Flows, functional inequalities and spectral estimates

Functional inequalities are a very important tool in global analysis, analysis on manifolds, in the study of nonlinear partial differential equations, in mathematical physics, in spectral theory, etc. There are many kinds of inequalities which arise in different contexts. Together with the classical ones, there has been recently a large effort to find new ones and to show how they can help to study very interesting problems in various areas.
In this talk I will discuss the general topic of functional inequalities, the existence of extremal functions for them, and their qualitative properties. I will present various methods that are useful in this topic and in particular I will show how the use of ad-hoc flows and entropy-entropy production methods helps immensely both in proving the inequalities, as in the study of their extremal functions.
Last I will discuss how a duality argument allows to obtain very accurate spectral estimates for differential operators from the optimal version of well adapted functional inequalities.