Description of the Course.

The classical Galois theory dates back to 1830, but it took more than 100 years for it to be reformulated (by Artin) in the language of module theory. Artin's criterion that decides when a field extension K/k is Galois allows to extend the classical Galois theory to Hopf algebras. This was initiated by Chase and Sweedler in 1969 in the commutative case, the general case being considered by Kreimer and Takeuchi in 1981. Today Hopf-Galois extensions appear in various branches of mathematics and physics, being also known as dual algebraic versions of non-commutative fiber spaces (the notion of quantum fiber space can be introduced as a module associated with a Hopf-Galois extension).

The purpose of this course is to make the transition from the classical Galois theory to the Hopf-Galois theory and to present some directions of study for the latter. Briefly, the content of the course is as follows:

- Classical Galois theory.

- Extensions of fields, extensions of separable Galois fields without groups, strongly graded rings, cross products, affine group schemes.

- Relevant examples of Hopf algebras.

- Hopf-Galois extensions and examples.

- Descent theory.

- Hopf-Galois theories in various categories.

Sono previsti interventi dei componenti il Comitato Tecnico Scientifico del progetto stesso.
Gli interessati sono invitati a partecipare.
Cordiali saluti

Mara Teresa Borgato

Prof. Michael Herty (RWTH Aachen University)

Abstract: We present recent work on interacting multi-agent optimization problems using meanfield limits. In particular, we are interested in the case of game aspects of possibly infinitely many players.A strategy for an efficient computation of the optimum is presented as well as meanfield limits.  Theoretical and numerical examples are given.

Gli interessati sono cordialmente invitati ad intervenire.

Luigi Pepe

Gli interessati sono cordialmente invitati ad intervenire.

Luigi Pepe

Title: Logarithmic bundles of multi-degree arrangements in $P^n$

Abstract: Let D = {D_{1},...,D_{l}} be a multi-degree arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let O^{1}_P^n(logD) be the logarithmic bundle attached to it. We show that O^{1}_P^n(logD) admits a resolution of length 1 which explicitly depends on the degrees and on the equations of D_{1},...,D_{l}. Then we prove a Torelli type theorem when D is made of a "sufficiently" large number of hypersurfaces: indeed, we recover the components of D as unstable smooth hypersurfaces of D_{1},...,D_{l}. Moreover we analyze the case of one quadric and a pair of quadrics, which yield examples of non-Torelli arrangements. In particular, through a duality argument, we prove that two pairs of quadrics have isomorphic logarithmic bundles if and only if they have the same tangent
hyperplanes. Finally we give a description of the conic case and of some line-conic cases.

The Gauss--Green formulas are of significant relevance in many areas of analysis and mathematical physics. This motivated several investigations to extend such formulas to more general classes of integration domains and weakly differentiable vector fields, and thus led to the definition of the divergence-measure fields. These are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose distributional divergence is a Radon measure, and so they form a new family of function spaces, which in a sense generalize the $BV$ fields. The divergence-measure fields were introduced at first by Anzellotti in 1983 for the case $p = \infty$, and then they have been rediscovered in the early 2000s by many authors interested in various applications.

In this talk, we shall present an overview of such researches, with a particular focus on the results concerning essentially bounded divergence-measure fields. For such a family of vector fields, Silhavy (2005), Chen, Torres and Ziemer (2009) and Comi and Payne (2017) showed that Gauss--Green formulas hold for sets of finite perimeter, by proving the existence of essentially bounded interior and exterior normal traces on the reduced boundary of the given set.

Subsequent extensions of these results and generalizations to non-Euclidean geometries will be also briefly discussed.