Training project for the XLI cycle of the PhD in Mathematical Sciences

PROFESSOR	COURSE	HOUR/CREDIT	ABSTRACT
Prof. Elena Beretta, New York University Abu Dhabi	A step into the world of nonlinear inverse problems	16 hours (3 credits)	The course aims to present nonlinear inverse problems for partial differential equations with important applications in imaging (medical, geophysical). The main prototype that will be addressed, exhibiting the ill-posedness features common to many inverse problems, is the model of electrical impedance tomography, from which significant analytical and numerical results have been derived over the past 40 years. The course will cover the ill-posed nature of inverse problems, issues of uniqueness, and estimates of continuous dependence on the data based on an infinite number of measurements. It will then introduce possible regularization techniques and proceed to recent results based on a finite number of measurements, highlighting, in this context, still open mathematical problems.
Dr. Giulio Colombo, University of Milan	Static spaces and related topics in mathematical General Relativity	15 hours (3 credits)	The aim of this course is to present some classical and more recent results concerning the classification and rigidity properties of static spaces, i.e., Riemannian manifolds arising as spatial cross-sections of static solutions to Einstein’s field equations of General Relativity. The topics treated in the course will include the uniqueness theorem for static black holes, the positive mass theorem and some recent uniqueness and rigidity results for static spaces with constant nonzero scalar curvature (corresponding to solutions to the Einstein’s field equations with nonzero cosmological constant) and for sub-static spaces, a generalization of the notion of static spaces that has been an active research theme in the last decade.
Dr. Emanuela Laura Giacomelli, University of Milan	Selected Topics in Mathematical Quantum Many-Body Systems	15 hours (3 credits)	This course will present functional analytic methods for the study of many-body quantum systems. These systems, composed of a large number of interacting particles, play a central role in various areas of physics, from condensed matter to nuclear physics. Within this context, Bogoliubov theory provides a powerful framework for capturing quantum effects arising from particle correlations. We will focus in particular on dilute quantum gases, which have attracted considerable attention in both the mathematical and experimental physics communities—especially in light of recent advances in the study of ultracold atomic gases. A central topic of the course will be recent applications of Bogoliubov theory to dilute Fermi gases, with the goal of developing a rigorous understanding of the role played by quantum correlations.
Prof. Lorenzo Luperi Baglini, University of Milan	Arithmetic Ramsey Theory	15 hours (3 credits)	Arithmetic Ramsey Theory is a branch of combinatorics concerned with the abundance of solutions to certain arithmetic problems on the natural numbers. A typical example of such a problem is determining whether a given configuration is ubiquitous enough that one is forced to find instances of it in each piece of every finite partition of the naturals. In the course, we will focus on how ultrafilters and model-theoretical methods can be used to give short and informative proofs of many fundamental results in this area, mostly related to Diophantine equations, including the theorems of Ramsey, Schur, Van der Waerden, Rado, Hindman, and various generalizations. If time allows, we will also present some results related to certain fundamental open problems in the area.
Prof. Massimiliano Morini, University of Parma	Level set and variational methods for curvature flows	15 hours (3 credits)	We review the classical viscosity level set methods for generalised mean curvature flows, the minimising movements variational approximation of level set flows, and a more recent distributional approach to deal with the well-posedness of crystalline curvature motions. We will also show how the latter distributional approach can be extended to some nonlocal motions. Finally, as time permits, we will present an anisotropic variant of the Ilmanen elliptic regularisation scheme and, relying on the aformentioned distributional formulation, we will show its convergence to level set solutions.
Prof. Marco Adamo Seveso, University of Milan	Hecke characters and their associated L-functions	15 hours (3 credits)	We will discuss Hecke charactes of number fields and their associated L-functions, fol-lowing the adelic approach introduced by J. Tate, with the subsequent reformulation by P. Deligne recognizing the L and epsilon factors. Time permitting, we will mention p-adic analogues of these results for the field of rational numbers.
Dr. Roberto Svaldi, University of Milan	Positivity in Algebraic Geometry	15 hours (3 credits)	0) Basics: line bundles, divisors, morphisms; linear, algebraic and numerical equivalence; fundamentals of intersection theory. 1) Ampleness and nefness: basic definitions; various classical characterizations of amplitude (cohomological, via Serre vanishing, and sheaf-theoretic, via global generation); Nakai-Moishezon criterion for amplitude; Kleiman’s theorem; the cone of effective curves; Klieman’s criterion for amplitude; Hodge index theorem and other Hodge-type inequalities. 2) Bigness: Iitaka theorem on stabilization of morphisms induced by linear systems; definition of Iitaka and Kodaira dimensions; definition of bigness and pseudo-effectiveness; the cone of large and pseudo-effective divisors; the volume function and its properties; finite generation of rings of sections and Zariski’s criterion. 3) Introduction to the ideas of higher-dimensional birational geometry: an excursus on ideas/techniques/results of the last 50 years on the birational classification of algebraic varieties over the field of complex numbers.
Dr. Lorenzo Tamellini, CNR-IMATI and Dr. Pietro Zanotti, University of Milan	Uncertainty quantification and surrogate models	15 hours (3 credits)	The description of a phenomenon of interest via a mathematical model often involves uncertain parameters, such as initial and boundary conditions, source and forcing terms, physical constants and geometric properties, that can be considered as random variables with some prior distribution. This course aims at introducing a set of techniques, known as Uncertainty Quantification (UQ), to assess and possibly reduce the impact of the uncertainty on the model prediction. More precisely, the forward UQ computes statistical indicators of the model output understood as a function of the uncertain parameters, as well as sensitivity indices that quantify the influence of each parameter. The inverse UQ estimates the posterior distribution of the parameters based on the available data. Both these tasks are computationally highly demanding, as they require the computation of the model output for many different combinations of the parameters. For this reason, the course introduces also the construction of surrogate models, to obtain fast and accurate approximations of the mapping of the parameters to the model output.