VERTEX OPERATOR ALGEBRAS AND MODULI SPACES OF ALGEBRAIC CURVES

(1)利用代数曲线上的算术Schottky-Mumford一致化理论，构造了算术几何范畴中的Teichmuller群胚。通过这种构造，我们部分地回答了Grothendieck关于相关Galois表示的猜想，并描述了由余弦场理论导出的单调表示。(2)推广了Ullmo-Zhang关于Bogomolov猜想的结果，给出了定义在数域上的阿贝尔簇的一个子簇同构于一个阿贝尔簇的条件.(3)我们描述了由纯虚指数超几何微分方程的单列表示定义的Riemann曲面的结构(与M.Yoshida共同工作).(4)我们确定了Kummer型代数数域的类群和单位群的结构，特别是四次Dirichlet域(与K.Katayama和C.Levesque共同工作).进一步，我们研究了代数整数环的幂积分基的Hasse问题(与Y.Motoda共同工作)。(5)在Wiener空间中定义了随机完整算子和规范不变的Wilson圈可观测量乘积的Chern-Simons积分。(6)研究了线性分辨的Buchsbaum Stanley-Reisner环，并用重数刻画了它们的重数。进一步，我们研究了偏差为2的单项理想的算术秩并确定了它。(7)我们研究了内部有柔性曲面的4-流形，证明了许多单连通的4-流形而不是4-球面有柔性曲面。此外，我们还引入了一个将单连通4-流形中的任何曲面变换为柔性曲面的操作。

(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.