A Study of Conjectures in Modular Representation Theory

Alperin的9988710号提案摘要在有限群的表示论中有许多深刻的理论，但其中之一，Broue猜想，现在已经获得了大量的关注，原因有很多。 首先，它意味着新的结果有关的字符，主要问题是结构的代数和派生类别，并有许多连接的表示理论群的李型和其他部分的数学。 主要研究者正在从六个不同的角度研究这个猜想和相关问题，包括特殊但重要的情况的证明，以及可能的推广，这将揭示完整解决方案所需的想法。 他还致力于其他几个问题的模块化表示理论，一个动机的一些工作代数拓扑。群论是一个古老的主题在数学，其根源在代数，但与应用各地的数学和科学。其中一个中心议题是群体的表征理论，它研究了该主题在其他领域出现的最常见方式。 调查人员正在研究一些关键的未决问题。 特定的主题是那些在密码学中确实有联系点的主题，例如，在所谓的群码中，即使这项工作正在寻求一般的新结果，而不是当前的应用。

ABSTRACT for Alperin's Proposal 9988710There are many deep conjectures in the representation theory of finite groups but one of them, Broue's conjecture, has now garnered a great deal of attention, for a number of reasons. First, it implies new results about characters, the main question is about structure of algebras and derived categories and there are many connections with the representation theory of groups of Lie type and other parts of mathematics. The principal investigator is studying this conjecture and related questions from six different points of view, including proofs of special but significant case, and possible generalizations which would shed light on the ideas needed for a complete solution. He is also working on a few other questions in modular representation theory, one motivated by some work in algebraic topology.Group theory is an old subject in mathematics, its roots in algebra but with applications all over mathematics and science. One of the central topics is the representation theory of groups which studies the most common way the subject arises in other areas. The investigator is working on some of the key open problems. The particular topics are ones that do have points of contact in cryptography for example, in so-called group codes, even though the work is seeking general new results rather than current applications.