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.

Back in 2008, Simon Willerton and Eugenia Cheng from Sheffield University decided to put together a series of videos (http://www.youtube.com/user/TheCatsters#p/u ) to teach the world about category theory. So impressed were we by their lucid and good humoured explanations that we decided to invite them both to come and give talks. To our great delight they both agreed!

Simon Willerton will be giving the first talk. This will take place in MS.02 this Wednesday, where as usual there will be food and drink available afterwards.

Simon's has told us his talk will be on " measuring the size of metric spaces - sets with a notion of 'distance' - and how that might lead to pure algebra having applications in ecology." The talk should be fun and will be accessible to all undergrads.

See you all on Wednesday!

Our first discussion group special of the term will be by everyone's favourite lecturer, Samir Siksek. In the talk we'll hear about some of the fascinating developments at the forefront of number theory, presented in a fun and accessible manner. Afterwards, there will be food and drinks available in the street along with a chance for some discussion. So come along!

Samir provided us with a short description of the talk:

Abstract: The generalized Fermat conjecture concerns the equation in coprime integers . This conjecture has been called the "new holy grail of number theory". In this fun talk we shall survey what is known and explore some recent directions.

The determination of how many independent vector fields there are on the sphere was one of the great successes of algebraic topology.

The story starts with the Hairy Ball Theorem: if is even, there are no nowhere zero (continuous) vector fields on . We must then turn to odd dimensional spheres, and wonder how many linearly independent vector fields can there be? On our way to answering this question, we will encounter many familiar structures: , , and even more general algebraic structures known as Clifford algebras, which will allow us to give a lower bound on the number of linearly independent vecor fields on spheres.

We will find many relations with K-Theory, similarities and common periodicities; these have been exploited by Frank Adams to prove that the lower bound provided by Clifford algebras is actually an equality, although we will not delve into the details.

Come tonight at MS.05 to hear about all this, and more! After which we find our way to the pub.

Update: This Discussion Group is postponed to Monday, March 8th.

Elliptic curves are, as any regular attendee of discussion groups knows by now, one of the richest and most interesting topics in mathematics.

Of particular importance over the rational numbers are elliptic curves with complex multiplication: they are those that come with additional endomorphisms. Over finite fields however, all elliptic curves have more endomorphisms than just the usual multiplication by maps: there is the beloved Frobenius that helps us count points and solve many other problems. But there are even more special elliptic curves over finite fields: supersingular curves. In this case, their ring of endomorphisms has the interesting structure of an order in a quaternion algebra, and so in particular is non-commutative.

All supersingular elliptic curves share many important properties, and it will be one of the aims of this talk to show the equivalence of many of these properties. These range from the description of the endomorphism ring to measuring -torsion, considerations of isogenies or just counting the number of points.

The talk will end with the consideration of the relation between elliptic curves with complex multiplication over and supersingular elliptic curves over finite fields; in particular, we deduce many easy estimates for the size of the torsion of an elliptic curve over (or over other number fields).

Come to MS.05 at 7:30 to hear about all this, and more! After which we reduce ourselves to the pub.

The theory of Riemann surfaces began (as one might expect) with Riemann wondering about the correct framework in which to study multivalued functions. Riemann realised he needed to consider 2-dimensional `domains' with some inscribed `geometry'; what we would now call Riemann surfaces and then of course the functions on these. As with complex analysis a number of remarkable theorems were then discovered with such beautiful examples as the Uniformization Theorem, Riemann-Roch Theorem and Riemann's Existence Theorem. This led naturally to wondering whether the analogues of these statements held for higher dimensional complex manifolds after which the theory was largely overhauled with the introduction of the language of sheaves, cohomology and the Serre Duality Theorem taking the place of the analytic tools previously developed with regards to harmonic functions and integrals.

If any of this has piqued your interest come along to MS.05 at the slightly unusual time of 7:00 to hear Callan McGill expound on this! After which, we will also have the opportunity to hear a bit about Algebraic K Theory from Joe Tait! Thereafter we will analytically continue ourselves to the pub!