Pure Postgraduate Seminars
The Pure Postgraduate Seminar Series provides an informal environment for pure maths postgrads to present mathematical ideas. The academic year 2007/2008 seminars were organised by Jacob George.
The seminars were held in the Alan Turing Building, Frank Adams Room (2). For the Autumn semester, the seminars were at Fridays from 2.00pm to 3.00pm. For the Spring semester, the seminars were at Fridays from 4.00pm to 5.00pm.
Autumn Semester 2007

21st September 2007The life and loves of Girolamo Cardano, mathematician adventurer.
Jacob GeorgeAbstract (click to view)Cardano was an ambitious, dishonest, hottempered, quarrelsome, conceited and humorless man, but capable of generosity, kindliness and merciless selfrevelation. His parents were an abominable pair; his favorite son was executed for murder; his other son was a scoundrel who managed to escape the gallows but brought Cardano nothing but unhappiness and disgrace. (J. R. Newman)
As a mathematician, Cardano is chiefly associated with the formula for the solution to the cubic and quartic equations which appeared in his book Ars Magna, or The Great Art. In this talk, I will detail his solutions as well as giving some account of his rather turbulent life. 
28th September 2007The Mathematics of Shuffling
Steve Clegg 
5th October 2007Oulipo  Aufurz Luv Maff 4 Eva!
John MacQuarrieAbstract (click to view)Oulipo is a propernounified abbreviation of 'Ouvroir de littrature potentielle', which roughly translates as Workshop of Potential Literature. The Oulipo are a group of primarily French writers who sought to reinvent literature, with a method analogous to that applied to mathematics by the Bourbaki group. In analogy to the rigorous axioms of Nicolas Bourbaki are formal constraints, which are adhered to throughout the text.
The Oulipo love maths, and use mathematical methods and axioms wherever they can in their constraints. We’ll outline what they’re all about, give a hundred thousand million examples, and maybe (so I get to say 'Profinite' somewhere) attempt an Oulipian poem, if I’m feeling brave enough. 
12th October 2007No Seminar
Cancelled due to illness 
19th October 2007An Invitation to Supermanifold and Superanalysis
Andrew BruceAbstract (click to view)We introduce the notion of odd variables from the necessity to understand (semi)classical fermionic fields such as the electron field. With this in mind we construct superfunctions and propose that these can be used to define the coordinate ring of a supermanifold. We then proceed to discuss calculus on the superdomain R^{nm}, in particular we introduce the Berezin integral of odd variables and show that it cannot be thought of as an integral in the usual sense. We will attempt to stress the similarities and striking differences between odd and even variables.

26th October 2007Coloured Linear Orders
Simon PereraAbstract (click to view)As Black History Month draws to a close it is important to reflect on the coloured version of events, be it in history or in mathematics, especially when it is not the standard, high profile version. This week’s seminar will include a celebration of the coloured version of the rationals.
I’ll define countable, 1transitive, coloured linear orders (CLO’s) and prove some elementary results involving them. I’ll end with a classification theorem for the case of finite colour sets. The classification is given by coding trees, which encode the CLO’s.
I’ll mention in passing the related notions of coloured shuffles and shuffle groups, but only to get pulses racing among the bridge playing fraternity. 
2nd November 2007Reading Week
No Seminar Scheduled 
9th November 2007Algebraic Automata Theory
Elaine RenderAbstract (click to view)Algebraic automata theory takes the theoretical computer science discipline of automata theory and adds aspects of group and semigroup theory, allowing the restatement of many important theorems in the area in group theoretic terms. It has also aided the study of many mathematical problems, such as the decision problems of combinatorial group theory, by reinterpreting them in terms of automata. I will give a basic introduction while trying to avoid anything too computer sciencey; covering the important definitions and a few simple results, as well as lots of examples (with pictures!).

16th November 2007Brouwer and Intuitionism
Richard SimmondsAbstract (click to view)At the beginning of the 20th century the socalled "Crisis in Foundations" pitted several schools of thought against each other on the right approach to the foundations of mathematics. Of these, formalism (lead by David Hilbert, regarding mathematics as symbolic) and intuitionism (lead by LEJ Brouwer, considering mathematics as mental) were foremost. Although Hilbert’s formalism became more famous, intuitionism is an interesting and unusual theory, and Brouwer was an equally unusual proponent.

23rd November 2007Rational functions over the Grassmann Algebra and hypersurfaces in superspace
Adam HaunchAbstract (click to view)There are two primary aims of this talk. The first is to expose some fundamental differences between rational functions over the Grassmann algebra and rational functions over the real numbers. With these distinctions clarified, we move to the second part in which we explore the concept of a supermanifold and show how the geometric properties of these surfaces are intimately related with the algebraic properties of the rational functions discussed previously.

30th November 2007Quadratic forms and the Witt Ring of a field
Erik PickettAbstract (click to view)I will introduce the basic terminology and results for quadratic forms over a field. I will use these to define the WittGrothendieck ring and Witt ring of a field and briefly talk about possible generalisations of these ideas.

