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

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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.