Tensor Categories and Applications

这个项目致力于张量范畴的研究，张量范畴是抽象的数学对象，用精确的数学语言表达了一种直观的对称概念。这些物体自然而然地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，其中一些特殊类型的张量范畴被用来描述任意子--受不同于玻色子或费米子类型的统计学支配的准粒子。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，这是设计量子计算机的一个潜在框架。这个项目的目的是增进对张量范畴的了解。该项目涉及以下领域：着眼于具体分类定理的一般理论的发展；张量范畴的一般理论在其他数学领域的应用；以及寻找张量范畴的新的有趣的例子。具体地说，工作包括：研究具有特定Grothendieck环的半单张量范畴；研究通过Wess-Zumino-Witten模型构造的著名张量范畴的可能作用(或者等价地，模范畴)；研究对称张量范畴的正特征；利用Kazhdan-Lusztig胞格和SoerGel双模理论寻找张量范畴的新例子；以及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.