De Rham Theory of Loop Spaces and Quantum Topological Invariants

研究了位形空间的环空间的同调的代数结构，阐明了它与辫子的有限型拓扑不变量的关系。特别地，我们研究了位形空间的圈空间的同调性和有限型拓扑不变量。本文主要研究复上半平面上Fuchsian群作用下轨道位形空间的圈空间的同调性。我们证明了这种loop空间的全同调同构于商曲面上的水平弦图代数。基于Browder运算，给出了轨道位形空间的迭代圈空间的同调的Poisson代数结构，并利用超几何积分给出了黎曼球面上共形场论的共形块空间的完整描述.特别地，我们阐明了积分圈是定义在补a disrciminantal安排上的某个局部系统中系数的局部有限链的同调中的可正则化圈。Morita研究了各种模空间的结构以及与之相关的模群，如黎曼曲面-映射类群的模空间和图-自由群的外自同构群的模空间。村上给了一个新的观点的猜想有关的渐近性琼斯不变量的纽结，双曲体积的纽结补，几何结构的3流形。

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.