QuIPS abstracts - Spring 2007
Elliot Costi - Constructive Recognition of Classical Groups
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
Mohan Panchanathan (and Stienlager Stongkerlard) - The Generalised Nottingham Group
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.
Reamonn O Buachalla - An Introduction to Compact Quantum Groups
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.
Colin Reid - Conjugacy Classes of Finite Groups
In my talk I will be discussing the following conjecture, taken from the
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.
Johanna Rämö - Products of two involutions in finite simple groups
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.
Emil Vaughan - The Four Colour Theorem
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
Andrew Curtis - Tie Hard : An Introduction to the Mathematics of Neckties
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.
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