The classical contraction principle is one of those basic results in Analysis with many fundamental applications. In this talk, we will examine a variational interpretation of it which turns out to be more flexible. In particular, it can be used to deal with situations where existence and uniqueness of solutions is known or expected. Though many classical situations can be treated, due to time restrictions we will focus on two representative examples: that of initial-value Cauchy problems for autonomous ODE systems, and the case of non-linear, non-variational monotone PDE equations in divergence form. It remains to be seen if this perspective could help in new situations.

The tone of the talk will be elementary. No specialized background is required.

Old questions in additive combinatorics and Ramsey theory ask what infinite patterns are unavoidable in sets of integer numbers with positive density (i.e. that account for a positive proportion of all numbers), but until recently, there were no positive answers. In the last few years a new technique was developed to address such questions, making use of ergodic theory and dynamical systems, which were previously restricted to handling finite patterns.

In this talk I will briefly survey the history of the subject, and explain the connection to ergodic theory. Then I will describe how we used this technique to answer a question of Paul Erdos regarding infinite sumsets. The talk will be based on joint work with Bryna Kra, Florian Richter and Donald Robertson.