Quantum topological structures in geometric representation theory

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The current research pursues new directions for geometric representation theory inspired by supersymmetric gauge theory. It also proposes new approaches to Fukaya categories of Lagrangian branes with consequences for mirror symmetry. Specific research includes a Fourier analysis of character varieties in terms of character sheaves, a theory of character sheaves for loop groups via bundles on elliptic curves, and new local and homotopical models for Lagrangian intersection theory. The methods are primarily algebraic and topological, but inspired by basic patterns found in harmonic analysis and microlocal analysis. Potential applications range from Langlands dualities for the cohomology of character varieties and categorical quantizations of bundles on elliptic curves to a sheaf-theoretic reformulation of Fukaya categories without appeal to holomorphic disks.The research aims to further interactions between mathematics and physics and to educate students in the new tools of homotopical geometry. Its focus includes objects at the crossroads of gauge theory, harmonic analysis, and the Langlands program. Activities include further exposition of important but difficult topics such as quantum field theory, as well as opportunities for students in diverse areas to interact with established researchers. It is difficult to estimate the impact outside of mathematics and physics, but the research has potential links to computational topology and its applications to understanding large data sets through small samples.