Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

研究的重点是量子群在1的根上的表示理论，在具有平坦连接的3-流形的不变量上，以及在统计力学的某些方面。量子群起源于量子场论中对可积模型的研究。这些模型虽然相对于它们的现实对应物来说相当简单，但它们表现出的特性很难用常规方法来研究。另一类简单但非常有趣的量子场论是拓扑量子场论。在这些理论中，“动力学”是缺席的，所有他们“知道”的拓扑结构的“underlyingspace-time”。这些理论提供了新的拓扑不变量。“物理”公式涉及无限维空间的连接的集成，数学公式涉及量子群的表示。该项目的主要方向之一是协调两种方法。另一个研究方向是随机离散表面和相关的结构，如随机划分，平面的随机平铺，平面图上的随机生成树等。这个方向的一些问题与拓扑量子场论（称为Chern-Simons理论）转变为拓扑弦理论时的极限密切相关。该项目的一部分集中在单位根量子群的研究。我们的目标是研究一般模范畴作为纤维化在相应群上的monoidal范畴，以及当是这种情况时，作为纤维化在辫群上的辫monoidal范畴。所得结果将用于构造和研究具有平坦联络的三维流形的不变量。这个方向与近几年文献中讨论的A-多项式与纽结的Jones不变量之间可能存在的关系密切相关。研究也将集中在二聚体模型和统计力学的相关模型。例如，6顶点模型的极限形状可能具有奇异边界，目标之一是研究这种奇异性的波动。

The research is focused on the representation theory of quantumgroups at roots of 1, on invariants of 3-manifolds with flatconnections, and on some aspects of statistical mechanics. Quantumgroups emerged from the study of integrable models in quantumfield theory. These models though quite simplistic relative totheir realistic counterparts, exhibit properties which are hard tostudy using conventional methods. Another class of simplistic butextremely interesting quantum field theories are topologicalquantum field theories. In these theories the "dynamics" is absentand all they "know" in the topology of the "underlyingspace-time". The theories provided new topological invariants."Physical" formulation involves integration over the infinitedimensional space of connections, mathematical formulationinvolves representations of quantum groups. One of the majordirections of the project is to reconcile two approaches. Anotherdirection of research is related to random discrete surfaces andrelated structures such as random partition, random tilings of aplane, random spanning trees on a planar graph etc. Some of thequestions in this direction are closely related to the limit when atopological quantum field theory known as Chern-Simons theoryturns into a topological string theory.Part of the project is focused on thestudy of quantum groups at roots of unity. The goal is toinvestigate the category of generic modules as a monoidal categoryfibered over the corresponding group, and when it is the case, asas a braided monoidal category fibered over a braided group. Theresults will be applied to construct and study invariants of3-manifolds with flat connections. This direction is closelyrelated to the possible relation between the A-polynomial andJones invariant of knots, which was discussed in the literature inthe last few years. The research will also be focused on dimermodels and on related models in statistical mechanics. Forexample, the limit shapes for the 6-vertex model may have singularboundaries, and one of the goals is to study the fluctuations suchsingularities.