Topological and geometric invariants from representation theory

表示论起源于世纪末的数学家Issai Schur和Ferdinand Frobenius的工作，研究代数对象（如有限群或李代数）的结构如何被更简单的线性对象（特别是矩阵）捕获。 这个项目的所有目标的一个共同主题是使用表示理论来研究几何空间。 三维拓扑量子场论、量子链接不变量和顶点算子都是在20世纪80年代和90年代使用各种代数输入发展起来的。该项目的一个主要目标是通过使用表示论的代数结构将这些结构提升到更高的维度。 这个目标是一个被称为分类的程序的一部分。分类程序是由Louis起重机和Igor Frenkel提出的。 一般来说，目标是向量空间应该被范畴所取代，向量空间上的作用应该被函子在范畴上的作用所取代。 在这个项目中涉及的类别是一个表示理论或组合的性质。 新的范畴理论的想法需要发展，以实现最雄心勃勃的目标的分类计划。 这个项目的另一部分是应用代表理论的技术，以分类问题的某些代数簇配备了一个组行动。 该项目的四个方面是：（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）理解仿射球面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.