Wonderful Varieties, Hyperplane Arrangements, and Poisson Representation Theory

几何表示理论研究几何对象的对称性所形成的代数结构。它与代数和几何的许多领域有联系，包括代数组合学、代数几何、数学物理和辛几何。本项目将通过在代数和辛几何中开发新的表示理论对象来探索这种丰富的相互作用。它还将为研究生提供研究培训机会。更详细地说，该项目将重点解决三个相互关联的问题。第一个项目是引入一类新的球形变种的加法类似物，它是由泊松-李群理论激发的简并构造的。第二个是探索拟阵舒伯特簇及其与环面几何的联系。第三，通过研究与辛分解有关的群胚，发展Poisson几何与辛表示理论之间的新联系。该项目由代数和数论计划和既定的激励竞争性研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Geometric representation theory studies the algebraic structures formed by symmetries of geometric objects. It has connections with many areas of algebra and geometry, including algebraic combinatorics, algebraic geometry, mathematical physics, and symplectic geometry. The present project will explore this rich interplay by developing new representation-theoretic objects in algebraic and symplectic geometry. It will also provide research training opportunities for graduate students. In more detail, the project will focus on three interrelated problems. The first project is to introduce a new class of additive analogues of spherical varieties, constructed using degenerations motivated by the theory of Poisson-Lie groups. The second is to explore matroid Schubert varieties and their connections to toric geometry. The third is to develop new connections between Poisson geometry and symplectic representation theory by studying groupoids associated to symplectic resolutions. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.