We give new examples of noncongruence subgroups $\Gamma\subset\SL_2(\Z)$ whose space of weight-3 cusp forms $S_3(\Gamma)$ admits a basis satisfying the Atkin--Swinnerton-Dyer congruence relations with respect to a weight-3 newform for a certain congruence subgroup.

This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces $H^1(\G,E_n)$, where $\Gamma$ is a lattice in $\SL(2,\C)$ and $E_n = \Sym^n\otimes \overline{\Sym}{}^n$, $n\in \N\cup \{0\}$, is one of the standard self-dual modules. In the case $\Gamma = \SL(2,\O)$ for the ring of integers $\O$ in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in $n$. We present a large amount of experimental data for this case, as well as for some geometrically constructed and mostly nonarithmetic groups. The computations for $\SL(2,\O)$ lead us to discover two instances with nonlifted classes in the cohomology. We also derive an upper bound of size $O(n^2 / \log n)$ for any fixed lattice $\G$ in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data.

In this paper we give exact values of the standard zeta function for cuspidal Hecke eigenforms with respect to $\Sp_2(\integers).$

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry from the theoretical, algorithmic, and practical points of view.

In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection segment functional. This heuristic was used to find geometric realizations in ${\mathbb R}^3$ for all vertex-minimal triangulations of the orientable surfaces of genera $g=3$ and $g=4$. Moreover, for the first time, examples of simplicial polyhedra in ${\mathbb R}^3$ of genus 5 with 12 vertices have been obtained.

We consider the problem of the numerical approximation of the linear controllability of waves. All our experiments are done in a bounded domain $\Omega$ of the plane, with Dirichlet boundary conditions and internal control. We use a Galerkin approximation of the optimal control operator of the continuous model, based on the spectral theory of the Laplace operator in $\Omega$. This allows us to obtain surprisingly good illustrations of the main theoretical results available on the controllability of waves and to formulate some questions for future analysis of the optimal control theory of waves.

A minimal permutation representation of a finite group $G$ is a faithful $G$-set with the smallest possible size. We study the structure of such representations and show that for certain groups they may be obtained by a greedy construction. In these situations (except when central involutions intervene) all minimal permutation representations have the same set of orbit sizes. Using the same ideas, we also show that if the size $d(G)$ of a minimal faithful $G$-set is at least $c|G|$ for some $c>0$, then $d(G) = |G|/m + O(1)$ for an integer $m$, with the implied constant depending on $c$.