Sphère de Riemann

Résumé

En mathématiques, la sphère de Riemann est une manière de prolonger le plan des nombres complexes avec un point additionnel à l'infini, de manière que certaines expressions mathématiques deviennent convergentes et élégantes, du moins dans certains contextes. Déjà envisagée par le mathématicien Carl Friedrich Gauss, elle est baptisée du nom de son élève Bernhard Riemann. Ce plan s'appelle également la droite projective complexe, dénoté \mathbb P^1(\mathbb C). Introduction La sphère de Riemann, obtenue en ajoutant au plan complexe un point à l'infini, est une variété complexe unidimensionnelle, également appelée une surface de Riemann. En analyse complexe, la sphère de Riemann permet une expression élégante de la théorie des fonctions méromorphes. La sphère de Riemann est omniprésente en géométrie projective et en géométrie algébrique comme exemple fondamental d'une variété complexe, d'un espace projectif, et d'une variété algébrique. Elle a également une uti

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https://fr.wikipedia.org/wiki/Sph%C3%A8re_de_Riemann

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We describe an injection from border-strip decompositions of certain diagrams to permutations. This allows us to provide enumeration results as well as q-analogues of enumeration formulas. Finally, we use this injection to prove a connection between the number of border-strip decompositions of the n x 2n rectangle and the Weil-Petersson volume of the moduli space of an n-punctured Riemann sphere.

2019

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Finite Element Method on Riemann Surfaces and Applications to the Laplacian Spectrum

Marc Maintrot

The objective of this PhD thesis is the approximate computation of the solutions of the Spectral Problem associated with the Laplace operator on a compact Riemann surface without boundaries. A Riemann surface can be seen as a gluing of portions of the Hyperbolic Plane made with suitable conditions to obtain a 2 dimensional manifold. The solutions of the Spectral Problem associated with the Laplace operator are to be understood as the eigenfunctions defined on the surface and their corresponding eigenvalues. This work is separated into two parts: the first part describes the method used to approximate the eigenvalues and eigenfunctions, the second focuses on the design of a program to compute these approximations. The approximation method is inspired by the Finite Element Method (FEM), in that it relies on the variational expression of the Spectral Problem and the definition of a finite subspace of functions in which the approximated eigenvalues and eigenfunctions are computed. However, it differs from the FEM in that it removes the euclidian basis of the FEM and is invariant under the isometries of the Hyperbolic Plane. To ful fill this objective, we begin by geodesically triangulating the surface as regularly as possible. This hyperbolic triangulation allows us to de ne the finite subspace of functions by using the concept of barycentric coordinates associated with each vertex of the triangulation (idea introduced by Whitney and taken up by Dodziuk). We then prove that the approximated solutions convergence to the exact ones when the diameter of the triangulation decreases, as well as the order of convergence. The program is a practical application of the theoretical work and allows the computation of the approximated eigenfunctions and eigenvalues.

EPFL2012

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Computational Approach to the Geometry of Compact Riemann Surfaces

Manuel Racle

The goal of this document is to provide a generalmethod for the computational approach to the topology and geometry of compact Riemann surfaces. The approach is inspired by the paradigms of object oriented programming. Our methods allow us in particular to model, for numerical and computational purposes, a compact Riemann surface given by Fenchel-Nielsen parameters with respect to an arbitrary underlying graph, this in a uniformand robust manner. With this programming model established we proceed by proposing an algorithmthat produces explicit compact fundamental domains of compact Riemann surfaces as well as generators of the corresponding Fuchsian groups. In particular, we shall explain how onemay obtain convex geodesic canonical fundamental polygons. In a second part we explain in what manner simple closed geodesics are represented in our model. This will lead us to an algorithm that enumerates all these geodesics up to a given prescribed length. Finally, we shall briefly overview a number of possible applications of our method, such as finding the systoles of a Riemann surface, or drawing its Birman-Series set in a fundamental domain.

EPFL2013

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