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.

## Instability and non-uniqueness in fluid dynamics

## Introduction to analytic number theory (Topics in Ad. Analysis 1)

This course considers the classical topics in analytic number theory, with a focus on tools coming from Fourier analysis. The list of topics to be covered is as follows:

1. Review of the basic elements of Fourier analysis: Fourier transform in L^1 and L^2; Plancherel's theorem; Tempered distributions; Fourier series; Convolution and approximations of the identity.

2. Diophantine approximations; Equidistribution of sequences; Notions of discrepancy; Erdös-Turán inequality; Irregularities of distribution.

## Advanced programming

Students will acquire a comprehensive understanding of advanced programming concepts, specifically in C++ and Python. They will become familiar with object-oriented and generic programming paradigms, as well as proficient in utilizing common data structures, algorithms, and relevant libraries and frameworks for scientific computing. Furthermore, students will be introduced to fundamental software development tools in a Linux environment, encompassing essential aspects like software documentation, version control, testing, and project management.

## Weak turbulence and wave kinetic equation

## Water waves

The water waves equations were introduced by Euler in the 18th century to describe the motion of a mass of water under the influence of gravity and with a free surface. The unknown of the problem are two time dependent functions describing how the velocity field of the fluid and the profile of the free surface (giving the shape of the waves) evolve. The mathematical analysis of the water waves equations is particularly challenging due to their quasilinear nature, and it has been (and still is) a central research line in fluid dynamics.

## Turbulent compressible fluid dynamics

Turbulence plays a fundamental role in many different applications varying from aeronautical to environmental simulations. This course aims at giving an overview of the phenomenology and mathematical modelling of compressible turbulent flows. We will start from a first part focused on gasdynamics (thermodynamics, compressible navier-stokes equations, speed of sound, shock waves). The second part of the course will instead be focused on turbulence phenomenology and modelling.

## Calculus of variations

Programme: Minumum problems for integral functionals in one independent variable: necessary and sufficient optimality conditions, solution to classical problems, problems invariant under reparametrization and geodesics. Minumum problems for multiple integrals: direct methods, lower semicontinuity and relaxation results for multiple integrals, quasiconvexity and polyconvexity.

Number of lectures: 30 two hour lectures

Period: November 15 - March 29