Higher Representations and Derived Equivalences

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This project will further the understanding of symmetries as a fundamental tool to organize structures in mathematics. The PI will bring new methods and results in representation theory, which is the mathematical study of symmetries via linear algebra, using higher categorical and homotopical methods. In the process, the PI will develop new computational approaches and new combinatorial objects are expected to arise. This project will topologically enhance classical structures in algebra in order to analyze symmetries and fixed points, with the ultimate aim of resolving some fundamental questions in representation theory. A key goal of the project is to make progress in the understanding of simple modules and decomposition matrices of finite groups of Lie type using perverse equivalences. The PI will work toward a proof of Broue's abelian defect group conjecture via Deligne-Lusztig varieties. This will require developing new homotopical methods to deal with certain fixed point constructions. The PI will also investigate genericity properties and look for combinatorial objects to encode numerical information.