Gauge theory and sympletic, contact geometry

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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.