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.