Floer homology, string theory and contact geometry

1. 代数曲面上孤立奇点的连杆具有自然接触结构。一般来说，链路的辛填充有各种拓扑类型。利用单极子方程和伪全纯曲线理论，证明了所谓简单奇点连杆最小填充的微分同胚型是唯一的。此外，我们还完全确定了简单椭圆奇点环的极小填充的微分同胚类型。通过乘积，我们还证明了包含有（2,3）点的正有理曲线的辛4流形一定是有理曲面。这些结果是与我们的一位研究员k.o o共同研究得出的。我们构造了一个与拉格朗日子流形L相关的滤波A_∞代数，给出了L的经典有理de Rham复形作为A_∞代数的量子变形。基于我们的A_∞代数，构造了拉格朗日交点Floer上同调中的阻塞和变形理论。从小对称猜想的观点出发，建立了辛边的数学基础。我们还将理论应用于辛几何中的Arnold猜想、Arnold- givental猜想和Maslov指数猜想等具体问题。这些结果是与k.f okaya， Y-G Oh和k.o ono共同研究得到的。建立高等属的理论是今后进一步研究的问题。

1. The link of an isolated singularity in an algebraic surface carries a natural contact structure. In general, there are various topological types of the symplectic fillings of the link. We proved that the diffeomorphism types of the minimal fillings of the link of so-called simple singularity are unique, by using the monopole equations and theory of pseudo-holomorphic curves. Moreover, we completely determined the diffeomorphism types of the minimal fillings of the links of simple elliptic singularities. By product, we also proved that symplectic 4-manifold which contains a positive rational curve with, a (2, 3)-cusp point must be rational surface. These results were obtained by joint works with K.Ono, one of our investigators.2. We constructed a filtered A_∞ algebra associated to a Lagrangian submanifold L. This gives a quantum deformation of the classical rational de Rham complex of L as A_∞ algebras. Based on our A_∞ algebra, we constructed an obstruction and deformation theories in Floer cohomology for Lagrangian intersections. From the point of view of minor symmetry conjecture, we established the mathematical foundations of the symplectic side. We also applied our theory to some concrete problems, for example, Arnold conjecture, Arnold-Givental conjecture and Maslov index conjecture, in symplectic geometry. These results were obtained by joint works with K.Fukaya, Y-G Oh and K.Ono. To establish a theory for higher genus cases is a problem for further research in the future.