The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The present themes are multiscale analysis, mechanics of materials, micromagnetics, modelling of biological systems, and problems related to control theory.The applications of mathematics developed in this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming from the real world, from industrial applications, and from complex systems.

## The periodic nonlinear Schrodinger (NLS) equations: deterministic and probabilistic approaches

In this course we will study different aspects of the nonlinear and periodic Schrodinger equation, mainly in 2D. We start with Strichartz estimates for the deterministic well-posedness, we then introduce the notion of transfer of energy, and we present few theorems concerning this very active area of research. We then move to the notion of Gibbs measure and its invariance. If we have time we introduce the wave kinetic equation associated to the NLS as well.