Springer Theory for Symmetric Spaces, Real Groups, Hitchin Fibrations, and Geometric Langlands

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This research project naturally sits at the intersection of representation theory and geometry. Representation theory is a branch of mathematics devoted to the study of symmetries that occur in nature using techniques from linear algebra, for example the study of symmetries in three-dimensional space or more generally the study of continuous symmetries of mathematical objects and structures (known as theory of Lie groups). Geometric methods have been very successful in solving problems in representation theory. The main goal of this project is to study various questions in representation theory using geometric methods. The PI will attack several longstanding problems concerning dualities for Lie groups. The PI will also investigate applications of representation theory to algebraic geometry, number theory, and related areas. Differential equations and integrable systems whose coefficients are residues modulo a prime are the subject of the other parts of the research project.In more detail, three projects will be pursued. In the first project, a generalized Springer correspondence will be developed for symmetric spaces. This project is closely related to deep questions in algebraic geometry, real groups, and harmonic analysis on p-adic groups. In the second project, the geometry of the so-called wonderful compactification of symmetric spaces will be used to prove Soergel's Koszul duality conjecture for real groups. In the third project, a theory of Hitchin fibrations for higher-dimensional varieties will be developed with the goals of constructing a non-abelian Hodge theory and establishing the geometric Langlands correspondence in positive characteristic for higher-dimensional varieties.