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.

## Advanced FEM techniques

This is an advanced monographic course on nonconcofming FEM, with particular focus on discontinuous Galerkin methods. The course, together with the complementary course “Advanced topics on the analysis of Finite Element Methods” lectured by Prof. L. Heltai during the same period, will cover in depth the formulation, analysis and practical implementation of nonconforming FEMs.

## Numerical Solution of PDEs

This course provides a high level overview on the numerical solution of Partial Differential Equations (PDEs). The course focuses on the Finite Element Methods (FEMs) but insights on all major classes of numerical methods will be discusses. Numerical methods will be presented and analysed within a rigorous mathematical setting. Key aspects such as consistency, stability, and convergence 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.

## Topics in advanced analysis 2

The course is focused on Partial Differential Equations. It will start from classical theorems up to some nonlinear PDEs under active research.

## Topics in nonlinear analysis and dynamical systems

The first part of the course deals with local and global bifurcation theory and applications to dynamical systems and PDEs, like the Lyapunov center theorem, Hopf bifurcation, traveling and Stokes waves for water waves, as well as other bifurcation problems in fluids. At the beginning we shall present the differential calculus and the implicit function theorem in Banach spaces. Later on I will deal with also cases in which the classical implicit function theorem can not be applied since the linearized operator has an unbounded inverse.

## Introduction to sub-Riemannian geometry

The aim of this course is to provide an introduction to the geometry of sub-Riemannian manifolds, and to illustrate some research directions in this domain. References:

## Topics in Continuum Mechanics

- Reminders on linear algebra and tensor calculus
- Kinematics of deformable bodies
- Eulerian and Lagrangian descriptions of motion
- Balance laws of continuum mechanics: conservation of mass, balance of linear and angular momentum, energy balance and dissipation inequality
- Constitutive equations
- Fluid dynamics: the Navier Stokes equations
- Solid mechanics: nonlinear and linearized elasticity
- Selected topics from the mechanics of biological systems

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