Combinatorial structures on Riemann surfaces and topological properties of the moduli space

We consider Riemann surfaces with abelian differentials constructed from lightning pairs. A lightning pair is a piecewise linear loop in the complex plane determined by a certain kind of combinatorial data. During the study of this method we obtained the following results: 1. 1. We gave geometrically a definition of unknotting numbers of a Haefliger (6,3)-knot, which means a smoothly embedded 3-dimensional sphere in the 6-dimensional sphere, and determined the unknotting number of each Haefliger (6,3)-knot. 2. We constructed explicitly a new infinite series of Einstein metrics on the S^3-bundles over S^2, which containing infinite numbers of inhomogeneous ones. 3. We investigated the localization theorem of Beilinson and Bernstein for D^^-^ on the projective spaces and SL_3 in positive character, and we showed that a tilting sheaf is obtained by taking the dual of the image of the structure sheaf by the Frobenius endomorphism. 4. We presented a short proof of Morley's theorem.