Categorical Algebra and Topological Field Theory

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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.