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.

## Sub-Riemannian Geometry

In this course I will introduce the basic concepts in sub-Riemannian geometry and I will discuss basic properties of the Carnot-Caratheodory distance and of the geodesics. Then I will discuss how to write the heat equation and Schroedinger equations on such manifolds. For the Schroedinger equation, In the case of almost-Riemannian geometry, I will show how different quantization procedures give rise to completely different phenomena.

## Regularity theory for minimisers of the perimeter

Program:

- Quick review of the theory of sets of finite perimeter.
- Partial regularity theory for the omega-minimizers of the perimeter:
- the epsilon-regularity theorem,
- the regularity of Lambda-minimizers and application to the prescribed mean curvature problem.
- Sketch of the proof of the full regularity in lower dimensions.

## Some aspects of mean curvature flow

Main arguments of the course:

- A review on properties of classical mean curvature flow.
- Local well-posedness.
- Comparison principle.
- Convexity preserving.
- Evolution of graphs.
- Formation of singularities and weak solutions.
- The minimizing movement method and the solution of Almgren-Taylor-Wang.
- The phase-field approximation and the Allen-Cahn equation.