Mathematical Sciences: Geometric Aspects of Representations of Quantum Groups and Kac-Moody Lie Algebras

本研究关注表征的研究 Kac-Moody李代数和量子群的上同调 它们与构形空间几何的关系， 无限的旗帜品种，以及一些相关的问题。 的 首席研究员将研究 量子群和反常层的表示 仿射李代数的上同调， 位形空间的上同调;顶点算子和 仿射李代数的上同调;与几何学的联系 无限旗空间; Selberg积分，结构 共形场论中算子代数的常数，以及 它们在有限域上的类似物;以及更高维的 概括。 量子群是一个新的研究领域， 数学家和物理学家 在数学方面，它 结合了“纯”数学的三个最古老的领域， 代数，分析和几何，但它是非常感兴趣的， 研究共形量子场论的物理学家

This research is concerned with the study of representations and cohomology of Kac-Moody Lie algebras and quantum groups and their relationship with the geometry of configuration spaces and infinite flag varieties, and some related questions. The principal investigator will study the connection between representations of quantum groups and perverse sheaves in configuration spaces; cohomology of affine Lie algebras and cohomology of configuration spaces; vertex operators and cohomology of affine lie algebras; connections with the geometry of infinite flag spaces; Selberg integrals, the structure constants of operator algebras in conformal field theory, and their analogues over finite fields; and higher dimensional generalizations. Quantum groups are a new area of research for both mathematicians and physicists. On the mathematical side, it combines three of the oldest areas of "pure" mathematics, algebra, analysis and geometry, yet it is of great interest to physicists working on conformal quantum field theory.