Quantitative isoperimetric inequalities for the eigenvalues of elliptic operators

Isoperimetric inequalities regarding the eigenvalues of the Laplace operator (for different types boundary conditions) are nowadays considered classical results. In this lecture we will discuss their quantitative versions, i.e., more refined inequalities that take into account "how far" the domain is from being the optimal one. Remarkably, all the main inequalities have been refined in just a few years, the last of which (that of the Robin problem) appeared only half a year ago. We shall consider the derivations of quantitative isoperimetric inequalities as well as the issue of their sharpness. We will then move to the biharmonic operator, for which very little is known in this regard. We will conclude with some open questions.