Mathematics

Aim of the course is to provide an introduction to the world of synthetic description of lower Ricci curvature bounds, which has seen a tremendous amount of activity in the last decade: by the end of the lectures the student will have a clear idea of the backbone of the subject and will be able to navigate through the relevant literature.

This an advanced monographic course on the numerical analysis of finite element techniques. Each year, the state-of-the-art of a research level topic is selected and presented with strong interaction with the students. Past topics have included: Virtual Element Methods (VEM), Nonconforming FEM.

This course provides a high level overview on the numerical solution of partial differential equations. All major classes of numerical methods will be analysed within a rigorous mathematical setting. Key aspects, such as consistency and stability, will be thoroughly investigated, providing the guidelines for the correct choice and implementation of numerical methods for a range of problems.

We consider Ising systems on a lattice involving antiferromagnetic interactions; that is, systems defined on spin functions (taking the values -1 or 1) with an energy favouring spins with alternating signs. Depending on the lattice geometry, ground states with alternating sign may (e.g., on the square lattice) or may not (e.g., on the triangular lattice) exist. In the latter case we say that the system is frustrated.

In the first part of the course I will review the main metric, embedding and structure theorem about Sobolev spaces (depending on the audience more or less in depth), and study the corresponding weak convergence. I will then study lower-semicontinuity conditions for local functionals (that is, integral functionals depending on the weak gradient), described by convexity conditions (convexity, polyconvexity, quasiconvexity, rank-1-convexity), and apply them to obtain solutions of minimum problems by the Direct Method of the Calculus of Variations.