Categorical Algebra and Topological Field Theory

研究了高阶亏格的莫尔斯同伦理论，发现它与Chern-Simons扰动理论是一致的，并特灵证明代数K-理论在拓扑学中的一个重要应用--余边定理.我们在Floer同调中发现了一个版本的Whitehead挠。作为一个应用，我们找到了一个具有平凡Floer同调的Lalangina子流形的例子福谷和小野证明了关于哈密顿系统周期轨道个数的Arnold猜想的一个同调版本。福谷和小野构造了Gromov-维滕不变量.福谷和Oh研究了具有Lagrange边界条件的余切丛中的伪全纯圆盘，发现它与有理同伦等价在此基础上，我们研究了A无穷范畴的同调代数，证明了Yoneda引理的一个版本，这个同调代数在几何上有应用，定义了带边界的3维流形的Foer同调，构造了一个与辛流形相联系的A无穷范畴（利用Lagrange子流形）.福谷完成了一份文件，包括宣布他们和充分的细节证明homolomocai代数部分。一个需要分析的非线性偏微分方程的几何应用。福谷写了一篇论文描述了它的主要部分，并正在准备论文描述分析的全部细节。

We studied morse homotopy theory of higher genus and find that it coicides with Chern-Simons Perturbation theory.We are tring to quantize S cobordism theorem, which is an important application of algebraic K-theory to topology. We find a version of Whitehead Torsion in Floer homology. As an application we find an example of a pair of Lalangina submanifold such has a trivial Floer homology (of Lagrangian intersection) but can not be made disjoint by Hamiltonian diffeomorphism.Fukaya and Ono proved a homology version of Arnold conjecture on the number of periodic orbit of Hamiltonian system.Fukaya and Ono constructed Gromov-Witten invariant for general syjmplectic manifolds.Fukaya and Oh studied pseudo holomorphic disks in cotangent bundles with Lagrangian boundary condition and find that it is equivalent to the rational homotopy theory on the base.We studied homological algebra of A infinity category and proved a version of Yoneda's Lemma.This homological algebra has a geometrid application, to the definition of the Foer homology of 3 manifolds with boundary and construction of an A infinity category associated to an Symplectic manifolds, (using Lagrangian submanifolds.)Fukaya completed a paper including anouncement of them and the full detail of the proof of homolomocai algebra part.One need analysis of nonlinear PDE for geometric application. Fukaya wrote a paper describing a main part of it and is preparing papers describing the full detail of Analysis.