We show that the minimal dilatation of pseudo-Anosov homeomorphisms of a closed oriented genus-two surface is equal to the largest root of $x^4-x^3-x^2-x+1$, which is approximately 1.72208.

The Schwarz map of the hypergeometric differential equation has been studied since the beginning of the last century. Its target is the complex projective line, the 2-sphere. This paper introduces the hyperbolic Schwarz map, whose target is hyperbolic 3-space. This map can be considered to be a lifting to 3-space of the Schwarz map. In this paper, we study the singularities of this map, and attempt to visualize its image when the monodromy group is a finite group or a typical Fuchsian group. General cases will be treated in forthcoming papers.

This paper describes a general algorithm for finding the commensurator of a nonarithmetic hyperbolic manifold with cusps and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this to find the commensurators of all nonarithmetic hyperbolic once-punctured torus bundles over the circle.

For hyperbolic 3-manifolds, the algorithm has been implemented using Goodman’s computer program Snap. We use this to determine the commensurability classes of all cusped hyperbolic 3-manifolds triangulated using at most seven ideal tetrahedra, and for the complements of hyperbolic knots and links with up to twelve crossings.

We describe our successful computation of a list of representatives of the 2,801,324 conjugacy classes of transitive groups of degree 32.

For several values of $m$, we show ways to construct some families of cyclic monic polynomials of degree $m$ with integer coefficients and constant terms $\pm 1$, and to express their roots in terms of Gaussian periods. We give several examples illustrating those techniques. With the aim to find other methods to construct such families of polynomials, we consider the question whether one of them can be obtained by means of rational transformations of a given ordinary family $F(t,X)\in\Z[t][X]$ (at the parameter $t$) of cyclic monic polynomials of degree $m$ (such families $F$ are easy to find). We show a method, due to René Schoof, that allows us to answer at least the simpler question whether a family $G(t,X)\in\Q[t][X]$ of cyclic monic polynomials of degree $m$ prime with constant term in $\Q^\times$ (independent of $t$ can be obtained from $F$} by means of rational transformations.

We prove, using interval analysis methods, that the $L^2$, $L^4$, and $L^8$ spectral radii of the traction double layer potential operator associated with the Lam\'e system on an infinite sector in $\mathbb R}^2$ are within $2.5 x 10^-3$, $10^-2$, and $10^-2$, respectively, from a certain conjectured value that depends explicitly on the aperture of the sector and the Lamé moduli of the system. We also indicate how to extend these results to $L^p$ for entire intervals of $p$, $p\geq2$.

We consider some multivariate rational functions that have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of series coefficients, and apply the inverse of this operator to the rational functions. We obtain new rational functions that seem to have only positive coefficients, whose positivity would imply positivity of the original series, and that, in a certain sense, cannot be improved any further.

We classify all cocompact torsion-free derived arithmetic Fuchsian groups of genus two by commensurability class. In particular, we show that there exist no such groups arising from quaternion algebras over number fields of degree greater than 5. We also prove some results on the existence and form of maximal orders for a class of quaternion algebras related to these groups. Using these results in conjunction with a computer program, one can determine an explicit set of generators for each derived arithmetic Fuchsian group containing a torsionfree subgroup of genus two. We show this for a number of examples.

We show that there are infinitely many nonisomorphic curves $Y^2 = X^5 + k$, $k \in {\mathbb Z}$}, possessing at least twelve finite points $k>0$, and at least six finite points for $k<$. We also determine all rational points on the curve $Y^2=X^5-7$.

The structure of the subring $J^{\Gamma \cap \Gamma^-1$ of the asymptotic Hecke algebra is described for $\Gamma$}a left cell of the Coxeter group of type $H_4$. A small set of generators over $\mathbb{Z}$ is produced. The subalgebras spanned by a subset of the basis $\left\{t_x\right\}_{x\in \Gamma\cap\Gamma^-1$ are determined.