We present a heuristic method for the reduction modulo 2 of a large, sparse bit matrix to a smaller, dense bit matrix that can then be solved by conventional means, such as Gaussian elimination. This method worked effectively for us in reducing matrices as large as 100,000${}\times{}$100,000 to much smaller, but denser square matrices.

Let $\cal G$ be a pseudogroup defined on a tree $Z$, and let $\Gamma$ be a finite set of generators for $\cal G$. The reduced fundamental group $\pibar(\Gamma)$ of $\Gamma$ is defined here. I give a new and experimentally inspired proof of a result of Levitt: If $\pibar(\Gamma)$ is a free group, there exists a finite set of generators $\Psi$ for $\cal G$ such that $\pibar(\Psi)$ is free on the set $\Psi$. If $\Psi$ has no dead ends, it is an interval exchange.

Like Gaboriau, Levitt and Paulin [Gaboriau et al. 1992], I prove that if $G$ is a finitely presented group acting freely on an $\R$-tree and $\Gamma$ is a corresponding set of pseudogroup generators, we're in one of the following situations: either $G$ splits as a free product with a noncyclic free abelian summand, or $\Gamma$ can be reduced to an interval exchange by normalizing and removing a finite number of dead ends, or the process of removing dead ends from $\Gamma$ does not terminate in a finite number of steps.

We study the complexity of performing Fourier analysis for the group $\SL_2(\fq)$, where $\fq$ is the finite field of $q$ elements. Direct computation of a complete set of Fourier transforms for a complex-valued function $f$ on $\SL_2(\fq)$ requires $q^6$ operations. A similar bound holds for performing Fourier inversion. Here we show that for both operations this naive upper bound may be reduced to $O(q^4\log q)$, where the implied constant is universal, depending only on the complexity of the "classical'' fast Fourier transform. The techniques we use depend strongly on explicit constructions of matrix representations of the group.

Additionally, the ability to construct the matrix representations permits certain numerical experiments. By quite general methods, recent work of others has shown that certain families of Cayley graphs on $\SL_2(\fq)$ are expanders. However, little is known about their exact spectra. Computation of the relevant Fourier transform permits extensive numerical investigations of the spectra of these Cayley graphs. We explain the associated calculation and include illustrative figures.

The Surface Evolver is a computer program that minimizes the energy of a surface subject to constraints. The surface is represented as a simplicial complex. The energy can include surface tension, gravity and other forms. Constraints can be geometrical constraints on vertex positions or constraints on integrated quantities such as body volumes. The minimization is done by evolving the surface down the energy gradient. This paper describes the mathematical model used and the operations available to interactively modify the surface.

To every elliptic curve over $\Q$ one can associate a function $L_E$, which is conjectured to be holomorphic and to satisfy a functional equation similar to the one satisfied by Riemann's zeta function. One can ask whether these functions satisfy an appropriate version of the Riemann Hypothesis (the traditional Riemann Hypothesis states that the nontrivial zeros of the zeta function lie on the "critical line'' of the functional equation).

In order to gather experimental evidence in support of this generalized Riemann hypothesis, I computed numerically a set of zeros for several hundred elliptic curves. All of the 7000 zeros computed for the various curves lie on the critical line.

A toute courbe elliptique définie sur $\Q$, on sait associer une fonction $L_E$, qui, conjecturalement, est holomorphe et vérifie une équation fonctionnelle analogue à celle de la fonction zêta de Riemann. On peut se demander, comme pour la fonction zêta, si les fonctions $L_E$ vérifient des conditions analogues à la "conjecture de Riemann'', qui affirme que les zéros non-triviaux de $\zeta$ se trouvent sur la "droite critique'' de l'équation fonctionnelle.

C'est pour tenter de donner un peu de crédit à cette hypothèse de Riemann généralisée que l'on a effectué des recherches numériques de zéros sur plusieurs centaines de courbes différentes. Ainsi, sur les 7000 zéros calculés, tous sont situés sur la droite critique.