In this talk, we will review the algorithms previously discussed for writing an arbitrary element of SL(d, q) as a word in its generating set and give an overview of how the algorithms work for the other classical groups.

A pro-p group is the inverse limit of some system of finite p-groups. Impetus on current research comes from four main directions, namely number theory, classification of finite p-groups, the theory of infinite groups(pro-p completions) and the broader area of profinite group theory. Structural and classification theorems are mostly based on the study of known examples of pro-p groups. I will give a brief introduction to profinite and pro-p groups, survey the 'universe' of countable pro-p groups and talk in some detail about the Generalised Nottingham Group, which is the group of formal power series in n variables, culminating in an explicit description of the lower central series and power structure of the group.

The talk will begin with a review of the necessary C*-algebra theory followed by a sketch of the Gelfand-Naimark Theorem. Woronowicz's definition of a compact quantum group will then be introduced and the prototypical example of SUq(2) presented. The generalisation of the Haar integral to the compact quantum group setting will be discussed, as will the relationship of compact quantum groups to Hopf algebras. Finally, (time permitting) the notion of a differential calculus over a quantum group will be discussed and the interaction of this structure with Connes' spectral triple approach to noncommutative geometry touched upon.

In my talk I will be discussing the following conjecture, taken from the Kourovka Notebook: Let G be a finite group with subgroups A and B such that G = AB and (|A|,|B|)=1. Let ccl(G) be the number of conjugacy classes of G. Then ccl(G) is at most ccl(A)ccl(B) (with equality if and only if G is a direct product of A and B). The focus will be on the case where G is soluble, and on some additional conditions which are sufficient to prove the conjecture in this case.

In what finite simple groups are all elements products of two involutions? (An involution is an element of order two.) The problem has been solved for most of the finite simple groups, and only some of the orthogonal and exceptional groups are left. In my talk I will descibe how I have been trying to tackle this problem.

The Four Colour Theorem was first conjectured in 1852. In its modern formulation it states that the vertices of any planar graph can be coloured with four colours such that no two adjacent vertices are coloured the same. We will look at the methods behind the successful 1976 proof by Appel and Haken.

This talk will be based on the work of two Cambridge mathematicians: Thomas Fink and Yong Mao. It was noticed by Mr Fink and Mr Mao that despite the myriad possibilities for different tie knots only four were in general usage. Rather than rely on trial and error to come up with new designs, they decided to use the power of mathematics to characterise and categorise all the possible knots. By making certain restrictions to exclude those which were aesthetically unappealing they were able to produce a further nine possibilities. The talk will involve a brief exposition of their methods and results. Neckties will be supplied for those of you wishing to try out some of the more recherche designs. Bruce Willis will not be appearing.

*Page last updated Feb 1, 2008*