Uryson width and volume

The Uryson width of an n-manifold gives a way to describe how closely it resembles an (n−1)-dimensional complex. It turns out that this is a useful tool to approach several geometric problems.

In this talk we will give a brief survey of some questions in ‘curvature-free’ geometry and sketch a novel approach to the classical systolic inequality of Gromov. Our approach follows up recent work of Guth relating Uryson width and local volume growth. For example we deduce also the following result of Guth: there is an ϵₙ>0 such that for any R > 0 and any compact aspherical n-manifold M there is a ball B(R) of radius R in the universal cover of M such that vol⁡(B(R)) ≥ ϵₙRₙ.