Dirac structures were first defined in the PhD thesis (1987) of Ted Courant, Alan Weinstein's student. Dirac geometry is a common setting for several geometric structures on smooth manifolds: Poisson structures, closed 2-forms and foliations, but also generalized complex structures, the "complexified" version of Dirac geometry. In this introductory mini-course we will discuss the basics of Dirac geometry: definition, examples, decomposition into pre-symplectic leaves, submanifolds of Dirac manifolds, operations with Dirac structures, presymplectic realizations and presymplectic groupoids. These concepts will be illustrated in the setting of Poisson geometry, where many of them originate; however, many classical constructions in Poisson geometry turn out to be more natural for Dirac manifolds.

Distributed methods for optimization in control.

Large-scale cyber-physical network systems have become ubiquitous in everyday life. Cooperative robots, sensor networks, smart grids and social-networks are just few examples. An important prerequisite for numerous estimation, learning, decision and control tasks arising in such complex systems is the solution of optimization problems in a distributed, peer-to-peer framework. In this new set-up, processing nodes have limited knowledge of the optimization problem and aim at solving it in a cooperative way.

In this course I will introduce the main mathematical set-up for distributed optimization by combining proper definitions from optimization, systems and graph theories. Then, I will recall some main concepts from convex optimization and duality theory, and show how they can be used to design distributed optimization algorithms. In particular, I will present some basic distributed primal, primal-dual and dual algorithms working in synchronous networks with fixed communication topology. In the second part of the course I will highlight some important challenges that distributed algorithms need to address as, e.g., asynchronous and unreliable communication, and present novel algorithms that can handle such issues. Finally, I will show some important control application in which these methods can be applied.

Introduction to discrete weak KAM theory.

The goal of these lectures will be to give an account of weak KAM theory through its discrete version. We will present the main topics of weak KAM theory: the notion of subsolutions and of weak KAM solutions, the definition and main properties of the Aubry set, regularity properties of subsolutions and minimizing measures.