Geometry of Langlands Duality

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The project is broadly directed towards geometry that is expected to underlie variousduality phenomena in mathematics (Langlands program) and physics (Gauge Theory).One goal is to reconstruct Loop Grassmannians and Mirkovic-Vilonen cycles from thepoint of view of Statistical Mechanics, and apply this to constructing two dimensionalloop Grassmannians. Another project is a to establish Langlands duality for categoriesof equivariant coherent sheaves on affine Steinberg varieties. A more technical projectis a geometric clarification of the Feigin-Frenkel description of the critical center Finally,the projects on Quantum Field Theory are on reformulating Costello?s mathematicalformulation of perturbative QFT in terms of flows and a speculation on the relation ofQFT to Grothendiecks function-sheaf dictionary. The most ambitious project attemptsto create geometric for the arithmetic case of Number Theory through a 'stochastic settheory'.The project aims towards relating and unifying developments in mathematics and theo-retical particle physics. The mathematical origin of these developments is the so calledLanglands program which is the modern view on a classical discipline of Number Theory. In time this program incorporated a number of central disciplines of mathematics, starting with Representation Theory, Algebraic Geometry and currently the Homotopy Theory. The relation to physics is a part of the current melting of barriers between mathematics and physics which arose in a period when the two subjects developed separately. The central impact on mathematics in the last quarter century was the import of the ideas from Quantum Field Theory (in particular String Theory), which is the part of physics that studies elementary particles. The proposed work attempts to work in both directions by applying ideas from physics to mathematics and mathematical constructions to physics. It also aims towards deeper understanding of the relation between Quantum Field Theory and Number Theory. The proposal also contains more standard topics within Representation Theory and Langlands Program.