2^nd Leeds And Manchester Event:
Joint University Mathematics Pure Postgraduate Seminar

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.

Everybody knows what a symplectic space is, right? What? What do you mean, "no"?

OK, so consider an n-dimensional 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.

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 Cⁿ to Cⁿ is surjective (Ax's theorem).

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 computer-assisted checking of almost two thousand special cases.

In my talk I will discuss the connections between the four colour theorem and the Temperley-Lieb 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.

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.

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 S¹ 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 1-1 correspondence between the different covers of X and the subgroups of the fundamental group of X.