We give presentations for the braid groups associated with the complex reflection groups $G_{24}$ and $G_{27}$. For the cases of $G_{29}$, $G_{31}$, $G_{33}$, and $G_{34}$, we give (strongly supported) conjectures. These presentations were obtained with VKCURVE, a GAP package implementing Van Kampen's method.

An overview is presented of Regina, a freely available software package for 3-manifold topologists. In addition to working with 3-manifold triangulations, Regina includes support for normal surfaces and angle structures. The features of the software are described in detail, followed by examples of research projects in which Regina was used.

There are 18 (and possibly 19) integers that are not of the form $ xy + yz + xz $ with positive integers $x, y, z$. The same 18 integers appear as exceptional discriminants for which no indecomposable positive definite binary quadratic form exists. We show that the two problems are equivalent.

The Ramanujan AGM continued fraction is a construct

{\small $${\cal R}_\eta(a,b) =\,\frac{a}{\displaystyle \eta+\frac{b^2}{\displaystyle \eta +\frac{4a^2}{\displaystyle \eta+\frac{9b^2}{\displaystyle \eta+{}_{\ddots}}}}}$$}

enjoying attractive algebraic properties, such as a striking arithmetic-geometric mean (AGM) relation and elegant connections with elliptic-function theory. But the fraction also presents an intriguing computational challenge. Herein we show how to rapidly evaluate ${\cal R}$ for any triple of positive reals $a,b,\eta$. Even in the problematic scenario when $a \approx b$ certain transformations allow rapid evaluation. In this process we find, for example, that when $a\eta = b\eta = $ a rational number, ${\cal R}_\eta$ is essentially an $L$-series that can be cast as a finite sum of fundamental numbers. We ultimately exhibit an algorithm that yields $D$ good digits of ${\cal R}$ in $O(D)$ iterations where the implied big-$O$ constant is independent of the positive-real triple $a,b,\eta$. Finally, we address the evidently profound theoretical and computational dilemmas attendant on complex parameters, indicating how one might extend the AGM relation for complex parameter domains.

The Ramanujan continued fraction

{\small $${\cal R}_\eta(a,b) =\,\frac{a}{\displaystyle \eta+\frac{b^2}{\displaystyle \eta +\frac{4a^2}{\displaystyle \eta+\frac{9b^2}{\displaystyle \eta+{}_{\ddots}}}}}$$}

is interesting in many ways; e.g., for certain complex parameters $(\eta, a,b)$ one has an attractive AGM relation ${\cal R}_{\eta} (a,b) + {\cal R}_{\eta}(b,a) = 2 {\cal R}_{\eta} \left((a+b)/2, \sqrt{ab} \right)$. Alas, for some parameters the continued fraction ${\cal R}_{\eta}$ does not converge; moreover, there are converging instances where the AGM relation itself does not hold. To unravel these dilemmas we herein establish convergence theorems, the central result being that ${\cal R}_1$ converges whenever $|a| \not= |b|$. Such analysis leads naturally to the conjecture that divergence occurs whenever $a=b e^{i\phi}$ with $\cos^2\phi \not = 1$ (which conjecture has been proven in a separate work) [Borwein et al. ] We further conjecture that for $a/b$ lying in a certain---and rather picturesque---complex domain, we have both convergence and the truth of the AGM relation.

We generate extensions of $\Q$ with Galois group $\SL_3(\F_2)$ giving rise to three-dimensional mod 2 Galois representations with sufficiently low level to allow the computational testing of a conjecture of Ash, Doud, Pollack, and Sinnott relating such representations to mod 2 arithmetic cohomology. We test the conjecture for these examples and offer a refinement of the conjecture that resolves ambiguities in the predicted weight.

The cut locus from a point on the surface of a convex polyhedron is a tree containing a line segment beginning at every vertex. In the limit of infinitely small triangles, the cut locus from a point on a triangulation of a smooth surface therefore tends to become dense in the smooth surface, whereas the cut locus from the same point on the smooth surface is also a tree, but of finite length. We introduce a method for avoiding this problem. The method involves introducing a minimal angular resolution and discarding those points of the cut locus on the triangulation for which the angle measured between the shortest geodesic curves meeting at these points is smaller than the given angular resolution. We also describe software based upon this method that allows one to visualize the cut locus from a point on a surface of the form $(x/a)^{n}+(y/b)^{n}+(z/c)^{n}=1$, where $n$ is a positive even integer. We use the software to support a new conjecture that the cut locus of a general ellipsoid is a subarc of a curvature line of the ellipsoid.

For $\ell =$ 3 and 5 it is known that every odd, irreducible, two-dimensional representation of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ and determinant equal to the cyclotomic character must "come from'' the $\ell$-torsion points of an elliptic curve defined over $\Q$. We prove, by giving concrete counter-examples, that this result is false for every prime $\ell > 5$.

Let $n$ be a positive integer. We consider the Sylvester resultant of $f$ and $g,$ where $f$ is a generic polynomial of degree 2 or 3 and $g$ is a generic polynomial of degree $n.$ If $f$ is a quadratic polynomial, we find the resultant's height. If $f$ is a cubic polynomial, we find tight asymptotics for the resultant's height.

Let {\small $\Delta = \{ G: g(G) =0, \Gamma_0(m) \le G \le N(\Gamma_0(m))$ $\mbox{ for some }m\},$} where {\small $N(\Gamma_0(m))$} is the normaliser of {\small $\Gamma_0(m)$} in {\small $PSL_2(\Bbb R)$} and {\small $g(G)$} is the genus of {\small $\Bbb H^*/G$}. In this article, we determine all the {\small $m$}. Further, for each {\small $m$}, we list all the intermediate groups {\small $G$} of {\small $\Gamma_0(m) \le N(\Gamma_0(m))$} such that {\small $ g(G) =0$}. All the intermediate groups of width 1 at {\small $\infty$} are also listed in a separate table (see www.math.nus.edu.sg/$\sim$matlml/).

Thompson has shown that up to conjugation there are only finitely many congruence subgroups of $\PSL(2,\R)$ of fixed genus. For $\PSL(2,\Z)$, Cox and Parry found an explicit bound for the level of a congruence subgroup in terms of its genus. This result was used by the author and Pauli to compute the congruence subgroups of $\PSL(2,\Z)$ of genus less than or equal to 24. However, the bound of Cox and Parry applies only to $\PSL(2,\Z)$. In this paper a result of Zograf is used to find a bound for the level of any congruence subgroup in terms of its genus. Using this result, a list of all congruence subgroups, up to conjugacy, of $\PSL(2,\R)$ of genus 0 and 1 is found.

This tabulation is used to answer a question of Conway and Norton who asked for a complete list of genus 0 subgroups, $\overline G$, of $\PSL(2,\R)$ such that

(i) $\overline G$ contains $\overline\Gamma_0(N)$ for some $N$.

(ii) $\overline G$ contains the translation $z\mapsto z+k$ iff $k$ is an integer.

Thompson has also shown that for fixed genus there are only finitely many subgroups of $\PSL(2,\R)$ which satisfy these conditions. We call these groups "moonshine groups.'' The list of genus 1 moonshine groups is also found. All computations were performed using Magma.