Baby geometri

The Jones polynomial introduced in 1984 was the first so-called quantum invariant in knot theory. It associates to every knot a polynomial, which is not defined in a geometric but rather in a diagrammatic way. Fifteen years later, Khovanov homology was introduced as a so-called categorification of the Jones polynomial. In spite of its combinatoric origins, Khovanov homology has an ever-growing number of geometric applications in knot theory. In this mini-course, we will first work through the detailed construction of Khovanov homology, following a modern version of Bar-Natan’s tangle setup. Then, we will see some of the applications, such as Rasmussen’s proof of the Milnor conjecture (regarding the unknotting number of torus knots). Previous knowledge of knot theory and homological algebra is helpful, but will not be required.

Hyperbolic spaces and groups were introduced by Gromov and have been studied extensively over the past few decades from a huge variety of points of view, and with applications to various fields such as the study of hyperbolic 3-manifolds. Many groups of interest in geometry and topology, most notably mapping class groups, are however not hyperbolic, and it is therefore natural to try to generalise the notion of the hyperbolic group to encompass as many of these examples as possible. The notion of hierarchical hyperbolicity, inspired by the deep work of Masur and Minsky, achieves just that. Roughly speaking, a hierarchically hyperbolic structure on a metric space or group is a coordinate system where the coordinates take values in hyperbolic spaces. I will give an introduction to hierarchically hyperbolic spaces, thereby describing the (coarse) geometry of mapping class groups, and explain what this kind of structure is good for and how to show that a given space or group is hyerarchical hyperbolic.