7th December 2007The Planar nBody Problem
Gemma LloydAbstract (click to view)We all know that the world revolves around me, but Smale in his paper "Problems on the nature of relative equilibria in celestial mechanics" seems to believe otherwise. So, we shall see what he thinks and give some examples. We will look at some mechanics, some groups, homology, representations,...
something for everyone! 
14th December 2007No Seminar
Cancelled for Christmas Lunch
Spring Semester 2008

1st February 2008Representation theory of psolvable groups.
Stavros ApostolouAbstract (click to view)We will go through some of the techniques used in the representation theory of psolvable groups, in particular we will see how Cliord theory applies in a group with such a structure mentioned above, making life easier. We will finish with an application of these methods to the proof of the FongSwan theorem.

8th February 2008Fibre bundles and Connections
Oliver LittleAbstract (click to view)Given a realvalued function on a manifold we can consider how it varies as we move along the manifold by taking the directional derivative. But if we generalise from a function to a "section" of a fibre bundle which takes as its values some more complicated geometrical object (tangent vector, tensor, element of a Lie group) do we still have a canonical derivative? Given a fibre bundle, we have a welldefined notion of whether a tangent vector is tangent to the fibre ("vertical") but not necessarily when it is "horizontal". Connections are an extra geomtrical structure in a fibre bundle which will help us answer these questions and more.

15th February 2008Examples in Ktheory
Hadi ZareAbstract (click to view)This is a summary talk not in a very precise way. The idea of Ktheory goes back to Grothendieck, where he introduced Kgroups of a ring. We start with this, and proceed to give example's of other versions of Ktheory, where most of them owe their root to Grothendieck. We like to show how this ideas are related, and one provides motivation for another one. In particular we will mention Quillen's version of higher algebraic Ktheories. We also may have time to mention definition of algebraic Ktheory of space, now known as Waldhausen' Ktheory. Among other examples we also have KKtheory, or Kasparov's K thoery of bimodules and also Ktheory of C*algebra, where we may give some remarks on them.
Notes are available for this seminar. 
22nd February 2008Six impossible rings
Lorna GregoryAbstract (click to view)Following Hodge's paper "Six impossible rings", I will construct some rings in transitive models of ZF to show that various implications between the Noetherian and Artinian conditions on a commutative rings no longer hold if we do not assume Zorn's lemma.

29th February 2008The Commuting Involution Graph for the Baby Monster
Ben WrightAbstract (click to view)Let G be a finite group, X a conjugacy class on involutions and t a fixed element of X. We define the Commuting Involution Graph C(G,X), having vertex set X with two points x and y in X being joined by an edge if and only if they commute in G.
We define the i^{th} disk of C(G,X) to be all the elements of X which have distance i away from t. It can easily be shown that the sizes of these disks are independent on the choice of t. We would like to calculate the disk sizes of the Commuting Involution Graph for the second largest sporadic simple group, the Baby Monster, for the class 2c of involutions. In this talk we will go through part of its construction and explain some of the results used in doing so. 
7th March 2008Six Impossible Rings (II)
Richard HarlandAbstract (click to view)Following on from Ms Gregory’s seminar a few weeks ago, I will be further dissecting Wilfred Hodges’ paper on Rings with properties contradicting Zorn’s Lemma. In particular, I will be showing how one can construct an atomless boolean ring in which all ideals are principal, with no infinite descending chains of ideals.
Attending Ms Gregory’s seminar is in no way a necessary prerequisite  the work does not follow on from her seminar, and I will cover any background material needed for the talk. 
14th March 2008The FacePolynomial of Nestohedra
Andrew FennAbstract (click to view)I will present calculations of the facepolynomials of many important series of nestohedra. A nestohedron is a simple polytope which arises from a connected graph. A series of nestohedron is a set of nestohedron arising from a series of graphs which is defined so that each successive graph is obtained from the previous one by the addition of a node that is connected to some subgraph or by replacing a node with two nodes connected by an arc. This raises the problem of how the face polynomials are related to each other both within and beween series. Examples of such series of nestohedron include the Permutohedron and the Associohedron or Stasheff polytope. Furthermore this raise the question of how natural operations on the graphs affect the polytopes and their facepolynomials. The problem of how to describe the faces of a nestohedron in terms of its graph is well known. It is usually attacked using fvectors. We show that an approach based on the facepolynomial of a polytope is a finer tool, since in the case of the facepolynomial we can transform operations on the graph into analytical techniques, which we cannot do with the fvector.

EASTER VACATION

11th April 2008Colouring Knots, A Simple Example in Algebraic Topology
Jerry Hopkinson 
18th April 2008Representation theory of Profinite groups
John MacQuarrieAbstract (click to view)The modular representation theory of a finite group G is the study of the (usually finitely generated) modules over the group algebra kG, for k a field of characteristic p dividing the order of G. We can (and usually do) organise modules by complexity using the concept of relative projectivity. Looking at relative projectivity works well for finite groups, and there’s no reason why we shouldn’t try and apply the results to profinite groups as well. We’ll define all these ideas and then do a proof. There may be no graph theory in this talk, but at least there’s vertices.

