Let X be a finite collection of sets. Given n>1 what is the maximal number of ways disjoint union of n sets (elements from X) is again a set in X? We will estimate this number in terms of n and the cardinality of X.
Nonlinear elliptic PDEs play a central role in many geometric and physical applications. To construct solutions and determine their qualitative behavior, our key tool is the maximum principle. We will discuss the role of elliptic PDE and the maximum principle in the solution of two important problems (the plateau problem, and optimal transport), and mention some interesting open questions.