Geometry of Langlands Duality

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This project is directed towards search for geometry that is expected to underlie the Langlands conjectures. One aspect is the interpretation of Langlands duality of reductive groups in terms of homotopy theory (in the framework of Derived Algebraic Geometry). Another aspect is a continuation of the work on developing a particular geometric Koszul duality (``Linear Koszul Duality'') and applying it to representation theory. In relation to physics, the project envisions a mathematical formalism of t'Hooft operators and a construction of a certain extended topological field theory in four dimensions, with applications to knot theory. The project also contains a speculation on the relation of number theory and physics. Here, a geometrization of number theory is expected to arise through a connection to quantum field theory. The project aims towards relating and unifying developments in mathematics and theoretical particle physics. The origin of these developments in mathematics is the so called Langlands 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, then 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. Recently, Witten, Kapustin and Gukov established a long sought bridge between Langlands program and quantum field theory (in particular gauge theory and string theory). 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.