Mathematics and theoretical physics (in particular, quantum and string theories) traditionally have a lot of connections and influence each other. This connections work in two sides. Mathematics is the main tool in theoretical physics. On the other side quantum and string theories produce new concepts and problems which become subjects of mathematical research. Most important and interesting results are obtained at the edge between these fields. The main aim of the project is to develop interdisciplinary connections between geometry and singularity theory on the one side and quantum field theory on the other. It consists of several adjoint themes, connected with quantum field theory: string theory, WDVV equations, Frobenius manifolds, Gromov-Witten invariants, mirror symmetry, integrable systems, Whitham hierarchies, Seiberg-Witten theory, matrix models, quantum topological invariants of knots and 3D-, 4D-manifolds, modern problems of theory of singularity and symplectic geometry. The goal of the project is to clarify connections between these theories and to use these connections for progress in them. The project includes, in particular: study of topology of moduli spaces of complex and real algebraic curves, corresponding Hurwitz spaces and their applications to quantum invariants of manifolds; development of tools for computing Gromov-Witten invariants of complex manifolds based on singularity theory; study of various spaces of mappings and their quantum invariants such as Vassiliev invariants of knots, 4-invariants of graphs, finite order invariants of complex mappings; mirror symmetry for Fano manifolds and especially 3D- and 4D- Fano manifolds, applications of integrable systems of KP/Toda/Calogero and Whitham type to study important problems of mathematical physics: Seiberg-Witten theory, Dijkgraaf-Vafa theory, Landau-Ginzburg models, matrix models etcetera; D-branes theory and associated deformations of open-closed topological field theory, Landau-Ginzburg models with boundaries, noncommutative Frobenius manifolds. |