Abstract: Gromov proved that every continuous map f: S^n \to R^q from the n-dimensional standard sphere admits a point y \in R^q such that the preimage f^{-1}(y) has (n-q)-dimensional Hausdorff volume at least vol(S^{n-q}).

Every continuous map f: T^n \to R^q from the n-dimensional torus admits a point y \in R^q such that the cohomology of the preimage f^{-1}(y) has rank at least (n-q).

We will explain why statements of this type are important and how they can be proven by the same technique.