Geometric Methods in Representation Theory

这是一个关于李理论和几何学的项目。李理论以数学家索菲斯·李的名字命名，可以被描述为研究微分方程组的解的空间的对称性。尤其重要的是建立经典和量子力学模型的方程组。解空间的几何性质编码了由微分方程式模拟的物理系统的性质。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。这项研究将集中在四个具体项目上。第一个项目将找到共形块和Verlinde公式的扭曲模拟，第二个项目是确定Kac-Moody李代数的饱和张量锥的一组无冗余的不等式，第三个项目将经典的Jacobson-Morosov定理推广到可对称化的Kac-Moody李代数，第四个项目是为群限制开发Fulton猜想的一般拓扑模拟。这个奖项反映了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。

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.