Mathematics

 Program:

• basic review of calculus in Banach spaces
• free and constrained critical points
• the Palais-Smale condition and the deformation lemma
• the Mountain-Pass theorem and other variational schemes
• Lusternik-Schnirelman category (time permitting)
• Krasnoselski genus (time permitting)

The exam will consist of a written part (about 1 hour and a half), and an oral part.References:

1. Elements of differential geometry
2. Definition of rank-varying sub-Riemannian manifolds
3. Continuity of the distance
4. Existence of minimizers
5. Normal and abnormal geodesics
6. End-Point and exponential map
7. Hausdorff dimension of a sub-Riemannian manifold
8. Intrinsic volume in a sub-Riemannian manifold
9. The Laplace-Beltrami operator on a sub-Riemannian manifold
10. Examples

Overview:
The course will be about recent advances on analysis over metric measure spaces with particular focus on thosewith Ricci curvature bounded from below. We will start from the definition of Sobolev space over a metric measure space and discuss:
- differential calculus on m.m.s.
- heat flow on m.m.s.
- definition of distributional Laplacian and Laplacian comparison estimates - some geometric properties of spaces with Ricci curvature bounded from below.

Systems of conservation laws are partial differential equations with several ap- plications coming from both physics and engineering, in particular from the fluid dynamics.

• 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.

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

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

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: