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.

## Optimal design of planar shapes with active materials

## Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems

## Markov diffusion semigroups and functional inequalities

After a brief introduction to general Dirichlet forms and their connections with contraction semigroups and self-adjoint operators, we will focus on the diffusive case,

aiming to investigate some analytic and geometric aspects of Markov diffusion semigroups and their infinitesimal generators.

A particular attention will be given to functional inequalities that can be studied in this framework, including but not limited to Poincaré, Sobolev and log Sobolev inequalities, and to relations with Ricci curvature bounds.

## Topics in mathematical epidemiology

In this course, we revisit the classical compartmental models of mathematical epidemiology and present some of their recent advances. The topics can be organized into three main modules. In the first module, we trace the history of epidemiological models starting from the pioneering work of Bernoulli (1766) up to the well-known Susceptible-Infected-Removed (SIR) model by Kermack and McKendrick (1927) and its subsequent variants. We study in detail the SIR-like models by using tools from stability theory and bifurcation theory of dynamical systems.

## Numerical Solution of PDEs Using the Finite Element Method

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

Topics:

## Optimal Transport

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

## Periodic Orbits of Hamiltonian systems through Variational Methods

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.

## Some aspects of mean curvature flow

Topics covered in the course: Main properties of classical mean curvature flow and anisotropic mean curvature flow. The minimizing movements method. Mean curvature flow as a limit of the Allen-Cahn equations.