学科联系起来的深刻见解，包括代数、几何、组合学和数学物理。至关重要的是，几何表示法建立了几何和代数之间的对应表。一方面，该表通过几何回答表示论中的问题；另一方面，它使用代数和组合学回答几何中的问题。该提案包括促进各级学生发展的项目，以及为妇女和其他在数学领域代表性不足的少数群体提供指导的项目。

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.