Theory of braids, hyperplane arrangements and applications to conformal field theory

1. Loop spaces of orbit configuration spaces}在与F.Cohen和M.Xicont\'encatl的合作中，我对配置空间的loop空间的同调的代数结构进行了研究。利用李代数描述了上半平面上与Fuchsian群作用相关的轨道位形空间的圈空间的同调，建立了与曲面上chid图代数的关系，并基于Chen迭代积分研究了位形空间的圈空间的de Rham上同调.作为应用，我发展了一种基于位形空间的环空间的上同调类构造链同伦不变量的系统方法.迭代积分和双曲体积}已知K。Aomoto指出，球面单形或双曲面单形的体积可以用基于Schl\'afli等式的对数形式的迭代积分来表示。利用这种方法，我们描述了从球面几何到双曲几何的体积函数的解析延拓和幂零型的可积联系，使得体积函数表现为水平截面。利用这类联络的奇性，证明了模空间边界上双曲体积的渐近性态。

1. Loop spaces of orbit configuration spaces}In collaboration with F.Cohen and M.Xicont\'encatl, I developed research on the algebraic structure of the homology of loop spaces of configuration spaces. We described the homology of loop spaces of orbit configuration spaces associated with actions of Fuchsian groups on the upper half plane by means of Lie algebras and established a relation to the algebra of chid diagrams on surfaces.We studied the de Rham cohomobgy of the loop spaces of the configuration spaces based on Chen's iterated integrals. As an application I developed a systematic approach to construct link homotopy invariants based on cohomology classes of the loop spaces of configuration spaces.2. Iterated integrals and hyperbolic volumes}It is known by K. Aomoto that volumes of spherical or hyperbilic simplices are expressed by iterated integrals of logarithmic forms based on Schl\'afli's equality. Using this method, we described the analytic continuation of the volume functions from spherical to hyperbolic geometry and an integrable connection of nilpotent type such that the volume functions appear as horizontal sections. By means of the singularities of such connections I investigatgd the asymptotic behavior of the hyperbolic volumes on the boundary of moduli spaces.