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.

## Variational models depending on curvatures in image segmentation

- Variational models for the reconstruction problem:
- the Mumford-Shah model, the Nitzberg-Mumford model,
- the model on labelled contour graphs.

- Three-dimensional scenes.
- Stable maps from a two-manifold to the plane.
- Apparent contours of embedded surfaces.
- Labelling an apparent contour.
- Visible contours

- A solution to the completion problem:
- completing a visible contour graph into an apparent contour.
- Examples of the algorithmic proof.

## Introduction to numerical analysis

The foundations of Numerical analysis

- Resolution of linear systems with direct methods
- Resolution of linear systems with iterative methods
- Polynomial interpolation and projection
- Numerical Integration
- Numerical solutions of ODEs

Numerical Methods for PDEs

- Finite Elements
- Elliptic Problems
- Parabolic Problems
- Hyperbolic Problems

## Geometric control theory

- Control systems on smooth manifolds; orbits and attainable sets.
- Linear systems: controllability test.
- Chronological calculus.
- Orbits theorem of Nagano and Sussmann
- Rashevskij-Chow and Frobenius theorems.
- Nagano equivalence principle.
- Control of configurations ("fallen cats").
- Structure of attainable sets; Krener's theorem.
- Compatible vector fields. Relaxation.
- Nonwandering points and controllability.

## Gamma-convergence

Direct methods in the calculus of variations:

- semicontinuity and convexity,
- coerciveness and reflexivity,
- relaxation and minimizing sequences,
- properties of integral functionals.

Gamma-convergence:

- definition and elementary properties,
- convergence of minima and of minimizers,
- sequential characterization of Gamma-limits,
- Gamma-convergence in metric spaces and Yosida transform,
- Gamma-convergence of quadratic functionals.

G-convergence: