Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这一建议的重点是在表示理论和数学物理的接口问题。更具体地说，它的目的是在1根的量子群的表示理论的问题，3流形的不变量，局部量子场论的某些问题（如摄动陈-西蒙斯理论的构造），以及在可积系统和统计力学的可解模型的问题。现代理论物理学的中心问题之一是如何建立与实验结果一致并在数学上适当的基本相互作用模型。局域量子场论的框架是标准模型背后的主要概念。然而，量子场论的框架在很大程度上仍然是一个数学难题。我的部分研究将集中于在半经典量化的背景下理解这个难题。提案其他部分的目标是组合构建拓扑和可积量子场论，并研究新出现的代数和分析问题。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The proposal is focused on problems which lie at the interface of representation theory and mathematical physics. More specially, it aims at problems in the representation theory of quantum groups at roots of 1, on invariants of 3-manifolds, certain problems in local quantum field theory (such as the construction of perturbative Chern-Simons theory), and on problems in integrable systems and solvable models of statistical mechanics.One of the central questions in modern theoretical physics is the construction of the model of fundamental interaction which is consistent with the experiment and mathematically adequate. The framework of local quantum field theory is main concept behind the standard model. However the framework of quantum field theory still largely remain a mathematical puzzle. Part of my research will focus on understanding this puzzle in the context of semi-classical quantization. The goal of other parts of the proposal is the construction of topological and integrable quantum field theories combinatorically and the study of emerging algebraic and analytical problems.