Mathematical Sciences: Cohomology and Modular Representations of Algebraic Groups

本文研究了半单代数群和李型有限群的上同调和表示理论中的一些密切相关的问题。一个主要的焦点是旗帜品种上的线丛的上同调，其中对模结构的研究将有助于解释消失行为。这项研究是在代数群的一般领域进行的。建议的研究分为两个大标题：旗簇上的线丛的上同调和内射模。沿着这些思路取得的进展既可以解释卢兹蒂希的猜想，也可以指出超越猜想的方向。

This research is concerned with a number of closely related questions about the cohomology and representation theory of semisimple algebraic groups and finite groups of Lie type. One major focus is the cohomology of line bundles on flag varieties, where the study of module structure would illuminate the vanishing behavior. This research is in the general area of algebraic groups. The proposed research falls under two broad headings: cohomology of line bundles on flag varieties, and injective modules. Advances along these lines would both shed light on Lusztig's conjecture, as well as, point the way beyond it.