Sheaves, Representations, and Dualities

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A duality in mathematics or theoretical physics is a correspondence between two theories which allows one to relate key phenomena in one theory to those in the other, despite their somewhat different, in a sense opposite, nature. A rough analogy is to the relation between a visual image and its reflection in a mirror. This project is in representation theory, that is, the study of algebraic structure of symmetries. Fundamental dualities have been playing an increasingly central role in the subject. One such duality is geometric Langlands duality, an outgrowth of reciprocity laws in number theory. Another is mirror symmetry, a phenomenon in algebraic geometry with origins in physics. The present project will derive further consequences of these fundamental ideas to representation theory. An additional goal is to study the common features of different dualities that appear in representation theory in order to find a unified approach resulting in a better understanding of the nature of the observed phenomena. The Principal Investigator will involve graduate and undergraduate students in projects thus promoting mathematical education and research at various levels. The project has several aims. One is to advance rapidly developing topological methods in the theory of modular representations. Another is to further develop etale sheaf methods in harmonic analysis on p-adic groups and finite groups of Lie type. Here we plan to work on uncovering the geometric phenomena underlying the theory of endoscopy in harmonic analysis on p-adic groups. Finally, the PI will continue to work on the theory of quantized symplectic resolutions, an exciting recent chapter in representation theory with surprising connections to physics and algebraic geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.