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.

## 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.

## Mechanics of biological systems

The course focusses on mathematical tools for the study of the mechanics of continuous media, with applications to mechano-biology and bio-robotics, with topics extracted from the following syllabus:

## 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

## Models and applications in CFD

The course provides an introduction to the numerical simulation of laminar and turbulent incompressible flows by using a finite volume method. Each topic will be corroborated by a set of numerical examples to be performed within the open source C++ finite volume library OpenFOAM.

**Module 0**

- Well posedness of an abstract problem
- Numerical approximation of an abstract problem: consistency, convergence and stability
- Lax–Richtmyer theorem

**Module 1**

## Semigroup theory and applications

**Program**

Bochner integral; Pettis and Bochner theorems; vector valued distributions and Sobolev functions.

Elements on unbounded operators: closed, dissipative and maximal dissipative operators.

Semigroups and their generators.

Cauchy problem for abstract equations, Duhamel formula.

Hille-Yosida, Lumer-Phillips and Stone theorems, construction of (semi)groups associated to Heat, Wave, Klein Gordon and Schrödinger equations.

## Advanced FEM techniques

## Topics in computational fluid dynamics

### Topics/Syllabus

- Introduction to CFD, examples.
- Constitutive laws
- Incompressible flows.
- Numerical methods for potential and thermal flows
- Boundary layer theory
- Thermodynamics effects, energy equation, enthalpy and entropy