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!

Just a short note to mention that today's Discussion Group is indeed happening; Ben Simpson will present us with his findings about category theory. Don't miss it!

Dirichlet's Theorem on primes in arithmetic progression states that for any coprime numbers and , we can find infinitely many primes congruent to modulo . A stronger version is in fact true: each residue class contains the same amount of prime numbers: if you write a table of all prime numbers up to and count how many end up in a given residue class (ie how many end up being congruent to modulo ), you will find that there are approximately . This means primes are equidistributed amongst the residue classes, as there are precisely integers between and that are coprime to , and hence only possible infinite families.

Something somewhat similar happens when looking at the splitting of primes in the ring of Gaussian integers : the behaviour is essentially described by Fermat's Theorem on primes expressible as the sum of two squares. Indeed, any prime congruent to mod factors as the product of two prime elements of (for example, ) whereas primes congruent to mod stay prime in this larger ring (lastly, ramifies in this ring: , which means that gains a repeated factor). So here we see that on average, half of the primes split into two factors, and half stay prime: this is related to the fact that the degree , as a Galois field extension, is 2.

This last observation actually generalises much further, and in fact this generalisation was formulated by Frobenius, who was unable to prove it. The Russian mathematician Chebotarev, however, was able to come up with a proof; this generalisation also gives us Dirichlet's Theorem as a special case. The method of proof, that Chebotarev allegedly thought of while carrying water from the lower part of his town to the higher part, has been invaluable in the development of class field theory. Indeed, when Artin first formulated his acclaimed general reciprocity law, he was unable to prove it; only later, after reading Chebotarev's proof, was he able to come up with a proof of his theorem.

Come to MS.05 to learn about all this, and more! After which we factor into the pub.

Discussion Groups are back for this term, starting with a talk about cryptography by Sir Alexander Thomas Aubrey Oakes, University Number 0624729.

Cryptography can be thought of as the science of keeping secrets. While you may not have any skeletons in the closet, you probably have credit cards, e-mail accounts, personal details and less than 5 passwords that you can remember. To protect these your computer uses algorithms that are publicly known and easily looked up by anyone. If people know exactly how your private information is hidden, how come no one knows that your password is a play on "Manchester City" and that you live in Chelsea?

It's surprising what information can be given away without fully compromising what's important. It is even possible to hold a conversation in the presence of a 3rd party who knows everything you do, and communicate without the 3rd party being able to understand what is being said. This in essence is public key cryptography.

After which, we secretly move on to the pub.

Classification of manifolds up to diffeomorphism is a hard problem (deciding whether two given 4-manifolds are diffeomorphic is at least as hard as the word problem for groups). Thom's brilliant idea (which had been considered, before him, by Poincaré and Pontryagin) was to introduce a weaker equivalence relation than diffeomorphism: the equivalence relation of cobordism.

Two -manifolds are (unoriented) cobordant if their disjoint union is the boundary of an -manifold with boundary. We can see that we can compose cobordisms end to end, and every manifold is cobordant to itself as we have , so it is easy to check it is indeed an equivalence relation. In each dimension we can form an abelian group, with operation disjoint union, the -th cobordism group.

There are other types of cobordism, that arise by putting structure on our manifolds and requiring that the cobordisms preserve this, for example oriented cobordism: we only study oriented manifolds, and the oriented boundary of the manifold with boundary has to give the correct orientation on the -manifolds.

Pontryagin's success was to realise that certain cobordism groups end up being equal to stable homotopy groups of spheres, helping the calculation of the latter.

Thom largely reversed the process and generalised Pontryagin's idea, making an explicit correspondence between cobordism groups and homotopy groups of certain spaces, now called Thom spaces.

Another useful way to study cobordism is through characteristic classes. Many cobordism theories have particular characteristic classes associated to them, and it turns out that two manifolds are cobordant if and only if all the characteristic numbers of these two manifolds agree. This lead to great insights into the theory of characteristic classes, for example the idea of a generalised genus (a topological invariant) as a homomorphism from the cobordism ring to some other ring, for example the rationals. This is then of great interest in classifying manifolds, and allowed things such as Milnor's construction of exotic -spheres.

If you're already impatient to hear more, then you should come to MS.05 this thursday at 7:30 pm; you'll hear about all this and then much more (such as topological quantum field theories!).

After which we realise that we are cobordant to our future selves that are in the pub.