A Study of Conjectures in Modular Representation Theory

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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.