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Mathematical Analysis, Modelling, and Applications

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 analysismechanics of materialsmicromagneticsmodelling 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.

Measure theory on Polish spaces

Polish spaces, i.e. topological spaces that are metrizable by a complete and separable metric, are a quite general and ubiquitous framework one might end up working on and well-known to be nice settings where to study measure theory. The course aims at giving an overview of this aspect. Among other things we will study duality theory between measures and functions, weak convergence, the disintegration theorem and Kolmogorov’s product theorem.

Nonsmooth differential geometry

In the first part we will show that on general metric measure spaces, a `Sobolev-like’ first-order differentiation theory is possible, with objects like differential forms and vector fields well defined. 
 
In the second part we will study spaces with Ricci curvature bounded from below, and see that on them the curvature bound makes it possible a second-order calculus, so that, among others, Hessian and covariant derivative are both well defined.

Functional analysis

Aim of the course is to introduce the basic tools of linear and nonlinear functional analysis, and to apply these techniques to problems in PDEs. The course is divided into two parts: the first one concerns spectral theory of linear operators, whose goal is to extend the classical notion of spectrum of a matrix to an infinite dimensional setting. The second part of the course introduces the methods of nonlinear analysis to find the zeros of a nonlinear functional on a Banach space. In particular it gravitates around the implicit function theorem and its variants.

Topics in continuum mechanics

This is a 60-hours introductory course on continuum mechanics and its applications. The aim is to provide first year students with a solid understanding of the fundamental principles of the subject.

Advanced geometry 2

 

Smooth manifolds and differential topology.

 

Introduction to geometric control

The course of 10 lectures will provide an introduction to geometric control theory. The first part of the course will be devoted to controllability, the second part will discuss stabilization, while the last part will focus on optimal control. No prior knowledge of control theory is required.

Course program:
1. Some basic questions in the control formalism, some examples of control systems.
2. Controllability of linear systems. Lie brackets and their relation with controlled motions.

Invariant manifolds for PDEs and some applications

Invariant manifolds are fundamental tools in the study of dynamical systems generated by differential equations. They provide coordinates in which the systems can be partially decoupled and can be used to track the asymptotic behaviors of the orbits. Therefore, starting with Poincare, Hadamard, Lyapunov, Perron and et al., people have studied extensively their existence, smoothness, and persistence under small perturbations (such as those due to the modelling procedure,  small noises, or computational round-off error, etc.).

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