Wonderful Varieties, Hyperplane Arrangements, and Poisson Representation Theory

奇妙的品种、超平面排列和泊松表示论

• 批准号: 2401514
• 负责人: Ana Balibanu
• 金额: $ 29.87万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401514&HistoricalAwards=false
• 关键词: Wonderful; Varieties; Hyperplane; Arrangements; Poisson

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.

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几何和辛几何中发展新的表征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，该项目将侧重于三个相互关联的问题。第一个项目是引入一类新的球形变体的加性类似物，它是由泊松-李群理论驱动的退化构造的。二是探索拟阵舒伯特变分及其与环面几何的联系。第三是通过研究与辛分辨相关的群类群，发展泊松几何与辛表示理论之间的新联系。该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。