Geometric and Cohomological Invariants in Modular Representation Theory

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Representation theory studies actions of groups and other algebraic structures on vector spaces. It has its origins in the study of symmetries and emerged as a subject in its own right more than a hundred years ago in the work of Frobenius and Schur. In its current stage of development, representation theory is intimately intertwined with numerous branches of mathematics, such as geometry, topology, combinatorics, and analysis, as well as with physics. In this project the PI will establish new connections between representation theory and geometry, and develop techniques that will shed light on longstanding problems in both areas. The project also contains several educational outreach initiatives aimed at elementary, middle and junior high school student in Seattle area. These initiatives range from an enrichment math program at a local elementary school, to a city wide math circle for middle schoolers, to public math lectures and a special Math Olympiad for students in grades 5-10. These opportunities are designed to attract more young people, particularly young women, to careers in the mathematical sciences, and to raise awareness and appreciation of the beautiful nature of mathematics in the next generation.This research focuses on four interrelated directions traceable to Quillen's fundamental work in group cohomology from 1971. A significant part of the project concentrates on interpreting categories associated to representations in geometric terms. In the first three projects, the PI will classify triangulated subcategories using support theories, find an analogue of the Beilinson-Bernstein localization theorem for complex Lie algebras in the case of infinitesimal neighborhoods of algebraic groups in positive characteristic, and investigate the Orlov correspondence for representation theoretic categories. In a fourth project, which is of a more algebraic nature and has geometric applications, the PI will establish the finite generation of cohomology for a new class of finite dimensional Hopf algebras.