Matzat has proved that the Mathieu group of degree 24 is a Galois group over the transcendent extension $\QQ(T)$. He does this by using a construction called nonrigid, proving the existence of a rational point in an appropriate Hurwitz space. Here we perform such a construction explicitly. We also deduce that, for any $\KK$ such that the equation $x^2+y^2+z^2=0$ has a nontrivial solution, there is a regular extension of $\KK(T)$ with Galois group $M_{23}$. To achieve this, we had to replace the usual tools of symbolic calculation by numerical constructions, and then to recover the algebraic objects by parametrizing certain curves of genus 0. This allows us to illustrate the power of covering map techniques developed in [Couveignes 1994; Couveignes et Granboulan 1994].

Matzat a prouvé que le groupe de Mathieu de degré 24 est groupe de Galois sur l'extension transcendante $\QQ(T)$. Il utilise pour cela une construction dite non rigide et prouve l'existence d'un point rationnel dans un espace de Hurwitz adéquat. Nous donnons ici une telle extension explicitement. Nous en déduisons aussi l'existence d'une extension régulière de $\KK(T)$ de groupe de Galois $M_{23}$ pour tout $\KK$ tel que l'équation $x^2+y^2+z^2=0$ ait une solution non triviale. Pour obtenir ces résultats, il a fallu remplacer les outils habituels du calcul formel par des constructions numériques et retrouver ensuite les objets algébriques en paramétrisant certaines courbes de genre 0. Cela nous permet d'illustrer la puissance des techniques de calcul de revêtements développées dans [Couveignes 1994; Couveignes et Granboulan 1994].

We survey what is known about spectra of combinatorial Laplacians (or adjacency operators) of graphs on the simplest finite symmetric spaces. This work is joint with J. Angel, N. Celniker, A. Medrano, P. Myers, S. Poulos, H. Stark, C. Trimble, and E. Velasquez. For each finite field $\Fq$ with $q$ odd, we consider graphs associated to finite Euclidean and non-Euclidean symmetric spaces over $\Fq$. We are mainly interested in three questions regarding the eigenvalues and eigenfunctions of the combinatorial Laplacian as $q$ goes to infinity: How large is the second largest eigenvalue, in absolute value, compared with the graph's degree? (The largest eigenvalue is the degree.) What can one say about the distribution of eigenvalues? What can one say about the "level curves'' of the eigenfunctions?

We show that, under mild nonflatness conditions, for any $r\ge 3$ and any $C^r$-immersion of a surface into $\R^3$ with an isolated umbilic point there exist an analytic surface with an isolated umbilic of the same index. The connection of this with Carathéodory's Conjecture on umbilics is discussed.

We present a method to construct irreducible rational matrix representations of finite groups, based on an efficient construction of fixed points of finite groups acting on complex vector spaces.

We describe a computational, heuristic approach to the problem of deciding whether or not a given finitely presented group has a free quotient of rank two or more. Our strategy is to construct a finite nilpotent quotient of the given group, to search for quotients that are free within a variety containing that quotient, and then lift to the original group. We give theoretical justification to our strategy, and describe successful computations with sections of the Picard group $\SL _2(\Z[i])$.

Let $\hz_{3}$ be three-dimensional hyperbolic space and $\Gamma$ a group of isometries of $\hz_3$ that acts discontinuously on $\hz_{3}$ and that has a fundamental domain of finite hyperbolic volume. The Laplace operator $\MinusDelta$ of $\hz_{3}$ gives rise to a positive, essentially selfadjoint operator on $L^2(\Gamma \backslash \hz_{3})$. The nature of its discrete spectrum $\dspec (\Gamma)$ is still not well understood if $\Gamma$ is not cocompact.

This paper contains a report on a numerical study of $\dspec (\Gamma)$ for various noncocompact groups $\Gamma$. Particularly interesting are the results for some nonarithmetic groups $\Gamma$.