The waist inequality is a topologist's version of the rank-nullity theorem in linear algebra. It states that for any continuous map from a ball of radius $R$ in $\mathbb R^n$ to $\mathbb R^q$ there is a point in $\mathbb R^q$ whose preimage is comparable in size to the ball of radius $R$ in $\mathbb R^{n-q}$.
There are now several proofs of this remarkable result. This talk will focus on a particular "coarse" version due to Gromov that lends itself to applications in coarse geometry and geometric group theory. I will formally introduce these new tools, explain the (few) things we already know about them, and give many suggestions for things we really ought to know.