(The links on the page will no longer be maintained. They were correct as of 8th September 2005 when this page was last maintained.)
Abstract
The feet of the altitudes of a triangle are the vertices of a new triangle called its pedal triangle. But what happens to its shape if we take the pedal triangle of that, and repeat the process indefinitely?
Following work of Kingston-Synge, Lax, Ungar, and Alexander we shall describe an expanding map of the tetrahedron whose orbits represent the angles of each triangle in such a pedal sequence. If the angles of the original triangle are an integer number of degrees then the sequence is periodic; but typically the pedal sequence approximates each possible shape and is only acute one quarter of the time!
Abstract
A real number x is said to be badly approximable if there exists a constant c(x) > 0, such that for any rational number p/q, |x - p/q| >= c(x) q-2. The set of such numbers is a null set of maximal Hausdorff dimension. This setup can be generalised to deal with a wide class of metric spaces, including certain "fractal" sets. We will describe such a generalised setup and give conditions under which the dimension of the set of badly approximable elements have positive (or indeed maximal) Hausdorff dimension. A range of applications will be given to number theory and dynamical systems.
Abstract
Recent results of Hall, Kublanovsky, Sapir, Margolis, Trotter and, independently, Steinberg, show that it is undecidable whether a finite cancellative category admits a faithful functor to a group. Consequently, it is undecidable whether a finite ample semigroup is a full subsemigroup of an inverse semigroup. On the other hand, we show that every cancellative category admits a faithful functor to a cancellative monoid, and deduce that every primitive ample semigroup is a full subsemigroup of a Rees matrix semigroup over a cancellative monoid.
Abstract
I will describe some recent developments related to various extensions of the Tits alternative (i.e., a finitely generated linear group is either virtually soluble or contains a free subgroup on two generators) and their applications in Riemannian geometry, ergodic theory, and geometric group theory.
Abstract
In this talk I discuss generating finite and infinite symmetric groups. No specialist knowledge is required.
Abstract
In this talk, I extend the classical arithmetic defined over the set of natural numbers N, to the set of all finite directed connected multigraphs having a pair of distinguished vertices. Specifically, I introduce a model F on the set of such graphs, and provide an interpretation of the language of arithmetic L={0,1,<=,+,x} inside F. The resulting model exhibits the property that the standard model on N embeds in F as a submodel, with the directed path of length n playing the role of the standard integer n. I will mostly discuss the graph theoretic aspect of this work as well as discussing some interesting open problems. The talk will be accessible to postgraduate and perhaps advanced undergraduate students.
Abstract
I will discuss new classes of fractals, intermediate in some ways between deterministic and random fractals, developed in joint work with Michale Barnsley and Örjan Stenflo. I will also show how one can compute their dimensions and estimate their degree of approximation to more standard random fractals.
Abstract
We will discuss the distribution of irreducible and primitive polynomials over finite fields. For example, for polynomials of degree n, if one specifies certain coefficients in advance, is there an irreducible, or even a primitive, polynomial of degree n over Fq with those prescribed coefficients? Is there a formula for the exact number of such polynomials? We will discuss a number of such problems as well as some recent progress on these kinds of problems.
Abstract
Noting that the full covering group of the Mathieu simple group M22 has surprisingly short efficient presentations, we study efficient presentations for both it and M22 itself. We use three different methods to obtain efficient presentations for both of these groups. We produce a number of new short presentations and also presentations which have nice structure.