Moduli space, Homotopical algebra, Field theory

1.拉格朗日Floer理论我们在Fukaya，oh，Ohta，Ono所著的《拉格朗日相交Floer理论--异常和障碍》一书中建立了拉格朗日子流形的Floer同调的基本故事。在这本书中，我们用函子的方式将过滤后的无限代数的结构与相对自旋的拉格朗日子流形的上同调群联系起来。我们给出了辛拓扑(见5)和镜像对称(见3)的几个应用。2与弦理论的关系我们将该理论推广到有内部标记点的情况。3同调镜像对称。文1所述的结果使精确地表述猜想成为可能。它的证明有两种方法，一种是用渐近分析的方法，一种是用Rigi解析几何的方法，第一种是Fukaye提出的。联络同调的关系利用拉格朗日Floer理论在环空间的表述，我们可以作为猜想来表述Floer理论到联络同调的正确关系。在辛拓扑中的应用Fukaya证明了素数定向的3流形L可以作为Lagrangian子流形嵌入C^3中当且仅当那里L有直接的S^1因子，他还用同样的方法证明了奥丁猜想及其在Kpi-one空间上的推广.在文[1]中，我们讨论了Hofer距离与Floer同调的挠度之间的关系，证明了Arnold Givental猜想是关于FOOO中单调拉格朗日子流形的，也没有证明了Flux猜想，即严格辛微分同态群在辛微分同态群中是C^1闭的。

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.