CAREER: Geometric category O and symplectic duality

本基金的目的是研究辛代数变种上某些类束的结构。当所讨论的变量是一个标志变量的余切束时，那么我们得到的范畴就相当于李代数表示的“O类”的一个块，该表示最初是由Bernstein， Gelfand和Gelfand引入的。因此，这一类别被称为“几何O类”。这一范畴被推测有许多美丽的性质，这些性质已经在某些特殊情况下得到了验证，这些特殊情况是由表征理论和多面体几何自然产生的。在这些猜想中最主要的是，这些范畴是科祖尔的，并且每一个辛变种都有一个“辛对偶”，而这个“辛对偶”的相关范畴是原始变种的科祖尔对偶。这一现象有望在李代数表示的各种分类方法和以前认为不相关的链接不变量之间形成一座桥梁。辛代数变体自然地出现在许多不同的数学领域。超曲变体的几何和拓扑性质为超平面排列的拓扑结构和拟阵的组合学提供了新的思路。颤振变体提供了无限维李代数作用的几何实现，导致正则基和在束的范畴上的作用。这项拨款在这幅图景中发挥了关键作用，为组合学中的Gale对偶和表示理论中的水平-秩对偶等现象提供了新的几何见解。这个项目将为这些独立的努力做出贡献，也将推进一种共同的治疗方法，将我们对各种个体现象的理解统一起来。除了研究部分，该资助还包括为研究生和博士后举办的年度研讨会，以架起这些不同领域的桥梁。

The purpose of this grant is to study the structure of certain categories of sheaves on symplectic algebraic varieties. When the variety in question is the cotangent bundle to a flag variety, then the category that one obtains is equivalent to a block of the "Category O" of representations of a Lie algebra, originally introduced by Bernstein, Gelfand, and Gelfand. For this reason, this category is referred to as "Geometric Category O". This category is conjectured to have many beautiful properties, which have been verified in certain special cases that arise naturally from representation theory and polyhedral geometry. Chief among these conjectures is that the categories are Koszul, and that each symplectic variety has a "symplectic dual" whose associated category is Koszul dual to that of the original variety. This phenomenon is expected to form a bridge between various approaches to categorification of Lie algebra representations and link invariants that were previously thought to be unrelated.Symplectic algebraic varieties arise naturally from many different areas of mathematics. Geometric and topological properties of hypertoric varieties have shed new light on the topology of hyperplane arrangements and the combinatorics of matroids. Quiver varieties provide geometric realizations of actions of infinite dimensional Lie algebras, leading to canonical bases and to actions on categories of sheaves. This grant plays a key role in this picture, providing new geometric insight to such phenomena as Gale duality in combinatorics and level-rank duality in representation theory. This project will contribute to each of these endeavors independently, and will also advance a common treatment that unifies our understanding of the various individual phenomena. In addition to the research component, the grant includes an annual workshop for graduate students and postdocs that bridges these various fields.