Pure Postgraduate Seminars
The Pure Postgraduate Seminar Series provides an informal environment for pure maths postgrads to present mathematical ideas. The academic year 2005/2006 seminars were organised by Marianne Johnson.
In the Autumn semester the seminars were held in the now nonexistant Newman Building. In the Spring semester, the seminars were held in the MSS Building, N6, Fridays at 4.00pm
Autumn Semester 2005

4 NovOrthonormal Bases for Cyclotomic Galois Extensions
2005
Erik PickettAbstract (click to view)If L/K is a Galois field extension with Galois group G, we know from the normal basis theorem that there exists an element x of L such that {g(x)g ∈ G} forms a basis for L over K. This talk will focus on such normal bases which are selfdual with respect to the trace form. A basis will be explicitly described for Cyclotomic field extensions.

11 NovLocal BRST Cohomology and Anomalies in Quantum Field Theory
2005
Andrew BruceAbstract (click to view)After introducing the basic notions of superalgebras we present the BRST algebra and its bicomplex. The local cohomology for this bicomplex is constructed and solved via the so called descent equations. The local cohomology is the used to tackle the issue of anomalies in quantum field theory. We consider the Bardeen anomaly as an explicit example.

18 NovA study of Ber+ and Ber Part I
2005
Adam HaunchAbstract (click to view)In GL(n—m), the natural analog of the determinant is the Berezinian. The Berezinian shares many properties of the determinant such as mutiplicativity. Unlike the determinant, however, the Berezinian of an expression such as:
ber(1  Az)
is not a polynomial, it is in fact a rational function. The aim of my talk is to discuss some properties of this rational function and to discuss some techniques for finding the "simplest" way of writing this function. 
25 NovA study of Ber+ and Ber Part II
2005
Adam HaunchAbstract (click to view)In GL(n—m), the natural analog of the determinant is the Berezinian. The Berezinian shares many properties of the determinant such as mutiplicativity. Unlike the determinant, however, the Berezinian of an expression such as:
ber(1  Az)
is not a polynomial, it is in fact a rational function. The aim of my talk is to discuss some properties of this rational function and to discuss some techniques for finding the "simplest" way of writing this function. 
9 DecThe free Lie algebra and Hall bases
2005
Marianne JohnsonAbstract (click to view)I will define the free Lie algebra of rank r and show how to construct the famous Hall bases. If I have time I will try to explain what is meant by Lie representations. This talk will be selfcontained and accessible to a general audience.
Spring Semester 2006

3 FebBanachTarski and the Axiom of Choice
2006
Jacob GeorgeAbstract (click to view)The Greeks, founders of the western mathematical tradition of rigourous proof from commonly held notions, would no doubt have been deeply uneasy about an argument which involved an (uncountably) infinite number of choices. So too were many 19th century mathematicians who realised that there was a serious cause for contention in what has now become the Axiom of Choice. Arguments as to the legitimacy of such an assumption raged for some time, but the divide has for the most part disappeared and most mathematicians consider the Axiom of Choice or any of its equivalents to be entirely legitimate. Presented with an infinite number of boxes, each containing a possibly infinite number of marbles, is it really possible to collect one from each box in any sort of finite time? The answer seems unclear. Although necessary for theorems as rudimentary as proving that any vector space has a basis, the Axiom throws up some strange and exotic consequences, one of which is the BanachTarski Theorem (formerly the BanachTarski paradox). The theorem demonstrates that it is possible to decompose a sphere into a finite number of parts, rearranging these through a series of rotations and translations, to obtain two spheres, each identical to the original. As groundbreaking as this theorem seems, greengrocers the world over have not sharpened their knives and produced infinite numbers of apples with a few strategically placed cuts. Here, definitions of the Axiom of Choice, Well Ordering Principle, Zorn’s Lemma and Tukey’s Lemma will be presented as well as a proof of their equivalence and the proof of the BanachTarski Paradox. Time permitting, a few results of mathematics sanschoice might creep in toward the end.

10 FebThe Happy End Problem
2006
Thomas GrubertAbstract (click to view)The Happy End Problem is a problem involving consideration of points in general position su±cient in number to guarantee the existence of a convex ngon. During this talk I will introduce the problem and the proof that such a sufficient number of points exists, some bounds on this value and my own work on the specific case where n=6.

17 FebUsing symbol space to count things
2006
Matthew HorshamAbstract (click to view)We address the problem of counting elements of the image set of a finitely presented group under maps with bounded deformation. We consider a finitely presented (infinite) group Γ and a quasimorphism φ mapping Γ to a subset of Euclidean space, possibly the Euclidean Lattice. We then consider the distribution of these points about the origin. You can think of this as measuring how much the distance from the identity element if changed on average by a controlled deforming map. In this talk we will present examples of finitely presented groups, their boundaries and limit sets and suitable quasimorphisms. We will introduce shift spaces (the symbol spaces from the title) and a method of coding groups and their boundaries to shift spaces. If there is time we will go through the proof of a central limit theorem in such a setting (with discussions of what a central limit theorem is and means if and when necessary) which involves introducing a function space on the symbol space, a particular class of operators on this function space called transfer operators and analysing some properties of their spectra (like eigenvalues with extra bits).
There probably won't be any pretty pictures. 
24 FebThe notion of algebra in homotopy theory
2006
Hadi ZareAbstract (click to view)I will talk about notion of algebra in homotopy thoery, it will give a simple motivation. This includes an example of infinite loop spaces.

31 MarInfinite Loop Spaces
2006
Hadi ZareNo Abstract 
5 MayFinding god with mathematics  and why the mathematics is found wanting
2006
Craig LaughtonAbstract (click to view)Creationists believe that the complex and diverse organisms that we see in the universe were created by a supernatural being, namely the god(s) of their religious choice. It is perhaps understandable why this seemed like a good explanation in the early nineteenth century, but after the publication of Darwin’s The Origin of Species in 1859, and because of the huge body of supporting evidence that has been amassed since, scientists now reject creationist ideas for the theory of evolution by natural selection. Recently there has been an attempt to rebrand creationism as "intelligent design" (ID) by toning down the religious content and presenting what seem like serious scientific reasons for rejecting our accepted theory of evolution. The aim is to have ID taught in American high schools, since the US constitution does not allow the religious content of creationism in the science class. This is only the beginning of a strategy (called The Wedge Strategy), which aspires to create an overtly Christian society in the US. Intelligent design really is nothing more than creationism dressed up in a shabby lab coat, the "designer" may be ambiguous, but there is no doubt they have a god in mind. And yet the ideas are gaining acceptance, not only in America but here in the UK. A recent BBC survey found that 39% of Britons feel creationism or ID best describe their view of the origin and development of life. So what has all this got to do with maths? Unfortunately some of the proponents of ID use mathematics to support their bogus claims. We will explore how they employ mathematical arguments and hijack valid theorems for their own pseudoscientific ends, and explain why the conclusions they draw are always found wanting. You might wonder if this is even a worthwhile exercise (indeed, I have often thought that I could be spending my time doing something more productive than learning about such nonsense), but science, and confidence in science, is being dangerously undermined, and people are being lied to  it is up to scientists to defend against these pernicious beliefs. Therefore I feel that I’m doing my bit in presenting this summary of how our own field is being misappropriated in the hope that if you ever encounter these ideas, you can fight them too!

12 May1^{st} Leeds And Manchester Event
2006
Joint University Mathematics Pure Postgraduate Seminar (The University of Manchester)
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)