One of the most important objects associated to a smooth manifold is its tangent bundle; for example many ideas in differential geometry are phrased in terms of various structures on the tangent bundle (connections, curvature, parallel transport, differential forms, ...).

For this it quickly becomes apparent that a study of vector bundles is very helpful, the tangent bundle being one particular kind. There are many objects to deal with this, for example the deep and powerful theory of characteristic classes. There is another very important theory that is very helpful in their description, and this is K Theory.

K theory also brings to light many surprising phenomena: it is a cohomology theory along with the more usual singular cohomology for example, but it has many very distinct properties: for example it is periodic (this is Bott's periodicity theorem, although slightly rephrased), and hence often infinite in infinitely many different dimensions. It also provides interesting ways of proving many well known theorems, for example it can be used to prove the Frobenius theorem stating that every finite dimensional division algebra over is either , or ; or to show that only the spheres () , and are parallelisable (that is, they admit as many everywhere linearly independent vector fields as their dimension; we already know from the Hairy Ball theorem that this means the sphere must be of odd dimension).

Join us at 7:30pm in MS.04 for an accessible overview of K-Theory. Then we go to the pub!

It's finally the end of the year and, to help you relax after all that hard work, the maths society is organising the annual WMS BBQ in week 10. It'll take place on Wednesday the 24th and it's open to all WMS members. We'll provide free drinks and buns, you just need to bring the meat. It's going to be awesome and the last WMS social of the year, so make sure you don't miss it: it all starts next Wednesday at 2 PM on BBQ site 2 (don't worry if you don't know where that is, you can just turn up to the Maths Department at 2 PM and someone will be there to guide you). We hope to see you all there!

The WMS Puzzle Trail is back for a third time next Wednesday! The concept is simple: go around campus in teams of 4 people or so solving puzzles (which won't be too hard or too academic) to get the location of the next ones. Blaze through it to get one of the prizes in store for the first few teams to finish or take it at a relaxed pace and just have some fun, it's completely up to you. So, if you're up for it, it's going to start at 12 PM on Wednesday the 24th in the Maths Department and you can sign up just before the beginning (and, of course, it's free for all WMS members). It should finish just in time for the WMS BBQ so you're more than encouraged to come to both!

DGs are back this Monday for a look at the fascinating world of algebraic number theory! If you feel like learning about some interesting mathematics in a relaxed setting after all those boring exams, feel free to join us in MS.04 this evening at 7:30 PM. Following the DG we'll, as usual, head on to the pub.

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

Elliptic Curves have motivated a large part of the mathematics of the last century: in particular, they have allowed for the famous proof of Fermat's Last Theorem. They still are an area of active research, in pure mathematics (for example, motivated by the Birch and Swinnerton-Dyer conjecture, one of the seven Clay Math problems) but also in applied mathematics in relation to many applications in cryptography.

Over the complex numbers, elliptic curves have a particularly simply description: an elliptic curve is just a complex torus. But this conceals a lot of their inner structure! Trying to classify them up to isomorphism leads into many interesting problems, such as the idea of moduli spaces, the mysterious j-invariant and doubly periodic functions on the complex plane.

These elliptic curves come with a group structure (either coming from their description as a torus or by considering a chord-tangent group law, the two being related by the Weierstrass function), and we can thus consider their endomorphisms (the group homomorphisms from the elliptic curve to itself). Every elliptic curve over the complex numbers comes with a multiplication by endomorphism for every integer , but for most of them, that's the end of the story. Special elliptic curves, with more endomorphisms, are said to have complex multiplication.

The study of elliptic curves with complex multiplication relates perhaps surprisingly with class field theory and the study of quadratic imaginary fields (fields of the form for positive integers ); there is a direct relation between isomorphism classes of elliptic curves with given endomorphisms and corresponding quadratic imaginary fields.

Join us in MS.04 at 7:30 to learn about all this and more! Don't miss it - this is probably the last Discussion Group for a while due to exams starting for many of us. We will of course translate ourselves to the pub afterwards.