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.

## A minimization approach to the wave equation on time-dependent domains

## Compactness by maximality

## Perturbation techniques applied to the real vanishing viscosity approximation of an initial boundary value problem

.## Harmonic Analysis

Harmonic Analysis is one of the most used tools in the study of Partial Differential Equation. Aim of the course is to provide the students with the basic instruments needed to address these problems. In particular I will cover the following topics:

- Maximal Functionals
- Singular Integrals
- $H^1$ BMO duality
- Littlewood Payley theory
- The $T1$ theorem.

## Free Boundary Problems

Free boundary problems are a class of problems in which in addition to the unknown function, there is also an unknown domain where the problem is settled. Aim of the course will be to discuss a few free boundary type problems and some of their applications. In particular I will focus on:

- The obstacle problem
- The one-phase Bernoulli problem
- The two-phase problem (if time will permit)

## Advanced Programming

Provide advanced knowledge of both theoretical and practical programming in C ++ and Python, with particular regard to the principles of object oriented programming and best practices of software development (advanced use of version control systems, continuous integration, unit testing).

**Syllabus:**