De Rham Theory of Loop Spaces and Quantum Topological Invariants

研究了位形空间的环空间同调的代数结构，并阐明了其与辫状体有限型拓扑不变量的关系。特别地，我们研究了组态空间的环空间与有限型拓扑不变量的同调性。研究了复上半平面上与Fuchsian群作用相关的轨道位形空间的环空间的同调性。证明了这种环空间的全同构与商面上的水平弦图代数同构。基于Browder运算，给出了轨道位形空间迭代环空间同调的泊松代数结构。本文用超几何积分的形式给出了黎曼球上共形场理论的共形块空间的完整描述。特别地，我们澄清了积分环是在补上的判别排列上定义的局部系统中具有系数的局部有限链的同调中的可正则环。Morita研究了各种模空间及其相关模群的结构，如Riemann曲面映射类群的模空间和图的模空间-自由群的外自同构群。关于结的琼斯不变量的渐近性、结补的双曲体积和3-流形的几何结构的猜想，村上给出了新的观点。

We investigated the algebraic structure of the homology of the loop spaces of configuration spaces and clarified its relation to finite type topological invariants for braids. Especially, we studied the homology of the loop spaces of configuration spaces and finite type topological invariants. We focused on the homology of the loop spaces of orbit configuration spaces associated with the action of Fuchsian groups on the complex upper half plane. We showed that the total homology of such loop space is isomorphic to the algebra of horizontal chord diagrams on the quotient surface. We introduced a structure of a Poisson algebra for the homology of the iterated loop space of the orbit configuration space based on the Browder operation.We gave a complete description of the space of conformal blocks for the conformal field theory on the Riemann sphere in terms of hypergeometric integrals. In particular, we clarified the integration cycles as the regularizable cycles in the homology of locally finite chains with coefficients in a certain local system defined over the complement a disrciminantal arrangement.Morita investigated the structure of various moduli spaces as well as their associated modular groups, such as the moduli space of Riemann surface -mapping class groups and the moduli space of graphs -outer automorphism group of free groups. Murakami gave a new point of view on a conjecture concerning the asymptotics of the Jones invariants for knots, the hyperbolic volume of the knot complement, and the geometric structure of 3-manifolds.