VERTEX OPERATOR ALGEBRAS AND MODULI SPACES OF ALGEBRAIC CURVES

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(1)Using arithmetic Schottky-Mumford uniformization theory on algebraic curves, we constructed Teichmuller groupoids in the category of arithmetic geometry. By this construction, we gave a partial answer to Grothendieck's conjecture on the associated Galois representations, and described monodromy representations induced from confbrmal field theory.(2)Extending Ullmo-Zhang's results on Bogomolov's conjecture, we gave a condition that a subvariety of an abelian variety defined over a number field is isomorphic to an abelian variety in terms of Neron-Tate's height functions.(3)We described the structure of Riemann surfaces defined from the monodromy representation of hypergeometric differential equations with purely imaginary exponents (joint work with M.Yoshida).(4)We determined the structure of the class groups and unit groups of algebraic number fields of Kummer type, specifically of quartic Dirichlet fields (joint work with K.Katayama and C.Levesque). Further, we investigated the problem of Hasse concerning power integral basis of the ring of algebraic integers (joint work with Y.Motoda).(5)We defined stochastic holonomy operator and the Chern-Simons integral of some product of gauge invariant Wilson loop observables in the Wiener space setting.(6)We studied Buchsbaum Stanley-Reisner rings with linear resolution and characterized them by their multiplicity. Further, we studied the arithmetical rank and determined it for monomial ideals of deviation two.(7)We studied 4-manifolds having flexible surfaces inside, and showed that a lot of simply connected 4-manifolds not the 4-sphere have flexible surfaces. Further, we introduced an operation to alter any surface in a simply connected 4-manifold into a flexible surface.