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.

## Theory and Practice of Finite Element Mehtods

This is shared course between the SISSA PhD track on Mathematical Analysis, Modeling, and Applications (math.sissa.it) and the Master in High Performance Computing (www.mhpc.it). It is a course that follows two parallel lines:

practice of finite element methods (mhpc students levels, ~20 hours), see the course (https://www.math.sissa.it/course/fem-with-dealii-2022)

theory of finite element methods (graduate students level, ~20 hours).

## An Introduction to modern tools for collaborative science

# An introduction to modern tools for collaborative science

## From CD to RCD spaces

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.

## Advanced FEM techniques

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.

## Numerical Solution of PDEs

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.

## Frustrated variational lattice 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.

## Local and nonlocal variational problems in Sobolev spaces

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.