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...

All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion, electric-magnetic fields on the surface. Good parameterization is crucial for simulation and visualization. This paper gives an explicit method for finding optimal global confor...

compact Riemann surfaces, e.g., two-dimensional compact manifolds with complex analytic structure, have been studied from very different points of view. On the one hand, they arise out of complex algebraic curves, and on the other hand as quotient spaces by the action of groups of Möbius transformations. These two settings are related in a highly obscure way, but the connection can sometimes be...

We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincaré dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we will define the discrete analogs of period matrices, Riemann’s bilinear relations, exponential of constant argument and series....

We organize and review some material from various sources about prepotentials, Riemann surfaces and kernels, WDVV, and the renormalization group, provide some further connections and information, and indicate some directions and problems.

We consider coalescing fermions on a Riemann Surface and derive generalized determinant formulas, complementing some results of 3].

This paper consists of mainly two parts. First it is a survey of some results on automorphisms of Riemann surfaces and Fuchsian groups. A theorem of Hurwitz states that the maximal automorphism group of a compact Riemann surface of genus 9 has order at most 84(g-1). It is well-known that the Klein quartic is the unique genus 3 curve that attains the Hurwitz bound. We will show in the second par...

For an open and dense subset in the Teichmüller space, we introduce a coordinate system which is (relatively) well-behaved under the action of the mapping class group. Surprisingly, the coordinates involve the set of projections and traces in a C∗-algebra. We argue that our approach might be helpful for representation of the mapping class group, and other issues.

In the earlier study of Riemann surfaces, one was interested in the embeddability of Riemann surfaces in Euclidean spaces. Complex Analysists showed that any Riemann surface can be embedded in Euclidean 3-space. But recently, this problem has been conveyed to the problem in the Riemannian manifolds. In this paper we investigate the history and very recent results of problem in affordable and sy...