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 in advanced analysis 2

The course is focused on Partial Differential Equations. It will start from classical theorems up to some nonlinear PDEs under active research.

## Topics in nonlinear analysis and dynamical systems

The first part of the course deals with local and global bifurcation theory and applications to dynamical systems and PDEs, like the Lyapunov center theorem, Hopf bifurcation, traveling and Stokes waves for water waves, as well as other bifurcation problems in fluids. At the beginning we shall present the differential calculus and the implicit function theorem in Banach spaces. Later on I will deal with also cases in which the classical implicit function theorem can not be applied since the linearized operator has an unbounded inverse.

## Introduction to sub-Riemannian geometry

The aim of this course is to provide an introduction to the geometry of sub-Riemannian manifolds, and to illustrate some research directions in this domain. References:

## Advanced Topics in CFD

The course deals with a list of advanced topics in computational fluid dynamics. In the first part, the governing equations governing fluid dynamics problems are reviewed and derived. Both compressible and incompressible flows are considered. The basics of the finite volume methods for the numerical discretization of fluid dynamics problems are reviewed and discussed starting from the basic example of an advection-diffusion equation.

## Topics in Continuum Mechanics

- Reminders on linear algebra and tensor calculus
- Kinematics of deformable bodies
- Eulerian and Lagrangian descriptions of motion
- Balance laws of continuum mechanics: conservation of mass, balance of linear and angular momentum, energy balance and dissipation inequality
- Constitutive equations
- Fluid dynamics: the Navier Stokes equations
- Solid mechanics: nonlinear and linearized elasticity
- Selected topics from the mechanics of biological systems

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