Geometric Methods in Representation Theory

这是一个李理论和几何的项目。李理论以数学家索菲斯·李命名，可以描述为研究微分方程系统解的空间对称性。尤其重要的是模拟经典和量子力学的方程组。解空间的几何性质编码由微分方程建模的物理系统的性质。这个项目将主要使用代数技术来发展这些解空间的几何性质和对称性之间的联系。研究将集中在四个具体项目上。第一个项目是寻找共形块和Verlinde公式的扭曲类比，第二个项目是确定Kac-Moody李代数的饱和张量锥的无冗余不等式集，第三个项目是将经典的Jacobson-Morozov定理推广到对称化的Kac-Moody李代数，第四个项目是为群限制发展一个富尔顿猜想的一般拓扑模拟。这个奖项反映了国家科学基金会的法定使命，并通过评估被认为是值得支持的使用基金会的知识价值和更广泛的影响审查标准。

This is a project in Lie theory and geometry. Named after the mathematician Sophus Lie, Lie theory may be described as the study of symmetries of spaces of solutions of systems of differential equations. Especially important are systems of equations that model classical and quantum mechanics. Geometric properties of solution spaces encode properties of the physical systems being modeled by the differential equations. This project will develop the connections between the geometric properties and symmetries of these solution spaces, using mainly algebraic techniques. The research will focus on four specific projects. The first project will find a twisted analog of conformal blocks and the Verlinde formula, the second project is to determine an irredundant set of Inequalities for the saturated tensor cone of a Kac-Moody Lie algebra, the third project will generalize the classical Jacobson-Morozov Theorem to symmetrizable Kac-Moody Lie Algebras, and the fourth project is to develop a general topological analog of Fulton's Conjecture for group restrictions.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.