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.

To all our friends in the Warwick Maths Society, we cordially invite you to join us for our annual Christmas get together this Tuesday.

We will be going out for a meal at The City Arms, in Earlsdon (Coventry), where we will be making the best use of Wetherspoons' steak night offer!

The plan is to gather in the pub at around 7.30, if you are on campus and are not sure how to get to Earlsdon then there will be a crew of helpers around the bus stop at 7pm. The 12 bus goes at 7.05, so don't be late!

We look forward to seeing you on Tuesday.

The WMS Exec.

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.