The concept of bundles permeates mathematics: It starts from simple set theory, where fiber bundles represent partitions of the set, explaining how to relate different partitions of sets together with a given map between sets; but it also explains many very surprising phenomena such as the Hopf fibrations which account for many nontrivial homotopy groups of spheres, for example there is a map whose image cannot be deformed to a point (in a similar way in which we can twist the circle around itself using the map of unit complex numbers).

It seems we have no hope of classifying all bundles, indeed we do not fully understand the homotopy groups of spheres, and it seems like classifying fiber bundles should be even harder. But there are surprisingly many canonical constructions that allow us to describe many bundles very efficiently; one of the most successful have been the characteristic classes for vector bundles, which have many equivalent interpretation and give much insight into the geometry and structure of such bundles.

In our journey towards understanding characteristic classes we will uncover many important aspects of the theory such as classifying spaces and surprising connections between homotopy groups and cohomology. The theory of characteristic classes, originally initiated by Chern, has allowed for the proofs of many very powerful results, and are related to many others.

Join us Monday in MS.04 for all of this, and more! Then we homotopy ourselves to the pub.

This term only, for all your revision and coffee based needs, Maths Café will be running twice a week under the guise of Revision Café. Every Wednesday and Friday from 1 PM to 3 PM, feel free to come have a cup of tea/coffee and get help with any course you need to revise for.

The idea of counting is one of the first mathematical ideas ever considered, and spawned many fundamental areas of mathematics such as number theory or combinatorics. The idea of counting is essentially that of putting some collection of objects in bijection with some finite set. However, we might also want to consider infinite sets, and in that case the idea of size of a set leads to different concepts for infinite sets: ordinals and cardinals. Ordinals and cardinals of finite sets are essentially the same thing, but differences appear when considering infinite sets. Their importance is then crucial to understanding how large infinite sets are, and allow us to formulate precisely how much larger some infinite sets are than others; for example we know that the set of real numbers cannot be put in bijection with the set of rational numbers; indeed, the first is uncountable whereas the second is countably infinite. But there also exists different sizes of uncountable sets, and this idea is fundamental to the concept of infinite cardinals.

There are many interesting problems that can be formulated in terms of ordinals and cardinals, probably the most famous one being the Continuum hypothesis, which has been proven to be independent for the currently accepted set of axioms for set theory (Zermelo-Fraenkel set theory).

Join us at 7:30 in MS.04 while Cosmin Davidescu explains all this and even some applications to some seemingly unrelated problems, for example in complex analysis.