Ferrara 2022/23

Lectures take place at Dipartimento di Matematica e Informatica, Sede: via Machiavelli 30, 44121 Ferrara, Sede distaccata: via Saragat 1, 44122 Ferrara.

Title and Credits: Geometry of principal frequencies

Teachers: Lorenzo Brasco

Syllabus: The first eigenvalue of the Laplacian on an open set, and more generally of a second order elliptic operator, is an important object both from an applied and theoretical point of view. In Mathematical Physics, it usually plays the role of the ground state energy of a physical system. Despite its importance, for general sets it is not easy to explicitly compute it: thus, we aim at finding estimates in terms of simple geometric quantities of the sets, which are the sharpest possible. The most celebrated instance of this kind of problems is the so-called Faber-Krahn inequality.
This course offers an overview of the methods and results on sharp geometric estimates for the first eigenvalue of the Laplacian and more generally of sharp Poincaré-Sobolev embedding constants (sometimes called "generalized principal frequencies"). In particular, we will present: supersolutions methods, symmetrization techniques, convex duality methods, the method of interior parallels, conformal transplantation techniques.

Dates 2022/2023: 24h lectures +  home assignments.

Title and Credits: An introduction to uncertainty quantification for PDEs, 4 CFU

Teacher: Lorenzo Pareschi, Giulia Bertaglia

Syllabus: The course aims to provide an introduction to numerical methods for uncertainty quantification with specific reference to PDEs. After defining the main concepts in the field of uncertainty quantification, including some references to probability theory, the course focuses on two main approaches. The Monte Carlo method, in its variants characterized by multi-fidelity techniques, and the methods based on generalized polynomial chaos expansions, both in intrusive and non-intrusive form. Specific applications to the case of hyperbolic systems with relaxation terms and reaction-diffusion equations will be considered. In-depth study by students through specific reading of articles will also be suggested.

Dates 2022/2023: 12h lectures + reading course + home assignments

Title and Credits: Recent topics in numerical methods for hyperbolic and kinetic equations, 4 CFU

Teacher: Lorenzo Pareschi, Giacomo Dimarco, Walter Boscheri

Hyperbolic and kinetic partial differential equations arise in a large number of models in physics and engineering. Examples of the applications area range from classical gas dynamics and plasma physics to semiconductor design and granular gases. Recent studies employ these models to describe the collective motion of many particles such as pedestrian and traffic flows, epidemiology and other dynamics driven by social forces. This course will cover the mathematical foundations behind some of the most important numerical methods for these types of problems. To this goal, the first part of the course will be devoted to hyperbolic balance laws where we will introduce the notions of finite-difference, finite volume, and semi-Lagrangian schemes. In the second part we will focus on kinetic equations
where, due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods requires a careful balance between accuracy and computational complexity. Finally, we will consider some recent developments related to the construction of asymptotic preserving methods.

Dates 2022/2023: 12h lectures + reading course + home assignments.

Title and credits: Computational intelligence and gradient-free optimization, 6 CFU

Teachers: Filippo Poltronieri, Mauro Tortonesi and Lorenzo Pareschi

Syllabus: This course provides an introductory overview of key concepts in computational intelligence with a focus on metaheuristic methods for global optimization. These include Genetic Algorithms (bitstring and integer vector genotype representations) and Particle Swarm Optimization (constrained PSO, quantum-inspired PSO, and a multi-swarm version of quantum-inspired PSO), extended with adaptation mechanisms to provide support for dynamic optimization problems. The main algorithms will be illustrated with the help of simple implementations in Matlab and/or R language. In the last part of the course, using a mean-field approach, rigorous convergence results for some of the methods will be presented.

Dates: 12h lectures + 4h assignments, May-June 2023

Title and Credits: (Modal) Symbolic Learning, 2CFU+2CFU (optional, for some research work)

Teacher: Guido Sciavicco

Syllabus: Symbolic learning is the sub-discipline of machine learning that is focused on symbolic (that is, logic-based) methods. As such, it contributes to the foundations of modern Artificial Intelligence. Symbolic learning is usually based on propositional logic, and in part, on first-order logic. Modal symbolic learning is the extension of symbolic learning to modal (and therefore, temporal, spatial, spatio-temporal) logics, and it deals with dimensional data. In this course we shall lay down the logical foundations of symbolic learning, prove some basic properties, and present the modal extensions of classical learning algorithms, highlighting which ones of those properties are preserved, and which ones are not.

Dates 2022/2023: September 2022, 4 lectures, 8 hours

Title and Credits: Plane Cremona transformations, 3 CFU

Teacher: Alberto Calabri

Syllabus: Rational and birational maps of the complex projective plane. Fundamental points and exceptional curves of a plane Cremona transformation. Quadratic transformations and De Jonquières maps. Properties like equations of conditions and Noether's inequality. Factorization of transformations and proofs of Noether-Castelnuovo theorem, Cremona equivalence of plane curves. Properties of the varieties parametrizing plane Cremona maps of fixed degree.

Dates 2022/2023: 20h-24h lectures +  home assignments.

Title and Credits: Galois Theory: from groups and forms to descent theory and extensions

Teacher: Prof. Daniel Bulacu, University of Bucarest

Syllabus:  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.

Dates 2022/2023: 4, 6, 11, and 13 July 2023, 16:00-18:00 (4 lectures, 8 hours).