Moduli space, Homotopical algebra, Field theory

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1. Lagrangian Floer theoryWe establish basic story of Floer homology of Lagrangian submanifolds in the book‘Lagrangian intersection Floer theory - anomaly and obstruction', written by Fukaya, Oh, Ohta, Ono.In this book, we aaaociate in a functorial way the structgure of filtered A infinity algebra to the cohomology group of relatively spin Lagrangian submanifolds.We gave several applications to symplectic topology (see 5) and to Mirror symmetry (see 3).2 Relation to string theoryWe generalized the theory to the case when there are interior marked points.3 Homological mirror symmetry.The result mentioned in 1 makes it possible to state the conjecture precisely. There are two approaches towerd its proof one is by asymptotic analysis and the other by rigi analytic geometry.The first one is proposed by Fukaya.4. Relation to contact homologyUsing the formulation of Lagrangian Floer theory by loop space we can state correct relation of Floer theory to contact homology, as a conjecture.5. Application to symplectic topologyFukaya proved that the prime oriented 3 manifold L can be embedded as a Lagrangian submanifold in C^3 if and only there L has a direct S^1 factor.He also proved Audin's conjecture together with its generalization to K pi one space using the same method. In FOOO we clafired the relation of Hofer distance to torsion of Floer homology.We proved Arnold Givental Conjecture for monotoneLagrangian submanifold in FOOO.Ono proved Flux conjecture which state that the group of exact symplectic diffeomorphisms are C^1 close in the group of symplectic diffeomorphisms.