Tensor Categories and Applications

这个项目致力于研究张量范畴，张量范畴是用精确的数学语言表达对称性的直观概念的抽象数学对象。这些对象自然出现在数学的各个领域。 此外，张量范畴与物理学的某些领域有关，例如凝聚态物理学，其中一些特殊类型的张量范畴用于描述任意子--由不同于玻色子或费米子类型的统计控制的准粒子。 张量范畴也出现在理论计算机科学中作为拓扑量子计算的框架，这是量子计算机设计的一个潜在框架。该项目的目的是推进有关张量范畴的知识。 该项目涉及以下领域：一般理论的发展，着眼于具体的分类定理;张量范畴的一般理论在其他数学领域的应用;以及寻找张量范畴新的有趣例子。 更具体地说，工作包括以下项目：研究具有特定Grothendieck环的半单张量范畴;研究可能的作用（或者，等价地，模范畴），研究正特征的对称张量范畴，利用Kazhdan-Lusztig胞腔和Soergel双模理论寻找张量范畴的新例子;探讨张量范畴在顶点代数和有限W-代数理论中的应用。

This project is devoted to study of tensor categories, which are abstract mathematical objects expressing an intuitive notion of symmetry in precise mathematical language. These objects naturally appear in various areas of mathematics. In addition, tensor categories are relevant to some areas of physics, such as condensed matter physics, where some special types of tensor categories are used to describe anyons -- quasi-particles that are governed by statistics different from bosonic or fermionic type. Tensor categories also appear in theoretical computer science as a framework for topological quantum computation, one potential framework for design of a quantum computer. The aim of this project is to advance knowledge about tensor categories. The project addresses the following areas: development of general theory with an eye on specific classification theorems; applications of general theory of tensor categories to other areas of mathematics; and search for new interesting examples of tensor categories. More specifically, the work includes the following projects: study of semisimple tensor categories with specific Grothendieck rings; study of possible actions (or, equivalently, of module categories) of well-known tensor categories constructed via the Wess-Zumino-Witten model; study of symmetric tensor categories in positive characteristic; search for new examples of tensor categories using theory of Kazhdan-Lusztig cells and Soergel bimodules; and search for applications of tensor categories in the theory of vertex algebras and finite W-algebras.