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.

## Mechanics and Geometry of Active Materials

- Introduction to the physics and applications of active materials: polymers, gels, natural and biological tissues
- A mechanical framework for active materials: metric description of elasticity with active strains
- Preliminaries: concepts and tools from three-dimensional differential geometry and differential geometry of surfaces
- Compatibility of active strains
- Dimension reduction: derivation of a Koiter-like plate theory from 3D elasticity with acrive strains
- Applications to shape morphing

## Introduction to sub-Riemannian geometry

- Isoperimetric problem and Heisenberg group.
- Sub-Riemannian length and metric.
- Rashevskii-Chow theorem.
- Existence of length-minimizers.
- Normal and abnormal geodesics.
- Hamiltonian setting; Hamiltonian characterization of geodesics.
- The endpoint map and the exponential map; conjugate and cut points.
- Nonholonomic tangent space.
- Popp volume and Hausdorff measure.
- Sub-Laplacian and sub-Riemannian heat equation.
- Lie groups and left-invariant sub-Riemannian structures.