几何表示论中的超局部方法

超局部方法是数学中很多领域中的有力工具。总的来讲，比起局部几何，即在空间中一点附近的研究，超局部几何是指在一个余向量（或等价的，在空间的余切从中的一点）附近的更精细的研究。比如说，给定一个函数或者层，超局部方法使我们不仅可以测量它在哪些点上是奇异的，而且还可以更精细的测量它沿哪些方向是奇异的。超局部几何自然的出现在几何表示论和几何朗兰兹纲领中，例如，由Lusztig引入的特征层，作为类李有限群特征理论的范畴化，可以定义成约化群上满足超局部支集在幂零锥中的层。又如几何Langlands纲领预测了两个范畴的等价，而他们都各自是由幂零的超局部条件定义的。在这个课题中，我将使用超局部几何的方法研究几何表示论当中的问题，三个主要的目标分别是：1. 椭圆曲线上的几何Langlands纲领，我们利用超局部支集将其关联至特征层和Hecke范畴的研究。2. 交换簇上的函数。3. 由特征层定义的拓扑场论。

The microlocal method has been a power tool in many part of mathematics. Broadly speaking, compare to local geometry, which means the study near a point in the space, microlocal geometry refers to a more refined study near a cotangent vector, or equivalently, near a point in the cotangent bundle. For example, give a function/sheaf, microlocal method allow one to measure not only on which point it is singular, but also alone which direction it is singular. In geometric representation theory and geometric Langlands program, the microlocal conditions naturally show up: for example, the character sheaves, which introduced by Lusztig as a categorification of the character theory of finite group of Lie type, can be defined as sheaves on a reductive group with nilpotent microlocal support (or singular support). Another example is that geometric Langlands program predict the equivalence of two categories, both of them are defined by nilpotent singular support condition. In this project, I will use microlocal method to study questions in geometric representation theory, three main goals are：1. geometric Langlands program on elliptic curves, we use microlocal support to related it to the study of character sheaves and Hecke categories. 2 Functions on commuting varieties. 3. The topological field theory defined by character sheaves.