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!

Morse Theory analyses manifolds by looking at the behaviour of differentiable functions on that manifold. We can gain a lot of insight into the topology of a manifold by looking at the critical points of a differentiable function on that manifold: different matrices of second partial derivatives (the Hessian) gives different local behaviours, like saddles, maxima or minima. Looking at what happens between different critical points, we can try to patch up what happens near each critical point to reconstruct our manifold somehow.

To do this, our smooth functions need to be sufficiently nice; the so called Morse functions.

We can in fact strengthen the approach by taking more care at what happens around critical points; we will then find a particularly neat way of packaging that information and passing from that information to topological information. In particular, a theorem of John Jones (with Graeme Seagal and Ralph Cohen) will make its appearance!

So make sure to come to MS.05 at 7:30 to learn about what Morse Theory is all about and why it's so amazing! After which we flow to the pub!

The Warwick Mathematics Society is hosting a special event - a maths Knowitalls contest!

If you're familiar with the BBC2 show Knowitalls, this is the same idea. In the different rounds, each contestant is given a mathematical topic and he has a small amount of time to say everything he knows about the topic.

In the first round, the idea is to come up with as much information as possible, and to mention some of the key points that the jury will have prepared in order to get some sweet bonus points.

In the second round, you will need to come up with one of the key points as fast as possible.

In the third round, you have to come up with as many examples as possible of a given type.

Everyone is more than welcome; we try to make things fair by giving harder topics to third/fourth years than to first or second years. The idea is simply to have fun by trying to remember some nice mathematical facts in familiar areas, not to embarass anyone. But giving impossibly hard questions to Cosmin is also always a fun thing to do; you'll find he doesn't know much about differential geometers after Riemann, for example. Also don't worry if you don't know anything about fiber bundles, we're reserving all those questions for one particular person.

We had a test run with a few regulars last Monday and it proved tremendously fun - as long as you take it lightly, you'll be sure to amuse yourself too! Please be sure to come!

A short notice: there is a discussion group tonight by Cosmin about some topics in the combinatorics of subdivision, for example on the subject of subdividing a square into even or odd numbers of congruent triangles. But it's going to be good! See you there!