Topological and geometric invariants from representation theory

表示理论起源于19世纪末的数学家Issai Schur和Ferdinand Frobenius的工作，它研究如何通过更简单的线性对象，特别是矩阵来捕捉代数对象(如有限群或李代数)的结构。这个项目的所有目标都有一个共同的主题，那就是使用表示理论来研究几何空间。三维拓扑量子场理论、量子链接不变量和顶点算符都是在20世纪80年代和90年代利用各种代数输入发展起来的。这个项目的一个主要目标是通过使用表示理论中的代数结构将这些结构提升到更高的维度。这个目标是被称为分类的程序的一部分。分类程序是由Louis Crane和Igor Frenkel提出的。一般说来，其目的是用范畴代替向量空间，用范畴上函子的作用代替向量空间上的作用。本项目所涉及的范畴具有代表性、理论性或组合性。为了实现分类计划最雄心勃勃的目标，需要开发新的范畴理论思想。本项目的另一个部分是将表示论技术应用于具有群作用的某些代数簇的分类问题。这个项目的四个方面是：(1)Reshetikhin-Turaev不变量的分类，(2)Turaev-Viro不变量的分类，(3)李超代数的分类，(4)了解仿射球簇的坐标环的乘法结构。

Representation theory, originating in the work of mathematicians such as Issai Schur and Ferdinand Frobenius at the end of the 19th century, studies how the structure of algebraic objects (such as finite groups or Lie algebras) can be captured by simpler, linear objects, in particular, matrices. A theme common to all of the goals of this project is the use of representation theory to study geometric spaces. Three dimensional topological quantum field theories, quantum link invariants, and vertex operators were all developed in the 1980s and 1990s using various algebraic inputs. One main objective of this project is to lift these constructions to higher dimensions through the use of algebraic structures from representation theory. This objective is part of a program known as categorification.The program of categorification was introduced by Louis Crane and Igor Frenkel. Generally speaking, the aim is that vector spaces should be replaced by categories and actions on vector spaces should be replaced by actions of functors on categories. The categories involved in this project are of a representation theoretic or combinatorial nature. New category-theoretic ideas need to be developed in order to achieve the most ambitious goals of the categorification program. Another part of this project is the application of representation-theoretic techniques to the classification problem of certain algebraic varieties equipped with a group action. The four areas of this project are the following: (1) categorification of the Reshetikhin-Turaev invariant, (2) categorification of the Turaev-Viro invariant, (3) categorification of Lie superalgebras, and (4) understanding the multiplicative structure of the coordinate rings of affine spherical varieties.