The main theme of the proposed research is the interaction among symplectic algebraic geometry, quantization, and geometric representation theory. Along the way the PI plans to establish several unexpected links between seemingly unrelated topics. Specifically, the PI will prove a 'quantum version' of an important equivalence of categories due to Bezrukavnikov. This 'quantum version' relates a category of D-modules on the base affine space to an equivariant derived category of sheaves on the affine Grassmannian. The PI then expects to find a natural interpretation of dynamical Weyl groups in terms of the algebra of differential operators on the base affine space. In a different direction, motivated by recent work of Braverman, Maulik, and Okounkov, the PI will address the problem of finding an interpretation of the canonical flat connection on the quantum cohomology of the Springer resolution as a Gauss-Manin connection. The idea here is that the correct mirror dual of the Springer resolution is a "noncommutative space" and the mirror dual of the quantum connection is the Gauss-Manin connection on periodic cyclic homology of that noncommutative space.
The proposed work opens up a wide variety of independent directions for research. One of the central features of the proposed projects is the application of methods and results developed in one branch of mathematics to a totally different area of mathematics. For example the research involves the systematic application of various algebraic techniques to problems in geometry and representation theory. Thus one of the goals of the project is to enhance the concept of the "unity of mathematics". The proposed work should also have far reaching applications not only in various areas of mathematics, but also in mathematical physics and theoretical physics, especially in quantum physics. The mathematical sciences, viewed as a whole, make a profound contribution to society. The current project goes a long way toward promoting the interactions between different disciplines and enhancing our scientific understanding.
