## Summary of Stefan Wenger’s course

Here is a summary of Stefan Wenger’s course, starting next Thursday at 10:30 am.

Isoperimetric filling problems appear in various areas of mathematics, notably in geometry, geometric group theory, and analysis. The purpose of the present lectures is to discuss several recent results concerning the possible growth of isoperimetric functions in a given metric space and relationships with large scale properties of the underlying space. In particular, we will explain the following two results and their proofs:

(i) a geodesic metric space in which every sufficiently long closed curve c bounds a Lipschitz surface of area at most ${(1-\epsilon)(4\pi)^{-1}length(c)^2}$ must be Gromov hyperbolic;

(ii) there exist nilpotent groups of step two whose isoperimetric Dehn function does not have the growth of a polynomial function.

The first result is optimal and generalizes results of Gromov, Olshanskii, Bowdich, Drutu, and Papasoglu. The second result answers in the negative a long-standing open question in geometric group theory about the possible growth of Dehn functions for nilpotent groups. The proof of both of the above results relies on newly established relationships between the growth of isoperimetric functions and the existence of suitable Lipschitz maps into asymptotic cones. We will explain these results and their proofs together with the necessary background. If time permits, recent results on the growth of higher isoperimetric functions will also be discussed.