Cohomology and Representation Theory: Reductive Algebraic Groups and Related Structures

首席研究员将研究涉及还原代数群表示论的问题。与还原群相关的结构包括李代数、Chevalley群、Weyl群和中心化代数。研究中涉及的方法和结构将是代数的和几何的。基本主题将是在对象的上同调和表示之间建立联系，以证明有关这些结构的新的、有趣的结果。建立这种关系也有助于提供具体的计算。被称为“群”、“环”和“李代数”的代数对象出现在生物学、化学和物理学的许多不同的物理应用中。这些代数对象通常具有复杂的内部结构和对称性。提取有关这些结构的信息可以提供重要信息，这些信息可用于诸如上述的一系列应用。 该项目属于表示论领域，该领域现在是数学的中心领域，因为它提供了研究复杂代数结构的系统方法。粗略地说，表示可以被认为是某些代数对象从不同视角的“快照”。这些快照是通过明确描述的矩阵提供的。通过将表示中的信息组合在一起，可以回答围绕这些复杂代数系统的许多问题。

The principal investigator will investigate problems involving the representation theory of reductive algebraic groups. The structures related to reductive groups include Lie algebras, Chevalley groups, Weyl groups and centralizer algebras. The methods and constructions involved in the study will be algebraic as well as geometric. The underlying theme will be to establish connections between the cohomology and representations of the objects in order to prove new and interesting results about these structures. Establishing such relationships also lends itself to providing concrete calculations. The algebraic objects known as "groups," "rings" and "Lie algebras" arise in many different physical applications in biology, chemistry and physics. These algebraic objects in general have complex internal structures and symmetries. Extracting information about these structures can provide vital information which can be used in a range of applicationssuch as those mentioned above. This project is in the areaof representation theory, which is now a central area of mathematics because it provides a systematic method for studying complicated algebraic structures. Roughly speaking, representations can be thought of as ``snapshots'' of some algebraic object fromdifferent viewing angles. These snapshots are provided via explicitly described matrices. By putting together the information from the representations, many questions surrounding these complicated algebraic systems can be answered.