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!

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!

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!