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Mathematical Analysis, Modelling, and Applications

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 methods in symplectic dynamics

Program:
 
- Introduction to critical point theory (Morse theory, Lusternik-Schnirelmann theory, equivariant methods and Fadell-Rabinowitz index)
- Introduction to symplectic dynamics: Hamiltonian flows, including geodesic flows and, more generally, Reeb flows
- The periodic orbits problems: introduction, and proof of a selection of results (multiplicity of periodic orbits, spectral characterization of Besse and Zoll Reeb flows, etc.).

Nash-Moser implicit function theorems and KAM theory

The main assumption of the classical implict function theorem in Banach spaces is that the linearized operator has a bounded inverse. This is sufficient for constructing bifurcation theory of periodic solutions of finite dimensional dynamical systems. On the other hand, there are several problems where this assumption is not satisfied, i.e. the linearized operator is unbounded, for example for the search of quasi-periodic solutions. To overcome this challenge, the Nash-Moser theory was developed. 
 

Topics in advanced analysis II

Topics in advanced analysis II

The course is focused on evolutionary PDEs of first oder: these will comprise

- first order PDEs and the method of characteristics, local existence and uniqueness

- linear transport equation, in particular continuity equation, Lagrangian representation, renormalization and mixing estimates

- hyperbolic systems of conservation laws, well posedness for scalar equation and systems in 1d, non uniqueness for Euler equation in multid

Advanced FEM techniques

This an advanced monographic course on the numerical analysis of finite element techniques. Each year, the state-of-the-art of a research level topic is selected and presented with strong interaction with the students. This year the course will be devoted to nonconcofming FEM, with particular focus on discontinuous Galerkin methods.

An introduction to 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 approximation,
• Gamma-convergence of quadratic functionals.

Mechanics of biological systems

The course focuses on mathematical tools for the study of the mechanics of continuous media, with applications to mechano-biology and bio-robotics, with topics extracted from the following syllabus:
 
 - Review on basic concepts from biology from a mechanics perspective.
 - Review of the mechanics of deforming solid bodies.
 - The notion of target metric and spontaneous curvature. Residual stresses. Active strains and active stresses. Morphogenesis and differential geometry.

Measure theory on Polish spaces

Polish spaces, i.e. topological spaces that are metrizable by a complete and separable metric, are a quite general and ubiquitous framework one might end up working on and well-known to be nice settings where to study measure theory. The course aims at giving an overview of this aspect. Among other things we will study duality theory between measures and functions, weak convergence, the disintegration theorem and Kolmogorov’s product theorem.

Nonsmooth differential geometry

In the first part we will show that on general metric measure spaces, a `Sobolev-like’ first-order differentiation theory is possible, with objects like differential forms and vector fields well defined. 
 
In the second part we will study spaces with Ricci curvature bounded from below, and see that on them the curvature bound makes it possible a second-order calculus, so that, among others, Hessian and covariant derivative are both well defined.

Functional analysis

Aim of the course is to introduce the basic tools of linear and nonlinear functional analysis, and to apply these techniques to problems in PDEs. The course is divided into two parts: the first one concerns spectral theory of linear operators, whose goal is to extend the classical notion of spectrum of a matrix to an infinite dimensional setting. The second part of the course introduces the methods of nonlinear analysis to find the zeros of a nonlinear functional on a Banach space. In particular it gravitates around the implicit function theorem and its variants.

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