Analogy is a powerful way used in philosophy, science, and mathematics to better understand one theory from similar ones. We will try to explain what we mean by analogy, we will make historical examples and we will present one of the most fruitful analogies: the analogy between number fields and function fields.

The Higher Infinite refers to the infinite cardinalities studied by set theory, as charted by large cardinal hypotheses known as large cardinal axioms. These axioms assert the existence of infinite cardinals so large that their existence cannot be proved within the standard ZFC system of set theory. Since the weakest of large cardinals, the "weakly inaccessible", were first defined and studied by Hausdorff over a century ago, a plethora of different and much stronger large cardinals have since then been identified in a great variety of contexts and taking many different forms. Indeed, after the groundbreaking results of Martin-Steel and Woodin in the 1980’s, establishing the tight connection between large cardinals and the determinacy of sets of reals, the theory of large cardinals has been expanding in multiple directions, yielding solutions to many well-known set-theoretic problems, as well as fertile applications to other areas of mathematics, from general to algebraic topology and homotopy theory, to abelian groups, etc. In this talk I will present some examples of large cardinals and will explain their role in mathematics by giving a number of examples in different areas where they have been applied to solve prominent open problems, some of them very recent.

The problem of existence and qualitative behavior of solutions of evolution equations is a classiacal one in the theory of PDEs. In this seminar I will focus on the use of Birkhoff normal form for the proof of the so called almost global existence results in Hamiltonian PDEs. such results deal with perturbation of linear hyperbolic equation (for example the wave equation) and ensure that solutions corresponding to smooth and small intitial data remain small and smooth for times of order . Here is the size of the initial datum.
Nowadays there exists a well established theory for semilinear equations in space dimension 1, but the theory for quasilinear equations and for equations in higher dimensional domains is only at the beginning. Furthermore some remarkable applications to physical relevant models like the water wave equations have been very recently obtained.
In this Colloquium, I will review the classical theory presenting the main ideas and then I will present some of the most recetn results in the domain trying to put into evidence the main tools that have been developed in order to deal with the problem.

Chemotactic blow up in the context of the Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that when the Keller-Segel equation is coupled with passive advection, blow-up can be prevented if the flow possesses mixing or diffusion-enhancing properties, and its amplitude is sufficiently strong. In this talk, we consider the Keller-Segel equation coupled with an active advection, which is an incompressible flow obeying Darcy's law for incompressible porous media equation and driven by buoyancy force. We prove that in contrast with passive advection, this active advection coupling is capable of suppressing chemotactic blow up at arbitrary small coupling strength: namely, the system always has globally regular solutions. (Joint work with Zhongtian Hu and Alexander Kiselev).