Systolic freedom

Systolic geometry is a subfield of quantitative topology, which started in the late 40s from questions of the following sort: given a non-simply-connected surface (or a higher-dimensional Riemannian manifold), what is the length of the shortest non-contractible loop? This quantity is called the systole; another example of a systolic invariant is the cosystole, which is the smallest area of a codimension-1 submanifold that does not separate the manifold into several pieces. Answering a question of Gromov, in 1999 Freedman exhibited first examples of Riemannian metrics in which the product of the systole and the cosystole exceeds the volume; this manifests the phenomenon of systolic freedom. In our joint work with Hannah Alpert and Larry Guth, we showed that Freedman's examples are almost as "free" as possible, by bounding the systolic product by the volume raised to the power of $1+\epsilon$. I will give an overview of the systolic freedom phenomenon, including the flavors of proofs in the field.