Pure Postgraduate Seminars
The Pure Postgraduate Seminar Series provides an informal environment for pure maths postgrads to present mathematical ideas. The academic year 2006/2007 seminars were organised by Stephen Clegg.
The seminars were held in the MSS building, O10, Fridays at 4.00pm
Autumn Semester 2006
6 OctOn Klein's solution to the quintic.
13 OctFrom Manifolds to Supermanifolds via Sheaves
Jacob GeorgeAbstract (click to view)
The origins of the concept of a manifold lie in differential equations as solution spaces. The generalisation is primarily due to Gauss and Riemann and resulted in the more modern concept of a space which is locally Euclidean. Smooth manifolds are the fundamental objects of differential geometry and have an additional smoothness condition which stitches together the various Euclidean neighbourhoods in some smooth manner. However, this is by no means the only characterisation. The much celebrated GelfandNaimark theorem established that for an arbitrary topological space, all of the topological information is to be found within its space of continuous complex valued functions. A similar construction is possible for smooth manifolds, the smooth structure implicitly defined by considering the algebra of smooth functions.
Here we define a manifold in terms of its sheaf of smooth real valued functions and show that this definition is indeed equivalent to the standard one. Supermanifolds, which are generally regarded as rather esoteric generalisations of manifolds will hereby arise naturally via an alteration of the structure sheaf.
20 OctVertices for irreducible characters of finite groups
Stavros ApostolouAbstract (click to view)
We would like to relate Green vertices for indecomposable modules, of the group algebra, of a finite group with the set of irreducible characters of the group. We cannot do this in general but when the group is nice, in the sense it has loads of normal subgroups, then we can associate vertices to irreducible characters.
We will define Brauer characters and blocks of finite groups and then we’ll show how the above idea applies to some good looking groups.
3 NovFormal groups and how they are used to construct abelian extensions of Local fields.
Erik PickettAbstract (click to view)
Local class field theory is the study of the maximal abelian extensions of local fields. One of the main tools in constructing these abelian extensons is a type of Formal group, first studied by Lubin and Tate. In this talk I will define local fields, give some motivation as to why we study them, define Lubin-Tate formal groups and use them to construct the maximal abelian extension of a given local field.
16 NovAn introduction to pro-finite groups
John MacQuarrieAbstract (click to view)
After making some topological demands, we can define a profinite group to be a categorical limit of a system of finite groups. Some profinite groups are outrageously big, but they all share some extremely useful properties with finite groups. This talk will define and discuss profinite groups with as few prerequisites as possible, and then prove a generalisation of part of Sylow's theorem, one of the coolest finite group theorems. At times the discussion will be necessarily categorical, but I promise (except as alternative to expletive) not to use the word "adjoint".
1 DecAn introduction to sphere packing.
8 DecModel Theory
Simon PereraAbstract (click to view)
Model theory aims to describe the behaviour of mathematical structures (fields, graphs, topological spaces, whatever) with sentences of first order logic. I will introduce the basic concepts of a structure, theory, model etc with simple examples. I will cover some core results and examples such as the Compactness Theorem and Lowenheim-Skolem Theorems, and the independence of both AC and Continuum Hypothesis from the ZF axioms, non-standard models of arithmetic. I will discuss the construction of mathematical structures by a two-player game between Abelard and Eloise (so-named by Wilfid Hodges). This approach may be new to those who have studied model theory. If I have time I hope to say a little about my research into the Grothendieck rings associated to R-modules.
15 DecA proof of a theorem of A. Klyachko using combinatorics on standard tableaux.
Marianne JohnsonAbstract (click to view)
In the last seminar of two thousand and six I will try to amuse you with my bag of tricks. A result of Klyachko I'll try to deduce with some poems inspired by a Doctor named Seuss.
Spring Semester 2007
2 FebAn axiomisation of geometry.
Stephen CleggAbstract (click to view)
I'll talk about Bachmann's axiomisation of Euclidean geometry which is defined solely in terms of group actions without the need for `lines' or `points'.
16 MarAn introduction to groups of finite Morley rank
23 MarGödel's incompleteness theorems
27 AprToric Topology and complex cobordism
Craig LaughtonAbstract (click to view)
Toric topology is the study of topological spaces with torus actions. In the past fifteen years the subject has quickly grown into a recognised subdiscipline of algebraic topology, and is now a very active area of research.
In contrast, the theory of complex cobordism stretches far back into twentieth century mathematics, for instance, the complex cobordism ring was determined by John Milnor and Sergei Novikov in the early sixties. More recent results have shown that the ideas of toric topology can also be applied to complex cobordism.
We will devote most of the talk to defining quasitoric manifolds, which are the main objects of study in toric topology, and several examples will be described. Complex cobordism will be introduced as a cohomology ring and we will explore its connections with toric topology.
The talk will be fairly basic with plenty of reassuring pictures, and the only prerequisites will be some basic algebra (groups, rings etc.) and the notion of a manifold.
4 MayH-spaces, Spaces with operations
Hadi ZareAbstract (click to view)
A group is a set with some operation which makes that set easier to understand. A similar point of view exists in topology and the relevant concept is the concept of an H-space.
These are spaces with operations. In particular loop spaces are H-spaces. We give an introduction on about H-spaces.
List of 2008/2009 seminars (Ali Everett)
List of 2007/2008 seminars (Jacob George)
List of 2006/2007 seminars (Stephen Clegg)
List of 2005/2006 seminars (Marianne Johnson)
List of 2004/2005 seminars (Matt Horsham)
List of 2003 seminars (Matthew Craven/Sara Santos)
List of 2002 seminars (Sarah Perkins/Sara Santos)