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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.

Analysis Seminars 2023-2024

Analysis Seminars are held on Thursday from 14:00 to 15:30.

Organizers: Alberto Maspero, Nicola Gigli.

Linear and nonlinear bifurcation problems (Topics in Ad. Analysis 2)

After introducing the theory of analytic functions between Banach spaces,  we shall present perturbative results for the spectrum of linear operators, in particular for separated eigenvalues of closed operators, with applications to the stability of traveling water waves. Then we shall present  bifurcation results  of periodic and quasi-periodic solutions of nonlinear dynamical systems as well as homoclinic solutions to hyperbolic equilibria of Hamiltonian systems.

Topics in high order accurate time integration methods

Ordinary differential equations (ODEs) describe many physical, biological and chemical phenomena. It is, thus, important to find approximations of ODEs which are highly accurate and, in order to obtain it within reasonable computational times, high order accurate time integration methods are very often chosen to proceed in time. In this course, we will revise ODEs and the theoretical results that guarantee their existence and uniqueness [1, 2].

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.

An introduction to Nekhoroshev theory

The present schedule may still be subject to little variations, mainly to minimize possible overlaps with other courses.

Instability and non-uniqueness in fluid dynamics

The incompressible Navier-Stokes system is a fundamental mathematical model used to describe the motion of fluid flows. Despite being very old, our comprehension of this system remains limited.
The question of whether Navier-Stokes solutions develop singularities in finite time is still unresolved, making it one of the seven millennium prize problems.  
However, thanks to Leray's contributions, we are aware of the existence of weak solutions in the energy class that persist globally in time.

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

The aim of this course is to present recent developments in the theory of non-equilibrium statistical physics for nonlinear waves, commonly known as wave turbulence.
A large system of weakly-interacting waves is generally governed by a large number of differential equations which describe the dynamics of each wave. Due to their complexity and the large number of waves,

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.

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