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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.

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.

Advanced Topics in Scientific Computing

This course provides a high level introduction to the numerical analysis of PDES and related high-performance computing techniques, focusing on problems in mechanics such as fluid dynamics. Students will acquire advanced understanding on Computational modelling techniques, both theoretical and practical. The course will utilise a combination of frontal lectures and live programming demonstrations using the C++ deal.ii (dealii.org) Finite Element Library.

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