Institut de recherche mathématique avancée

Résumé : We present a compact scheme whose core concept involves decomposing it into two subsystems of equations. The Physical Equations utilise the function and its K derivatives at a node by implementing physical relations. These equations operate locally, with no exchange of information with other nodes,as the physics involved are governed by local operators. The Structural Equations depend on linear relationships between the function and its derivatives across a stencil of R points, which we call a R-block, establishing complete connections between a node and its neighbours. These relationships are independent of the physics involved since they are established regardless of the specific problem. In this presentation we address in particular the accuracy and stability of these methods. Based on joint work with S. Clain, G. J. Machado and Ricardo Costa

Résumé : The number of homomorphisms from a graph G to graphs G_q indexed by a positive integer q defines an invariant of G in the parameter q. We shall take G_q to be a Cayley graph on an Abelian group. (For the cognoscenti: because we are then effectively counting tensions and can then define a dual invariant counting the corresponding flows.) Of particular interest have been those sequences for which a polynomial in q results, not least because you can evaluate a polynomial invariant in q at values of q that are not positive integers and sometimes make sense of what this invariant says about G. The classical case is where (G_q) is the sequence of complete graphs on q vertices, for which we obtain the chromatic polynomial of G, and if for instance I evaluate the chromatic polynomial at q = -1, then I get the number of acyclic orientations of G. In this talk I will try to explain why we might wish to cast our net further out and seek those sequences (G_q) for which the homomorphism counts from G have a rational generating function (which includes the case where the counts are polynomial in q). To help do so, I will use two examples, the first where the Abelian group on which G_q is defined is cyclic order q, and the second where the Abelian group is a q-fold direct sum of the group of order 2. Joint work with Delia Garijo (Univ. Seville) and Anna de Mier (UPC Barcelona)

Résumé : The Mordell-Weil group of abelian varieties over complex function fields can be studied using techniques from differential geometry and algebraic topology. Specifically, each rational point corresponds to an algebraic section of the associated abelian scheme. While such sections admit local logarithms, the obstruction to the existence of a global logarithm is encoded in a lattice known as the relative monodromy of the section. Intriguingly, this object appears to be deeply linked to the arithmetic properties of the section itself. For example, under certain hypotheses, the non-triviality of the relative monodromy is equivalent to the section being non-torsion. Furthermore, it is conjectured that the rank of the relative monodromy relates to the dimension of the minimal abelian subscheme containing the section’s image. In this talk, I will present partial results toward this conjecture, along with applications to transcendence problems. This is joint work with F. Tropeano (Università Roma Tre), extending earlier work of Corvaja and Zannier in the setting of elliptic surfaces.

Résumé : L'application d'un champ électromagnétique externe permet de contrôler l'évolution d'un système quantique. Les applications associées, telles que la manipulation du spin nucléaire des protons en imagerie médicale ou le contrôle des états internes d'atomes piégés en information quantique, exigent de piloter un système d'un état initial donné vers un état cible, ou un voisinage de celui-ci, en minimisant le temps de transfert. Un modèle usuel pour cette classe de problèmes s'appuie sur le formalisme de l'équation de Schrödinger intégrant un terme de contrôle bilinéaire. Conçu pour être accessible sans prérequis en mécanique quantique, cet exposé dressera un panorama des méthodes fondamentales du domaine sur les 25 dernières années, allant des théorèmes pionniers de Beauchard et Coron jusqu'aux contributions récentes de Nersesyan et Duca sur la contrôlabilité en temps arbitrairement petit.

Résumé : The study of non-abelian cohomology was pioneered by Simpson in the 90s, and is, broadly speaking, the study of representations of the fundamental groups of algebraic varieties; on the other hand, many questions on this subject were already studied in the early 1900s by mathematicians such as Painlevé, Garnier, and Schlesinger. As for usual cohomology, there are many different realizations of non-abelian cohomology, such as Betti, de Rham, étale, and even crystalline, leading to a rich interplay between dynamics, differential equations, and arithmetic. I will give an introduction to this topic, focusing on the mod p aspects, and the proof of a conjecture of Bost/Ekedahl--Shepherd-Barron--Taylor in many new cases. This is based on joint work with Daniel Litt.

Résumé : The Mandelbrot set is one of the most iconic objects in mathematics due to its visual appeal. It has its origin in the complex dynamics of quadratic polynomials. In this talk, we introduce Julia sets and the Mandelbrot set from scratch. We will show that a Julia set is connected if it is parametrized by a point inside the Mandelbrot set and completely disconnected otherwise. Time permitting, we will show that the Mandelbrot set itself is connected.