Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces

The minicourse will start with a quick introduction, essentially from scratch, to currents  in metric spaces in the sense of Ambrosio-Kirchheim. This will be followed by a proof  of the isoperimetric filling inequality of Euclidean type for cycles in CAT(0) spaces. This important inequality is due to Federer-Fleming for Euclidean space and to Gromov and Wenger in the general case. Some applications will be discussed. If time permits,  an improvement of the isoperimetric inequality for cycles of dimension greater than or  equal to the asymptotic rank of the underlying CAT(0) space, also due to Wenger, will be sketched. This pertains to notions of higher-rank hyperbolicity studied recently in work of Kleiner, the lecturer, and others.

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