Teichmueller groupoids and monodromy in conformal field theory

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1. We described the monodromy representation of Teichmueller goupoids associated with conformal field theory. Extending Ullmo-Zhang's result on the Bogomolov conjecture, we gave a condition that a subvariety of an abelian variety is isomorphic to an abelian variety in terms of the value distribution of a Neron-Tate height function on the subvariety. We described the Riemann surfaces associated with the monodromy representation of hypergeometric equation with purely imaginary exponents.2. We gave an explicit formula of the Hasse unit index for the unit group of quadratic fields, and considered a Problem of Hasse for the ring of integers in certain abelian fields.3. We tried to justify the perturbative Chern-Simons theory using the asymptotic expansion theory via infinite dimensional stochastic analysis, and derived a simple Homfly polynomial.4. We showed that for certain algebraic geometry codes, the minimum distance are equal to the Fang-Rao bound, and found an algebraic geometry code of other type with same property.5. We gave the upper bound for the average number of connected components of the induced subgraphs of the graphs for simplicial polytopes, and proved that the arithmetical rank is equal to the projective dimension for the almost complete intersection Stanley-Reisner ideals.6. We calculated the virtual cohomological dimension and the Euler number of the mapping class group of a three-dimensional handlebody.