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.

## Introduction to the calculus of variations

- Elements of convex analysis, polar and bipolar function and their properties, convex envelopes. Semiclassical theory, Euler-Lagrange equations and relation with elliptic PDE’s. Regularity of minimizers.
- Direct method, quasiconvexity, polyconvexity, rank-one convexity and their relations. Semicontinuity theorems for scalar and vectorial functionals; existence of minimizers.
- Relaxation theorems, representation of relaxed functionals; convex, quasiconvex, polyconvex and rank-one convex envelopes.

## Semilinear equations in singularly perturbed domains with applications to phase transitions and geometrically constrained walls

- Phase transitions models and the Cahn-Hilliard functional.
- The Modica-Mortola Gamma-convergence result.
- Structure of local minimizers of the Cahn-Hilliard functional in convex domains.
- Construction of nonconstant local minimizers in nonconvex domains.
- Extreme geometry and geometrically constrained magnetic walls:
- construction of nontrivial local minimizers in singularly perturbed domains,
- asymptotic analysis of the behavior of local minimizers in the singular limit.

## Mathematical Theory of Elasticity

- Nonlinear elasticity: derivation of the equilibrium equations.
- Existence theory via the implicit function theorem (Korn inequality, existence theory for linearized elasticity) and by variational methods (polyconvexity, weak continuity of minors, Ball's Theorem).
- Properties of solutions: global invertibility, singularities (cavitations).
- Rigorous relation between nonlinear and linearized elasticity in terms of Gamma-convergence.
- If time permits, lower dimensional models.

## Morse theory

- Introduction and motivation.
- Basic review of homology theories.
- Morse functions, the deformation lemma, attaching cells, Morse inequalities.
- A newer approach: Morse-Smale functions, counting the flow lines.
- Some infinite dimensional generalizations and applications to critical points theory.

## Topics in Nonlinear Analysis

In this course I will present the Nash-Moser hard implicit function theorem in scales of Banach spaces with different applications, as, for example, the Siegel linear conjugacy theorem or the KAM (Kolmogorov-Arnold-Moser) theorem concerning quasi-periodic solutions of nearly integrable Hamiltonian systems.

## Variational Methods

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:

## Introduction to sub-Riemannian geometry

- Elements of differential geometry
- Definition of rank-varying sub-Riemannian manifolds
- Continuity of the distance
- Existence of minimizers
- Normal and abnormal geodesics
- End-Point and exponential map
- Hausdorff dimension of a sub-Riemannian manifold
- Intrinsic volume in a sub-Riemannian manifold
- The Laplace-Beltrami operator on a sub-Riemannian manifold
- Examples

## Introduction to Measure Metric Spaces

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 in One Space Variable

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

Despite recent progress, the mathematical understanding of these equations is still incomplete. In particular, no general well-posedness theory is presently available for systems of conservation laws in several space variables.