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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 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.

Optimal Transport

Location and selected dates:

Lectures will take place in Room 133-Ambrosetti.

  • October: 2, 3, 9, 10, 23, 24, 30, 31
  • November: 6, 13, 20, 21, 27, 28
  • December: 4, 5, 12, 18, 19
  • January: 8, 9, 15, 16, 22, 23, 29, 30
  • February: 5, 6

The lecture of Friday, November 14 is suspended to allow students to participate in the SISSA Orientation Day 2025.

Topis in advanced analysis II

Content:

The course is focused on Nonlinear Partial Differential Equations. It will start from classical theorems up to some nonlinear PDEs under active research.
1) First order PDEs
2) Transport equations and weakly regular vector fields
3) Hyperbolic conservation laws
4) Euler Equations and convex integration techinques
5) Hamilton-Jacobi equation

Schedule:

23/9/2025 - 15/12/2025

Tuesday 9-11, room 133

Wednesday 14-16, room 133

 

Topics in Hamiltonian PDEs and critical point theory

Abstract: 
In this course I will present existence and multiplicity results of periodic solutions of Hamiltonian systems, finite and infinite dimensional, like wave equations, which can be obtained by variational methods and critical point theory. I will also present other results regarding the long time dynamics of Hamiltonian PDEs on tori, addressing both stable and unstable dynamics.

First lecture:

Tuesday  13 January, 11-13, room 133

 

 

Geometric Control and sub-Riemannian geometry

The course of 20 lectures will provide an introduction to geometric control and sub-Riemannian geometry. The first part of the course will be devoted to controllability, the second part will discuss stabilization and Optimal Control, while the last part will focus sub-Riemannian geometry. No prior knowledge of control theory is required.
 
Course program:
 
PART 1.

Introduction to analytic number theory (Topics in Advanced Analysis I)

Contents:

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. 

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