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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.

Computational mechanics by reduced order methdos

Theoretical lectures:

  • 31/3, 16-18, Room A-133
  • 01/4, 10-12, Room A-129
  • 15/4, 16-18, Room A-133
  • 23/4, 14-16, Room A-134
  • 24/4, 11-13, Room A-133
  • 29/4, 16-18, Room A-133
  • 30/4, 11-13, Room A-134

 Practical session:

  • 15/4, 14-16, Room A-134
  • 24/4, 14-16, Room A-133
  • 29/4, 14-16, Room A-134
  • 30/4, 14-16, Room A-134

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.).

Singular perturbations in fractional Sobolev spaces

Singular perturbations are a classical way to tackle problems where either a solution is not ensured by a lack of coerciveness, or there are too many solutions, typically due to a lack of strict convexity.

Nekhoroshev theory for PDEs

Among finite dimensional Hamiltonian systems, the integrable ones are characterized by the existence of special coordinates (action-angle variables) in which the dynamics is particularly explicit: the angles evolve linearly in time and the actions remain constant for all times.
Nekhoroshev theorem guarantees, under suitable regularity and non-degeneracy hypotheses, that when a small perturbation is added to an integrable Hamiltonian, the action variables are quasi-conserved for exponentially long times.

Wave kinetic equations: Global solutions and long-term behavior

Course description:
 
Recent progress in the theory of non-equilibrium statistical physics for nonlinear waves has brought much attention to the study of solutions to wave kinetic equations. These solutions, which capture the average evolution of large wave systems undergoing weakly nonlinear interactions, present a variety of asymptotic behaviors connected to interesting physical phenomena, such as energy cascades and Bose Einstein condensation. 
 

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.

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