This project studies symplectic topology, which is a mathematical foundation of Hamiltonian dynamics. The research is also related to the mathematical foundation of string theory in theoretical physics. The project includes a systematic study of the "transversality problem" in those theories. We may regard this problem as a mathematical equivalent to the problem of infinity in quantum field theory in physics. The work is expected to answer essential questions in the mathematical foundation of theoretical physics. The project also studies several mathematical problems appearing in differential equations of classical mechanics.
In this project, the PI and collaborators will study moduli spaces of holomorphic maps from bordered Riemann surfaces to a symplectic manifold with Lagrangian boundary conditions. This is a generalization of the theory of Floer homology which provides a mathematical formulation of topological open string theory. At the same time, the project will deepen the understanding of the virtual fundamental chains, which has important applications in the study of moduli spaces in differential geometry and topological field theories. Applications to Mirror symmetry and concrete problems in symplectic topology will also be investigated.
