2^{nd} Leeds And Manchester Event:
Joint University Mathematics Pure Postgraduate Seminar
The 2^{nd} Leeds And Manchester Event: Joint University Mathematics Pure Postgraduate Seminar took place in Leeds on Friday the 27^{th} of March 2009. Details of the talks are given below.
All talks were in the School of Mathematics at The University of Leeds, directions can be found here and the University of Leeds campus map can be found here.
Some funding was available for travel expenses and subsidised a postseminar meal. This event was organised by: Ali Everett (Manchester), Lotte Kestner (Leeds) and Davide Penazzi (Leeds).
Programme

12:00Welcome
Tea/Coffee 
12:30Hyperplanes and Symmetry
Sam Elliott (Leeds)Abstract (click to view)In his famous book of 1932, Stephan Banach proved many of the early fundamental results in the area of functional analysis and Banach spaces, but also left a number of open problems. Over the years, many of the problems were solved, but at the beginning of the 1990s two questions on the symmetry of these spaces remained.
In this talk I aim to describe how these two questions were finally answered, and how the two apparently different problems turned out to be related. 
13:00Alternating and Symplectic Spaces
Ali Everett (Manchester)Abstract (click to view)Everybody knows what a symplectic space is, right? What? What do you mean, "no"?
OK, so consider an ndimensional vector space over a field, and associate an alternating, bilinear form to it. This is a alternating space. One more condition on this form can turn my space into a symplectic space. So in the short time I have, I will run through a few properties of these spaces, hopefully proving something interesting at the end. Or everybody will fall asleep  I'm OK with either. 
15:00Model Theory and an Application
Phil Ellison (Leeds)Abstract (click to view)Model theory uses formal languages to describe mathematical structures. Insight into a structure can be gained by investigating the subsets of it which are definable in a given language. In this talk, I will give a brief introduction to the basic concepts of model theory (languages, structures, models, completeness, etc.) and sketch the proof of a result which doesn't appear obviously model theoretic, namely: any injective polynomial function from C^{n} to C^{n} is surjective (Ax's theorem).

14:00Tea/Coffee Break

13:30The TemperleyLieb Algebra and the Four Colour Theorem
Andrew Reeves (Leeds)Abstract (click to view)First proposed by Guthrie in the middle of the 19th century, the Four Colour Conjecture was for over a century one of the most famous unsolved problems in mathematics. The eventual proof of the conjecture, by Appel and Haken in the 1970s, is almost as famously difficult, requiring the computerassisted checking of almost two thousand special cases.
In my talk I will discuss the connections between the four colour theorem and the TemperleyLieb algebra, a structure originally arising from the study of phase transitions in statistical mechanics. In particular, I hope to outline a proof of the fact that the four colour theorem is actually equivalent to a proposition similar to the word problem for this algebra. 
14:30Differential Geometry for Algebraists: Operators, Brackets and Divergence on Algebras
Jake George (Manchester)Abstract (click to view)Vector fields, differential operators, div, grad, Laplace operators  all of these are concepts which seem completely inseparable from geometry. On closer inspection, all of these are defined in terms of the algebra of smooth real valued functions on a manifold. As such, the underlying manifold can be more or less jettisoned, differential geometry now being the study of the structure of this algebra.
One may now pose the question  why was the manifold required in the first place? Couldn't some abstract vestigial notions of vector fields etc, still be defined for a wider class of algebras? In this talk, I'll tackle this question and give some constructions. If there's time at the end, I'll show how these definitions can then be applied to the algebra of densities on a manifold, an algebra generalising smooth functions and volume forms. 
15:30The Fundamental Group and Classification of Covering Spaces
Bev O'Neill (Manchester)Abstract (click to view)In this talk I will be introducing the simplest and most important functor of algebraic topology: the fundamental group. As the first of the homotopy groups, the fundamental group gives us information about the one dimensional structure of a topological space by investigating homotopy classes of maps from S^{1} into that space. The theory of covering spaces is intricately linked to that of the fundamental group. We will see that for a sufficiently "nice" space X there is a 11 correspondence between the different covers of X and the subgroups of the fundamental group of X.

16:30FiveASide Football Match
Leeds Pure Mathematics v Manchester Pure Mathematics 
18:00Dinner
Session One (The Mall)
Session Two (The Mall)
Post Seminar
Previous Meetings

12^{th} May 2006The 1^{st} Leeds And Manchester Event

27^{th} March 2009The 2^{nd} Leeds And Manchester Event