Sobolev and Poincar\'{e} inequalities play a central role in the regularity theory of elliptic operators. This theory can be adapted to the case when ellipticity degenerates, but one has to work in a certain metric measure space associated to the operator instead of the Euclidean space. Investigating geometric properties of these metric measure spaces can provide powerful tools to tackle regularity questions. One of the central questions for PDEs is what types of Sobolev inequalities (if any) such a space supports. On the other hand, if a metric measure space is known to support a Sobolev inequality, one can deduce some interesting geometric properties, such as the doubling condition. In this talk I will discuss connections between geometric properties of metric measure spaces, functional inequalities, and regularity theory for degenerate elliptic operators.