The Schwarz map defined by the ratio of two solutions of the hypergeometric equation has been studied mainly when the exponents are real. In this paper, we study this map when the exponents are purely imaginary, a case that has been neglected for over a hundred years. A fundamental domain in the source plane and that in the target plane are constructed; the Schwarz map restricted on these domains is conformally isomorphic and the whole map can be recovered by this restriction through repeated use of the Schwarz reflection principle. We investigate the shape of these fundamental domains both analytically and numerically, and conclude with open questions.

For a real square matrix $M$, Hadamard's inequality gives an upper bound $H$ for the determinant of $M$; the bound is sharp if and only if the rows of $M$ are orthogonal. We study how much we can expect that $H$ overshoots the determinant of $M$, when the rows of $M$ are chosen randomly on the surface of the sphere. This gives an indication of the "wasted effort'' in some modular algorithms.

Let $P_s(n) = \sum_{j=1}^{n} j^{-s}$. For fixed $s$ near $s=\frac12$, we divide the unit interval into bins and count how many of the partial sums $P_s(1)$, $P_s(2)$, \dots, $P_s(N)$ lie in each bin $\mone$. The properties of the histogram are predicted by a random model unless $s=\frac12$. When $s=\frac12$ the histogram is surprisingly flat, but has a few strong spikes. To explain the surprises at $s=\frac12$, we use classical results about Diophantine approxmation, lattice points, and uniform distribution of sequences.

We prove two lemmata about Schubert calculus on generalized flag manifolds $G/B$, and in the case of the ordinary flag manifold $\GL_n/B$ we interpret them combinatorially in terms of descents, and geometrically in terms of missing subspaces. One of them gives a symmetry of Schubert calculus that we christen descent-cycling. Computer experiment shows these two lemmata are surprisingly powerful: they already suffice to determine all of $\GL_n$ Schubert calculus through $n=5$, and 99.97\%+ at $n=6$. We use them to give a quick proof of Monk's rule. The lemmata also hold in equivariant ("double'') Schubert calculus for Kac--Moody groups $G$.

Bailey and Crandall recently formulated "Hypothesis A'', a general principle to explain the (conjectured) normality of the binary expansion of constants like $\pi$ and other related numbers, or more generally the base $b$ expansion of such constants for an integer $b \geq 2$. This paper shows that a basic mechanism underlying their principle, which is a relation between single orbits of two discrete dynamical systems, holds for a very general class of representations of numbers. This general class includes numbers for which the conclusion of Hypothesis A is not true. The paper also relates the particular class of arithmetical constants treated by Bailey and Crandall to special values of $G$-functions, and points out an analogy of Hypothesis A with Furstenberg's conjecture on invariant measures.

This article proposes to represent subgroups of finitely presented groups by their image in a quotient. It gives algorithms for basic operations in this representation and investigates how iteration of this approach can be used to extend known quotients groups with a solvable normal subgroup.

We use Zagier's method to compute the critical values of the symmetric square $L$-functions of six cuspidal eigenforms of level one with rational coefficients. According to the Bloch--Kato conjecture, certain large primes dividing these critical values must be the orders of elements in generalised Shafarevich--Tate groups. We give some conditional constructions of these elements. One uses Heegner cycles and Ramanujan-style congruences. The other uses Kurokawa's congruences for Siegel modular forms of degree two. The first construction also applies to the tensor product $L$-function attached to a pair of eigenforms of level one. Here the critical values can be both calculated and analysed theoretically using a formula of Shimura.

This paper formulates some conjectures about the amplitude of resonance in the General Standard Map. The main idea is to expand the periodic perturbation function in Fourier series. Given any rational rotation number, we choose a finite number of harmonics in the Fourier expansion and we compute the amplitude of resonance of the reduced perturbation function of the map, using a suitable normal form around the resonance, which is valid for asymptotically small values of the perturbation parameter. For this map, we obtain a relation between the amplitude of resonance and the perturbation parameter: the amplitude is proportional to a rational power of the parameter, and so can be represented as a straight line on a log-log graph. The convex hull of these straight lines gives a lower bound for the amplitude of resonance, valid even when the perturbation parameter is of the order of 1. We find that some perturbation functions give rise a phenomenon that we call collapse of resonance; this means that the amplitude of resonance goes to zero for some value of the perturbation parameter. We find an empirical procedure to estimate this value of the parameter related to the collapse of resonance.

We address the problem of topologically characterising simple closed geodesics in the figure-eight knot complement. We develop ways of finding these geodesics up to isotopy in the manifold, and notice that many seem to have the lowest-volume complement amongst all curves in their homotopy class. However, we find that this is not a property of geodesics that holds in general. The question arises whether under additional conditions a geodesic knot has least-volume complement over all curves in its free homotopy class.

We also investigate the family of curves arising as closed orbits in the suspension flow on the figure-eight knot complement, many but not all of which are geodesic. We are led to conclude that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.

Kigami has shown how to construct Laplacians on certain self-similar fractals, first for the Sierpiński gasket and then for the class of postcritically finite (\pcf\) fractals, subject to the solution of certain algebraic equations. It is desirable to extend this method to as large a class of fractals as possible, so in this paper we examine a specific example that exhibits features associated with finite ramification, but which does not fall into the class of \pcf\ fractals. We show by a method of deconstruction that this fractal is a member of a family of three fractals for which the \pcf\ condition holds in a generalized sense. We then study the algebraic equations whose solution is required for the actual construction of the Laplacian. We obtain experimental evidence for the existence and uniqueness of solutions. This experimental work uncovers two symmetries that were not initially apparent, only one of which has a natural explanation. By exploiting the symmetries, we give a nonconstructive proof of existence.

Let $X$ be a curve of genus $g \ge 2$ over a field $k$ of characteristic zero. Let $X\box0 A$ be an Albanese map associated to a point $P_0$ on $X$. The Manin--Mumford conjecture, first proved by Raynaud, asserts that the set $T$ of points in $X(\kbar)$ mapping to torsion points on $A$ is finite. Using a $p$-adic approach, we develop an algorithm to compute $T$, and implement it in the case where $k=\funnyQ$, $g=2$, and $P_0$ is a Weierstrass point. Improved bounds on $\#T$ are also proved: for instance, in the context of the previous sentence, if in addition $X$ has good reduction at a prime $p \ge 5$, then $\#T \le\ 2 p^3 + 2 p^2 + 2p + 8$.

We analyze the $128$-dimensional Mordell--Weil lattice of a certain elliptic curve over the rational function field k(t), where k is a finite field of $2^{12}$ elements. By proving that the elliptic curve has trivial Tate--Šafarevič group and nonzero rational points of height $22$, we show that the lattice's density achieves the lower bound derived in our earlier work. This density is by a considerable factor the largest known for a sphere packing in dimension 128. We also determine the kissing number of the lattice, which is by a considerable factor the largest known for a lattice in this dimension.

We construct elliptic curves over $\Q$ of high Mordell--Weil rank, with a nontrivial torsion subgroup. We improve the rank records for the cases $\Z/2\Z \times \Z/2\Z$, $\Z/3\Z$, $\Z/4\Z$, $\Z/5\Z$, $\Z/6\Z$, $\Z/7\Z$ and $\Z/8\Z$.