Floor homology, singularities and deformation theory

我们从理论基础和应用两个方面研究了辛几何中的Floer理论。我们将哈密顿系统的Floer理论推广到拉格朗日交的Floer理论。即引入了滤子A_∞-代数、滤子A_∞-双模等概念，并给出了几何结构。作为应用，我们研究了拉格朗日嵌入的Maslov类的非平凡性、哈密顿变形下的拉格朗日交等，并用Floer理论证明了通量猜想。这一猜想是理解哈密顿群在辛同态群中的微分同态的基础。这些结果被写成研究论文，在项目期间预印。与Ohta一起，我们试图通过对链接的辛填充来理解复杂曲面上的孤立奇点。特别地，我们建立了极小辛填充的唯一子，并研究了它与Brieskorn关于简单奇点的结果之间的关系。对于简单椭圆型奇点，我们给出了光滑性的分类，并解释了Pinkham关于光滑性存在条件的结果，这些结果发表在研究期刊上。

We studied Floer theory in symplectic geometry in both theoretical foundation and applications. We extend Floer theory for Hamiltonian systems to that for Lagrangian intersections. Namely, we introduce the nation of filtered A_∞-algebras, filtered A_∞-bimodule etc. and give geometric constructions. As applications, we studied non-triviality of the Maslov class of Lagrangian embeddings, Lagrangian intersection under Hamiltonian deformations, etc. We also prove the flux conjecture using Floer theory for symplectomosphisms. This conjecture is basic for understanding how the group of Hamiltonian diffeomorphisms in the group of symplectomosphisms. These results are written up as research papers, preprints during the term of project.With Ohta, we tried to understand isolated singularities on complex surfaces through symplectic fillings of their links. In particular, we established uniquener of minimal symplectic fillings and studied its relation to Brieskorn's results for simple singularities. In the case of simple-elliptic singularities, we gave classification and interpreted Pinkham's result concerning the condition for existence of smoothings.These results are published in research journals.