9322519 Kontsevich The project deals with interrelated subjects in differential topology, homotopical algebra and perturbative methods of quantum field theory. The central model is the quantum Chern-Simons theory with various generalizations to higher-dimensional and non-commutative settings. The method of Feynman diagrams gives new invariants of knots, manifolds and new cohomology classes of classifying spaces of diffeomorphism groups. A non-commutative version of the Chern-Simons theory produces cohomology classes of moduli spaces of complex algebraic curves. It will be applied to algebraic construction of Gromov-Witten invariants of symplectic manifolds and to mirror symmetry. The project applies a computational scheme developed several decades ago in particle physics and thermodynamics (Feynman diagrams) in various domains of modern mathematics. Feynman graphs (visualized as spider webs) turn out to be a very powerful tool which relates many abstract algebraic structures to geometric and topological objects of different dimensions. One of the recent applications was a new series of knot invariants (i.e. quantities distinguishing closed lines in the space without self-intersections). These new invariants are the first ones which are really computable via fast working algorithms, and they can be applicable in hydrodynamics and chemistry of long molecules. In general, the project gives a unified view on several parts of mathematics. There are many strong evidences that it leads to an understanding of so called "mirror symmetry," one of the deepest discoveries of recent years in string theory, which is a mathematical theory treating gravitation in a quantum mechanical setting. ***
