Symplectic structures and singularities

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It is not always the case that Floer homology for pairs of Lagrangian sumanifolds can be defined. We constructed the obstruction theory for defining Floer homology for pairs of Lagrangian submanifolds in order to clarify when it is defined. When all the obstruction classes vanish, Floer homology can be defined. However, it depends on a choice of so-called bounding chains. Dependence of Floer homology over bounding chains can be understood in the framework of filtered A_∞-algebra associated to Lagrangian submanifolds. This algebra controls the deformation (extended moduli) of unobstructed Lagrangian submanifolds and is important in itself. These results are presented in a preprint by Fukaya, Oh, Ohta and Ono.Ono and Ohta classified diffeomorphism types of minimal symplectic fillings of links of simple singularities and simple elliptic singularities (complex dimension 2). For an isolated singularity, the minimal resolution and the Milnor fibe, if it exists, give typical example of minimal symplectic fillings. But they a not diffeomorphic in general. In the case of simple singularity, they turn out diffeomorphic thanks to existence of the simultaneous resolution by Brieskorn. We studied this phenomenon from contact/symplectic viewpoint. Kanda also contributed in a course of this research.