For the last discussion group of the year, we take a look at the famous solution of Fermat's Last Theorem, arguably one of the greatest achievements of 20th century mathematics. The proof involves an amazing amount of fascinating mathematics and, among other things, led to the groundbreaking "modularity theorem". Join us this Thursday at 7:30pm as we explore the underlying concepts and unfold the brilliant ideas that formed the proof of the theorem (in an informal and accessible manner, while still maintaining the amount of detail needed to actually get an idea of what's going on).

In this DG, Sam Playle will give us an introduction to some of the mathematical aspects of classical and quantum electromagnetism. These beautiful theories were some of the motivating examples behind the study of gauge theories whereby considerations of the local symmetries of a field theory give rise to conservation laws. Due to the work of Atiyah and Donaldson these ideas in fact had a number of surprising applications to mathematics, in particular on the topology of 4-manifolds. If this gets your electrons excited then come along to MS.05 at 7:30 to hear more.

Two major areas of combinatorics are enumeration (counting), and graph theory (drawing): in this presentation, we hope to guide the audience through the art of enumerative graph theory (counting what you have drawn). In particular we will look at enumerating spanning trees of graphs, a problem which can be tackled from numerous directions, showcasing many of the combinatorists favourite weapons. Finally we will see some applications of such enumeration in other areas. The presenter is prefers projects involving (easy) problem solving rather than exposition, so the talk should be followable by all.

So join Owen Daniel, and his graphs of many shapes and sizes on Monday night, before we follow a Hamiltonian path to the pub...

Braid Groups can arise naturally in a number of different ways. Intuitively one may view braid groups as 'strands' between some finite sets of points with some naturally defined composition. More formally, one can define braid groups as the fundamental group of some configuration spaces. In turn, these are isomorphic to the mapping class group of the n-punctured disc.

In tonight's discussion group Paul Mortimer will give an introduction to braid groups along with what he covered in his project. This talk may unfortunately not be for the faint hearted. The material was described to us by Paul as "PhD level". He did go on to clarify, "the most difficult aspect of the project was presenting it in a fashion understandable to fellow fourth years." There is therefore some hope that the talk will be an understandable account of Paul's wonderful achievements over the past two terms. Stay Categorical!

In the first half of this discussion group we will hear about elliptic curves with multiplication. These are elliptic curves where the endomorphism ring contains more than just a copy of the integers. The theory is extremely interesting and has a number of deep connections with, amongst other things, questions regarding number fields.

The second half will concern the Weil Conjectures. Originally stated and proved for algebraic curves (by Weil himself), the Weil Conjectures offer a wonderful analogy between the geometry of projective varieties over finite fields and various ideas from algebraic topology.

Come along to MS.05 to hear Cosmin Davidescu and Sam Derbyshire expound on these topics. After which we will head to the pub. Stay categorical.