Representation theory via topological field theory

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The proposed research program addresses topological questions in algebraic geometry and representation theory. The focus is on the structure of categories of sheaves on moduli spaces arising in gauge theory. The goal is to apply topological field theory and derived algebraic geometry to the representation theory of real and complex Lie groups. The approach exploits two complementary viewpoints on the Geometric Langlands Program, taking advantage of its rich structure as a 4d topological field theory and as a geometric analogue of harmonic analysis. Another aim is to build upon the emerging dictionary between symplectic geometry and singularity theory, obtaining new insights into the structure of the Fukaya category and constructible sheaves. Applications include quantization of integrable systems, and the possibility of an elementary theory of perverse sheaves.The proposed program contributes to the rapidly expanding study of quantum analogues of classical geometry. It draws inspiration from both the Langlands program relating the harmonic analysis of symmetric spaces with the Galois symmetries of number fields and the mathematical physics of mirror symmetry and its higher dimensional antecedents. The techniques in play are of interest to a broad range of mathematicians including those that study symmetries and quantum systems. The program also offers many points of interface with high energy physicists working on strings and D-branes. The primary broader impact of the proposed research is educational, with an emphasis on the preparation of students to work in related areas.