Abstract:
A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees that the Euclidean structure on the polygons induces a unique conformal structure on the quotient surface, making it into a closed Riemann surface. In this case, a modulus of continuity for uniformizing coordinates is found which depends only on the geometry of the polygons and on the identifications. An application is presented in which a uniform modulus of continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it possible to prove that they converge to a Teichm\"uller mapping on the Riemann sphere.

Abstract:
In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi surfaces. And in this construction we can control the geometry of the compact Riemann surface by the geometry of the graph. We show that almost all such surfaces have large first eigenvalue and large Cheeger constant.

Abstract:
This is a commentary on Teichm\"ullers' paper "Ver\"anderliche Riemannsche Fl\"achen" (Variable Riemann Surfaces), published in 1944. This paper is the last one that Teichm\"uller wrote on the problem of moduli. At most places the paper contains ideas and no technical details. The author presents a completely new approach to Teichm\"uller space, compared to the approach he took in his first seminal paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" and its sequel "Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl\"achen" in which he completed some of the the results stated in the former. In the paper "Extremale quasikonforme ...", Teichm\"uller led the foundations of what we call today Teichm\"uller theory (but without the complex structure), defining its metric and introducing in that theory the techniques of quasiconformal mappings and of quadratic differentials as essential tools. In the present paper, the approach is more abstract, through complex analytic geometry. Teichm\"uller space, equipped with its complex-analytic structure, is characterized here by a certain universal property. Among the other ideas and results contained in the paper, we mention the following: (1) The existence and uniqueness of the universal Teichm\"uller curve, rediscovered later on by Ahlfors and by Bers. At the same time, this introduced the first fibre bundle over Teichm\"uller space. (2) The proof of the fact that the automorphisms group of the univeral Teichm\"uller curve is the extended mapping class group. (3) The idea of a fine moduli space. (4) The idea of using the period map to define a complex structure on Teichm\"uller space.

Abstract:
The known (explicit) examples of Riemann surfaces not definable over their field of moduli are not real whose field of moduli is a subfield of the reals. In this paper we provide explicit examples of real Riemann surfaces which cannot be defined over the field of moduli.

Abstract:
This paper is the third in a sequel to develop a super-analogue of the classical Selberg trace formula, the Selberg supertrace formula. It deals with bordered super Riemann surfaces. The theory of bordered super Riemann surfaces is outlined, and the corresponding Selberg supertrace formula is developed. The analytic properties of the Selberg super zeta-functions on bordered super Riemann surfaces are discussed, and super-determinants of Dirac-Laplace operators on bordered super Riemann surfaces are calculated in terms of Selberg super zeta-functions.

Abstract:
We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear, Poisson-bracket, obtainable from a single polynomial C(onstraint) function. For a continuous class of quartic constraints, we explicitly work out finite dimensional representations of the corresponding C-Algebras.

Abstract:
In this paper, by using the Kuranishi coordinates on the Teichm\"uller 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.

Abstract:
In this paper we classify all Riemann surfaces having a large abelian group of automorphisms, that is having an abelian group of automorphism of order strictly bigger then $4(g-1)$, where $g$ denotes as usual the genus of the Riemann surface.

Abstract:
To the spectral curves of smooth periodic solutions of the $n$-wave equation the points with infinite energy are added. The resulting spaces are considered as generalized Riemann surfcae. In general the genus is equal to infinity, nethertheless these Riemann surfaces are similar to compact Riemann surfaces. After proving a Riemann Roch Theorem we can carry over most of the constructions of the finite gap potentials to all smooth periodic potentials. The symplectic form turns out to be closely related to Serre duality. Finally we prove that all non-linear PDE's, which belong to the focussing case of the non-linear Schr\"odinger equation, have global solutions for arbitrary smooth periodic inital potantials.

Abstract:
In this paper, we have extended S.S. Chern's second basic theorem about holomorphic mapping between two Riemann surfaces to more general case, and also obtained two similar results.