Moduli space, Homotopical algebra, Field theory

1. 在Fukaya， Oh, Ohta， Ono所著的《拉格朗日交集花理论-异常与阻碍》一书中建立了拉格朗日子流形花同调的基本故事。在本书中，我们用泛函的方式将滤波a无穷代数的结构与相对自旋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）的几个应用与弦理论的关系我们将理论推广到内部有标记点的情况同调镜像对称。第1节中提到的结果使得精确地陈述这个猜想成为可能。对它的证明有两种方法，一种是用渐近分析的方法，另一种是用解析几何的方法。第一个是fukaya提出的。与接触同调的关系利用拉格朗日花理论在环空间上的表述，我们可以将花理论与接触同调的正确关系表述为一种猜想。在辛拓扑中的应用fukaya证明了当且仅当L有直接的S^1因子时，面向素数的3流形L可以作为拉格朗日子流形嵌入到C^3中。他还用同样的方法证明了奥丁猜想，并将其推广到K π 1空间。在FOOO中，我们阐明了Hofer距离与Floer同调的扭转的关系。证明了单调拉格朗日子流形的Arnold给出猜想。证明了Flux猜想，该猜想证明了精确辛微同态群在辛微同态群中是C^1接近的。

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.