The strong ABCABC conjecture over function fields (after McQuillan and Yamanoi)

The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the two independent proofs of the strong $abc$ conjecture over function fields given by McQuillan and Yamanoi. The first proof relies on tools from differential and algebraic geometry; the second relies on analytic and topological methods. They correspond respectively to the Nevanlinna and the Ahlfors approach to the Nevanlinna Second Main Theorem.Comment: 35 pages. This is the text of my Bourbaki talk in march 200

On some differences between number fields and function fields

The analogy between the arithmetic of varieties over number fields and the arithmetic of varieties over function fields is a leading theme in arithmetic geometry. This analogy is very powerful but there are some gaps. In this note we will show how the presence of isotrivial varieties over function fields (the analogous of which do not seems to exist over number fields) breaks this analogy. Some counterexamples to a statement similar to Northcott Theorem are proposed. In positive characteristic, some explicit counterexamples to statements similar to Lang and Vojta conjectures are given.Comment: To appear in the "Atti del Terzo Incontro Italiano di Teoria dei Numeri - Pisa - Settembre 2015". Comments are welcom

Colored and Dissipative Continuous Spontaneous Localization model and Bounds from Matter-Wave Interferometry

Matter-wave interferometry is a direct test of the quantum superposition principle for massive systems, and of collapse models. Here we show that the bounds placed by matter-wave interferometry depend weakly on the details of the collapse mechanism. Specifically, we compute the bounds on the CSL model and its variants, provided by the the KDTL interferometry experiment of Arndt's group [Phys. Chem. Chem. Phys., 2013, 15, 14696-14700], which currently holds the record of largest mass in interferometry. We also show that the CSL family of models emerges naturally by considering a minimal set of assumptions. In particular, we construct the dynamical map for the colored and dissipative Continuous Spontaneous Localization (cdCSL) model, which reduces to the CSL model and variants in the appropriate limits. In addition, we discuss the measure of macroscopicity based on the cdCSL model.Comment: 9 pages, 5 figures; accepted for publication in Physics Letters A (2017

Virtual and rapid prototyping of an underactuated space end effector

A fast and reliable verification of an initial concept is an important need in the field of mechatronics. Usually, the steps for a successful design require multiple iterations involving a sequence of design phases-the initial one and several improvements-and the tests of the resulting prototypes, in a trial and error scheme. Now a day’s software and hardware tools allow for a faster approach, in which the iterations between design and prototyping are by far reduced, even to just one in favorable situation. This work presents the design, manufacturing and testing of a robotic end effector for space applications, realized through virtual prototyping, followed by rapid prototyping realization. The first process allows realizing a mathematical model of the robotic system that, once all the simulations confirm the effectiveness of the design, can be directly used for the rapid prototyping by means of 3D printing. The workflow and the results of the process are described in detail in this paper, showing the qualitative and quantitative evaluation of the performance of both the virtual end effector and the actual physical robotic hand