We develop algorithms for three problems. Starting with a complex torus of dimension $g\ge 2$, isomorphic to a principally polarized, simple abelian variety $A$/$\C$, the first problem is to find an algorithmic solution of the hyperelliptic Schottky problem: Is there a hyperelliptic curve C of genus g whose jacobian variety $\mathcal{J}_C$ is isomorphic to A over $\C$? Our solution is based on [Poor 1994]. If such a hyperelliptic curve C exists, the next problem is the construction of the Rosenhain model $\medmuskip1mu C : Y^2=X\mskip1mu (X-1)(X-\lambda_1)(X-\lambda_2)\,$\dots$(X-\lambda_{2g-1})$ for pairwise distinct numbers $\lambda_j \in \C\setminus\{0$, $1\}$. Applying the theory of hyperelliptic theta functions we show that these numbers $\lambda_j$ can easily be computed by using theta constants with even characteristics. If the abelian variety A is defined over a field k (this field could be the field of rational numbers, an algebraic number field of low degree, or a finite field), we show only in the case $k=\Q$ for simplicity, how the method in [Mestre 1991] can be generalized to get a minimal equation over $\Z\!\left[\frac{1}{2}\right]$ for the hyperelliptic curve C with jacobian variety $\mathcal{J}_C \cong_{\C} A$. This is our third problem. For some hyperelliptic, principally polarized and simple factors with dimension $g=3$, 4, 5 of the jacobian variety $J_0(N)=\mathcal{J}_{X_0(N)}$ of the modular curve $X_0(N)$ we compute the corresponding curve equations by applying our algorithms to this special situation.

A question of Gordon, mistakenly attributed to Erdős, asks if one can start at the origin and walk from there to infinity on Gaussian primes in steps of bounded length. We conjecture that one can start anywhere and the answer is still no. We introduce the concept of periodic Gaussian moats to prove our conjecture for step sizes of $\sqrt 2$ and 2.

This paper completes the construction of the Brauer tree of the sporadic simple Thompson group in characteristic $19$. Our main computational tool to arrive at this result is a new parallel implementation of the DirectCondense method.

We present a new method, based on generalizations of Shiffman's variational principle, [Nowak 1993; 1994], for the construction of minimal surfaces on Schwarzian chains in curved space forms. The main emphasis of our approach is on the computation of all minimal surfaces of genus zero (disks with holes) that span a given boundary configuration - even unstable ones. For many boundary configurations we derive numerical finiteness results on the number of minimal surfaces spanning a given boundary configuration. We use graphs of Shiffman's function to illustrate bifurcation phenomena and the Morse index of minimal surfaces. We also present some convergence results for the numerical method.

Based on previous results of the two first authors, it is shown that the combinatorial construction of invariants of compact, closed three-manifolds by Turaev and Viro as state sums in terms of quantum $6j$-symbols for $\SL_q(2,\C)$ at roots of unity leads to the unitary representation of the mapping class group found by Moore and Seiberg. Via a Heegaard decomposition this invariant may therefore be written as the absolute square of a certain matrix element of a suitable group element in this representation. For an arbitrary Dehn surgery on a figure-eight knot we provide an explicit form for this matrix element involving just one $6j$-symbol. This expression is analyzed numerically and compared with the conjectured large $k=r-2$ asymptotics of the Chern-Simons-Witten state sum [Witten 1989], whose absolute square is the Turaev-Viro state sum. In particular we find numerical agreement concerning the values of the Chern-Simons invariants for the flat $\SU(2)$-connections as predicted by the asymptotic expansion of the state sum with analytical results found by Kirk and Klassen [1990].