Homotopy theory is the study of continuous deformation of topological spaces ("continuous" spaces). One especially important idea in that area is that of a homotopy group: it encodes all the ways to map a sphere continuously into any space, up to continuous deformation of such maps.

Homotopy groups turn out to be particularly difficult to describe in general - even the homotopy groups of spheres (which just represent continuous maps from spheres to spheres) are very complicated and their structure is still not entirely explained. Several patterns emerge however, some quite obvious but many more subtle.

Of particular interest is that some homotopy groups eventually stabilise, giving rise to stable homotopy groups. These are also very mysterious, and are only entirely calculated up to . Many interest patterns emerge here too, especially some glimpse of a periodicity. To explain this, we must first study the homotopy groups of some matrix groups, which are of particular importance in algebraic topology. This reveals a beautiful pattern known as Bott Periodicity, which will turn out to explain many of the features we have encountered so far.

Join us in MS.05 at 7:30pm as Sam Derbyshire explains this - starting from the basic definitions of homotopy groups, which just requires an intuitive understanding of continuity.

After this, we homotopy ourselves to the pub!