CAREER: Realizing Derived Equivalences

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DMS-0134938Brooke E. ShipleyThe research component of this proposal involves several questions which fall into two overall projects. One project is to consider when an equivalence between derived categories is induced by a more richly structured equivalence. The goal here is to develop an obstruction theory for realizing derived equivalences by underlying structured equivalences. In particular, this may have applications to the question associated to Broue's Conjecture in representation theory of which stable equivalences lift to derived equivalences. The second project is to develop homotopy theoretic algebraic models. One piece of this project is to continue work with John Greenlees on algebraic models for rational equivariant cohomology theories. More generally, the investigator proposes to develop algebraic models for various stable homotopy theories. The proposed research projects involve the interplay beween the study of algebraic structures and topology, the study of shapes or spaces. Algebraic topologists use algebraic structures to describe and simplify topological phenomena. In a project onrealizing derived equivalences, the investigator hopes to use techniques developed in algebraic topology to attack questions which originate in algebra. These techniques include the use of a theory of obstructions, which determines whether certain constructions are possible. In another project the investigator plans on extending existing algebraic models to include structures involving symmetries. The educational component of this proposal also includes several parts. The investigator will be involved in the summer research programs for undergraduates at the University of Chicago and Purdue University. With Lucho Avramov, she will organize a workshop on topics related to derived categories of interest to a broad range of algebraically related fields. With Jim McClure and Guershon Harel, the investigator will develop a new course for mathematics education majors based on Euler's ``Elements of Algebra". The goals of this course are to help mathematical education students master skills in algebra, develop confidence for teaching algebra, and understand the motivation for material contained in a standard abstract algebra course. The investigator will also participate in several aspects of the Women in Science Program at Purdue University.