25th April 2008The Nonnegative Inverse Eigenvalue Problem
Antony Cronin (University College Dublin)Abstract (click to view)The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining a set of necessary and sufficient conditions such that the set σ = {&lambda_{1}, λ_{2}, . . . , λ_{n}} is the spectrum (set of eigenvalues) of a nonnegative n×n matrix A. In this talk I will give a brief history of the problem including progress to date before surveying some recent results and detailing some subproblems and applications.

2nd May 2008Noncommutative Projective Algebraic Geometry
Chelsea WaltonAbstract (click to view)Geometry: there are many variants of this term, yet due to Alain Connes, Noncommutative Geometry may be initially realized as a deeply differential geometric field. However in the late 1980s, Noncommutative Projective Algebraic Geometry emerged on the scene through Michael Artin and William Schelter’s attempt to completely classify geometric objects of certain noncommutative rings. In this talk I will describe the very beginnings of this journey. In particular I will give examples of noncommutative projective spaces associated to noncommutative rings and will further illustrate how the geometry of these spaces yield vital information of the rings of interest.

9th May 2008A Permutation Construction of the First Janko Group
Ali EverettAbstract (click to view)The Finite Simple Group Classification Theorem states that every finite simple group may be classified into one of eighteen disjoint infinite families, or is a "sporadic" group. There are 26 sporadic groups, so called as they cannot be classified into the other 18 infinite families.
In 1966, Janko proved the existence of a nite simple group (which was the first modern sporadic group), showing that it was unique satisfying three properties. This group was denoted J_{1}, and explicitly constructed as a subgroup of GL_{7}(11). In 1967, Livingstone constructed this group as a permutation representation of degree 266, by starting with the projective special linear group L_{2}(11) as a point stabiliser. This seminar will consist of a "hands on", explicit construction of this permutation group. We will start by briefly glossing over some group theoretical notions, and then defining a set of 266 vertices for L_{2}(11) to act on. We will then construct an undirected graph (without loops) on this set and finally construct the group of symmetries that preserve the edges of this graph. The final step is to show that this group satisfies the three properties required and thus defining J_{1}. 
16th May 2008No Seminar
Cancelled due to illness 
23rd May 2008Why maths isn't fun or is it!?
Juergen LandesAbstract (click to view)In my opinion maths is considered (by the wider public) to be boring. I want to argue that this is quite unfortunate. I think it can and should! be fun. I’ll talk about some mathematical "puzzles" we can explain to nonmathematicians and give them a flavor of what we are doing. So if you expect some "serious" mathematics you will be disappointed. If you’re up for a laugh and some nonsense then you’re very welcome.

30th May 2008An introduction to de Rham cohomology
Rabah Djabri 
6th June 2008Projective Connections and Densities
Jacob GeorgeAbstract (click to view)Projective connections were first considered by Cartan, Veblen and Thomas in the 1920s with respect to two different viewpoints  the first as a special case of a Cartan (later Ehresmann) connection and the second as an equivalence class of affine connections. The main impetus for the study of such objects came from the theory of general relativity which had been published not long before.
The algebra of densities on the other hand is a formal algebraic tool, beloved of physicists for its natural scalar product structure and by mathematicians for leading to a cohomological interpretation of classes of differential equations.
Here I will present some recent results on the link between the seemingly unrelated structures and will try to give some indication of why considering Poisson structures on the algebra of densities is related to considering projective structure on the manifold itself. 
13th June 2008Framed Cobordism and the Homotopy Groups of Spheres: the Pontryagin Construction
Stephen MillerAbstract (click to view)Higher homotopy groups are a natural extension of the fundamental group, one of the basic tools of algebraic topology. They carry a lot of information about the topology of a space. For example, a map between "nice" spaces that induces an isomorphism on all homotopy groups must be a homotopy equivalence. Unfortunately, homotopy groups are not as well behaved as the fundamental group or the well known homology groups. Even calculating the homotopy groups of spheres, seemingly the simplest of topological spaces, is a major outstanding problem in topology. The Pontryagin construction gives some insight by translating the problem into another setting. Calculating the (n + i)^{th} homotopy group of the nsphere turns out to be equivalent to classifying framed idimensional manifolds in codimension n, up to an equivalence relation known as framed cobordism. In this talk I plan to give an overview of homotopy groups and framed manifolds, and explain how the Pontryagin construction allows us to pass from one to the other. I will then show how to classify one and twodimensional framed manifolds in any codimension, which gives us some of the homotopy groups of spheres. Unfortunately this is only a small part of the problem, and in higher dimensions working with framed manifolds becomes difficult very quickly. For that reason, more results have been produced using algebraic methods such as spectral sequences. One goal of current research is to bridge the gap between the algebraic and geometric approaches by interpreting the algebraic sequences in terms of manifolds and related geometric objects.
Previous Seminars
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)