Mathematics

Advanced course dedicated to the Numerical Solution of Partial Differential Equations through the deal.II Finite Element Library.

Topics:

The course intends to be a broad introduction to Optimal Transport theory and some of its applications in geometry and analysis. 

Hamiltonian systems give a very good description of those physical phenomena where the energy is (approximately) conserved: from planetary orbits to the motion of particles.

Typically, however, the dynamics is highly sensitive to the initial conditions and therefore it is difficult to find specific orbits in the systems such as those connecting two subsets of phase space or those which are periodic.

Program:

• Quick review of the theory of sets of finite perimeter.
• Partial regularity theory for the omega-minimizers of the perimeter:
• the epsilon-regularity theorem,
• the regularity of Lambda-minimizers and application to the prescribed mean curvature problem.
• Sketch of the proof of the full regularity in lower dimensions.

The aim of this course is to show some connections between the asymptotic behaviour of the parabolic scalar Ginzburg-Landauequations (also called Allen-Cahn equations) and mean curvature flow of a hypersurface. We shall discuss also theasymptotic behaviour of the stationary points of these equations and some connections with minimalsurfaces and prescribed mean curvature surfaces.Contents of the lectures.

"Geometric Control Theory” by Ugo Boscain (CNRS, LJLL, Sorbonne Université, and INRIA), Paris)

The course refers to the use of computational fluid dynamics techniques to address advanced applications in environmental, cardiovascular and industrial contexts. Each topic will be corroborated by a set of numerical examples to be performed within the open source C++ finite volume library OpenFOAM.

 The course concerns the basic arguments in the theory of Partial differential equations of the main families: elliptic, hyperbolic, parabolic, Hamilton-Jacobi.

Program