Microlocal Geometry in Gauge Theory

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The research supported by this grant lies at the crossroads of mathematics and physics. It aims to apply ideas originating in quantum field theory to problems in representation theory. A main theme is the study of Fourier-type symmetries not evident from traditional viewpoints. Another theme is the description of complicated homotopical constructions in intuitive combinatorial terms. The primary tools developed and applied belong to microlocal geometry, a subject with a rich history going back to classical mechanics. A broad goal is the education of students in the new frontiers of this rapidly developing direction. There will also be ample opportunities for outreach to other fields and for increased public engagement with mathematics. Specific projects include an elliptic theory of character sheaves for loop groups, a categorical Verlinde formula along with marked genus zero calculations, and a combinatorial model of microlocal sheaves. The methods involve the traditional microlocal geometry of differential equations and constructible sheaves, as well as the new homotopical microlocal geometry of coherent sheaves found within derived algebraic geometry. Applications include new geometric Langlands correspondences in genus zero and genus one, and calculations with combinatorial models of microlocal sheaves.