Gauge theory and sympletic, contact geometry

1. We proved that the intersection form of any symplectically filling 4-manifold of the link around the simple singularity C-D12-D1/-is negative definite。为了证明这一点，我们展示了一个关于塞伯格-维顿入侵者的消失理论。In the case = EイイD28イD2， we proved the intersection form of any minimal symplectically filling 4-manifold is equivalent to EイD28イD2。我们发现了一些拓扑限制，对简单的椭圆形单体填充4-单元。这些都是与K一起工作的。Ono.2。我们研究了一个结构来定义Langrangian的交集，在一般情况下的浮动同源性。我们构建了一个结构类的系统，它是拉格朗日亚群的Q-同源类。如果我们的作品类别消失了，那么我们就可以定义Floer同源性。对于建筑，我们研究了J-全息盘模块空间的方向问题。在某些方面，我们发现了一种适合模块化空间的良好条件，使之成为有意义的。我们将进一步应用我们的创作理论和修改的Floer同源理论，用于Lagrangian交集、Arnold-Givental概念和Maslov指数概念。这些都是与K一起工作的。Fukaya， M. Kontsevich，Y. - G..哦，还有K. Ono。

1. We proved that the intersection form of any symplectically filling 4-manifold of the link around the simple singularity CィイD12ィエD1/Γ is negative definite. For the proof, we showed a vanishing theorem on the Seiberg-Witten invariants. In the case Γ = EィイD28ィエD2, we proved the intersection form of any minimal symplectically filling 4-manifold is equivalent to EィイD28ィエD2. Moreover we obtained some topological restriction of symplectically filling 4-manifolds for the simply elliptic singularities. These are joint works with K. Ono.2. We investigated obstructions to define Langrangian intersection Floer homology in general situation. We constructed a system of the obstruction classes, which are Q-homology classes on the Lagrangian submanifold. We proved that if our obstruction classes vanish, then we can define the Floer homology. For the construction, we studied orientation problem of moduli spaces of J-holomorphic disks. In particular, we obtained a sufficient condition for the moduli spaces to be orientable. Furthermore we applied our obstruction theory and modified Floer homology theory to the Arnold conjecture for Lagrangian intersections, the Arnold-Givental conjecture and the Maslov index conjecture. These are joint works with K. Fukaya, M. Kontsevich, Y.-G. Oh and K.Ono.