Some classical theorems on open Riemann surfaces

Ideal Theory on Open Riemann Surfaces

Introduction. The theorems of the classical ideal theory in fields of algebraic numbers hold in rings of analytic functions on compact Riemann surfaces. The surfaces admitted in our discussion are closely related to algebraic surfaces; we deal either with compact surfaces from which a finite number of points are omitted or, more generally, with surfaces determined by an algebroid function. The ...

On local nature of some classical theorems on Hamilton cycles

Correlation Functions for Some Conformal Theories on Riemann Surfaces

We discuss the geometrical connection between 2D conformal field theories, random walks on hyperbolic Riemann surfaces and knot theory. For the wide class of CFT’s with monodromies being the discrete subgroups of SL(2,R I ) the determination of four–point correlation functions are related to construction of topological invariants for random walks on multipunctured Riemann surfaces.

New Proofs of the Torelli Theorems for Riemann Surfaces

In this paper, by using the Kuranishi coordinates on the Teichmüller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential geometric proofs of the Torelli theorems, both local and global, for Riemann surfaces.

Computing on Riemann Surfaces

These notes are a review on computational methods that allow us to use computers as a tool in the research of Riemann surfaces, algebraic curves and Jacobian varieties. It is well known that compact Riemann surfaces, projective algebraiccurves and Jacobian varieties are only diierent views to the same object, i.e., these categories are equivalent. We want to be able to put our hands on this equ...