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.

## Topics on Nash-Moser and KAM theory

The course deals with Hormander's approach to implicit function theorems of Nash-Moser type and to its recent extensions. We will consider scales of function spaces of both Holder and Sobolev classes, and their two different mechanisms leading to sharp regularity solutions. Strong relations with the Littlewood-Paley decomposition will be emphasized, as well as the role of orthogonality in the L^2-based case. We will discuss the algebraic part of the scheme, also in comparison with the (more standard) Moser's approach.

## Sub-Riemannian Geometry

In this course I will introduce the basic concepts in sub-Riemannian geometry and I will discuss basic properties of the Carnot-Caratheodory distance and of the geodesics. Then I will discuss how to write the heat equation and Schroedinger equations on such manifolds. For the Schroedinger equation, In the case of almost-Riemannian geometry, I will show how different quantization procedures give rise to completely different phenomena.