Abstract
Jensen's inequality says that the differential inequality defining convexity is equivalent to an integral inequality. This is a special instance of a more general phenomenon, an idea with manifold ramifications in analysis, PDEs, and probability: subharmonicity methods in integral inequalities; otpimal stopping and optimal control in stochastic processes, and, more recently, the "Bellman function technique" in harmonic analysis, developed since the late 90s by Nazarov, Treil, and Volberg. The seminar is an excursion around this topic, following one among many possible paths.