Mathematical Sciences: Representations of Semisimple Groups over Finite Fields and Quantum Groups

本研究关注的是 有限域上的半单群和量子群。 的 首席研究员将研究有限 量子群的维不可约表示. 根1。 他还将研究字符束及其 与有限李群的不可约特征标的关系 类型. 最后，他将研究包络的规范基 代数 其中一位博士后同事将研究朗的 正特征猜想与某些群的Chow群 曲线的模空间 另一位博士后助理将 关于代数的Lefschetz迹公式的证明 Stacks和Weil关于Tamagawa数的猜想 功能字段。 另一位博士后助理将研究这两个 分析工具的应用，以了解 量子群和使用量子群来接近 分析性的问题。 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 representations of semisimple groups over finite fields and quantum groups. The principal investigator will study characters of finite dimensional irreducible representations of quantum groups at roots of 1. He will also study character sheaves and their relation with the irreducible characters of finite groups of Lie type. Finally, he will study canonical bases of enveloping algebras. One of the postdoctoral associates will work on Lang's conjecture in positive characteristic and Chow groups of some moduli spaces of curves. Another postdoctoral associate will work on a proof of the Lefschetz Trace Formula for algebraic stacks and the conjecture of Weil on Tamagawa Numbers over function fields. Another postdoctoral associate will study both the applications of analytic tools to the understanding of quantum groups and the use of quantum groups to approach questions of analytic interest. 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.