Categorical Algebra and Topological Field Theory

我们研究了高亏格的Morse同伦理论，发现它与Chern-Simons扰动理论是一致的，我们正在尝试量子化S余边定理，这是代数K-理论在拓扑学中的一个重要应用。我们在Floer同调中找到了怀特黑德扭转的一个版本。作为应用，我们找到了一对具有平凡Floer同调(拉格朗日交)但不能被哈密尔顿微分同态分解的Lalangina子流形的例子。Fukaya和Ono证明了关于哈密尔顿系统的周期轨道数的Arnold猜想的同调版本。Fukaya和Ono构造了一般光滑流形的Gromov-Witten不变量。Fukaya和oh研究了具有拉格朗日边界条件的余切丛中的伪全纯圆盘，发现它在基础上等价于有理同伦理论。我们研究了无穷大范畴的同调代数，并证明了Yoneda引理的一个版本Fukaya(利用拉格朗日子流形)定义了3个有边界流形的Foer同调，并构造了一个与辛流形相关的A无穷范畴。Fukaya完成了一篇论文，包括对它们的说明和同伦代数部分的详细证明。Fukaya写了一篇论文，描述了它的主要部分，并正在准备描述分析的全部细节的论文。

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.