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

本文研究了有限域上半单群和量子群的表示。首席研究员将研究1根处量子群的有限维不可约表示的特征。他还将研究李型有限群的特征束及其与不可约特征的关系。最后，他将研究包络代数的规范基。其中一名博士后将研究曲线模空间的正特征群和周群中的Lang猜想。另一名博士后将研究代数堆栈的Lefschetz迹公式的证明和多摩川数在函数域上的Weil猜想。另一名博士后将研究分析工具在理解量子群中的应用，以及使用量子群来解决分析感兴趣的问题。量子群对于数学家和物理学家来说都是一个新的研究领域。在数学方面，它结合了“纯粹”数学的三个最古老的领域：代数、分析和几何，但它对研究共形量子场论的物理学家非常感兴趣。

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.