Mathematical Sciences: Cohomology and Modular Representations of Algebraic Groups

本课题主要研究半单代数群的上同调和模表示以及它们的李代数和相关的李型有限子群的一些相互关联的问题。我们将特别关注FLAG变种上的线丛的上同调，其中有许多关于模结构和消失行为的公开问题。对于有限群，研究将集中于内射模和普通表示的分解行为mod p。群是定义了乘法的代数结构。这些结构普遍存在于数学、化学、物理和计算机科学的许多领域。有限群可以看作是向量空间变换的代数集合，通过这些表示，可以确定它们的性质和结构。

This project is concerned with a number of interrelated questions about the cohomology and modular representations of semisimple algebraic groups, along with their Lie algebras and related finite subgroups of Lie type. Special attention will be given to the cohomology of line bundles on flag varieties, where there are many open questions about module structure and vanishing behavior. For the finite groups, the research will center on injective modules and on the decomposition behavior mod p of ordinary representations. A group is an algebraic structure with a multiplication defined on it. These structures occur commonly in many areas of mathematics, as well as, chemistry, physics and computer science. Finite groups may be viewed as algebraic sets of transformations of vector spaces and through these representations, it is possible to determine their properties and structure.