Finding multidimensional configurations in subsets of the integer grid

Furstenberg and Katznelson's multidimensional Szemer\'edi theorem says that any subset of $\mathbf{Z}^d$ having positive upper density must contain a homothetic copy of any fixed finite collection of points. For example, any such subset must contain the four vertices of an axis-aligned square. It is a major open problem to prove a strong quantitative version of this theorem. I will introduce this problem and its history, as well as explain an unexpected connection with group cohomology.