Dans cet exposé nous discuterons des résonances pour un graphe quantique dont sa partie compacte est attachée en un sommet à une arête infinie. Les conditions de transmission à ce sommet dépendent d’un petit paramètre et nous démontrons sous certaines hypothèses sur la géométrie du graphe l’existence d’une famille de résonances dont la partie imaginaire tend vers l’infini.

Ce travail est motivé par une question issue de la physique expérimentale où de telles familles de résonances ont été observées. Je montrerai comment avec des outils mathématiques élémentaires il est possible de montrer l’existence et la localisation de ces résonances.

Il s’agit d’un travail interdisciplinaire en collaboration avec Maxime Ingremeau, Ulrich Kuhl, Olivier Legrand, Junjie Lu (Univ. Nice).

 Monsters populate mathematics : topologist's sine, Vitali set, Weierstrass function… These counter-examples to naive intuitions often have in common that they are defined either in a convoluted way, either with an oscillating function like the sine. The o-minimal paradigm allows us to forget those oddness and to make our first intuitions true, by considering only objects that have a "reasonable" definition in a way. What is an o-minimal structure? What examples do we know of? What is happening there? How do they act in complex geometry, in number theory, in optimization?

In this talk, I will present the results of a collaboration with Benjamin McKenna on the injective norm of large random Gaussian tensors and uniform random quantum states and, time allowing, describe some of the context underlying this work. The injective norm is a natural generalization to tensors of the operator norm of a matrix and appears in multiple fields. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, known as geometric entanglement. In our recent preprint, we provide high-probability upper bounds in different regimes on the injective norm of real and complex Gaussian random tensors, which corresponds to lower bounds on the geometric entanglement of random quantum states, and to bounds on the ground-state energy of a particular multispecies spherical spin glass model. Our result represents a first step towards solving an important question in quantum information that has been part of folklore.

Simuler numériquement de manière précise l'évolution des interfaces séparant différents milieux est un enjeu crucial dans de nombreuses applications (multi-fluides, fluide-structure, etc). La méthode MOF (moment-of-fluid), extension de la méthode VOF (volume-of-fluid), utilise une reconstruction affine des interfaces par cellule basée sur les fractions volumiques et les centroïdes de chaque phase. Cette reconstruction d'interface est solution d'un problème de minimisation sous contrainte de volume. Ce problème est résolu dans la littérature par des calculs géométriques sur des polyèdres qui ont un coût important en 3D. On propose dans cet exposé une nouvelle approche du calcul de la fonction objectif et de ses dérivées de manière complètement analytique dans le cas de cellules hexaédriques rectangulaires et tétraédriques en 3D. Les résultats numériques montrent un gain important en temps de calcul.

L'existence de métriques kählériennes canoniques (Kähler-Einstein, à courbure scalaire constante, etc...) dans une classe de cohomologie donnée d'une variété kählérienne compacte admet une formulation variationnelle comme équation d'Euler-Lagrange de certaines fonctionnelles. Grâce aux travaux profonds de Darvas-Rubinstein et Chen-Cheng, on sait que de plus qu'elles admettent des points critiques (donc des métriques canoniques) ssi elles satisfont une condition de croissance linéaire. Après avoir passé en revue ces objets fondamentaux, j'expliquerai comment cette caractérisation permet de généraliser des travaux d'Arezzo-Pacard et Seyyedali-Szekelyhidi portant sur la stabilité de telles métriques par éclatement de la variété. Il s'agit d'un travail en collaboration avec Mattias Jonsson et Antonio Trusiani.

Algebraic curves over a finite field $\mathbb{F}_q$ have been a source of great fascination, ever since the seminal work of Hasse and Weil in the 1930s and 1940s. Many fruitful ideas have arisen out of this area, where number theory and algebraic geometry meet, and many applications of the theory of algebraic curves have been discovered during the last decades. 

A very important example of such application was provided in 1977-1982 by Goppa, who found a way to use algebraic curves in coding theory. The key point of Goppa's construction is that the code parameters are essentially expressed in terms of the features of the curve, such as the number $N_q$ of $\mathbb{F}_q$-rational points and the genus $g$. In this light, Goppa codes with good parameters are constructed from curves with large $N_q$ with respect to their genus $g$.

Given a smooth projective, algebraic curve of genus $g$ over $\mathbb{F}_q$, an upper bound for $N_q$ is a corollary to the celebrated Hasse-Weil Theorem,

$$N_q \leq q+ 1 + 2g\sqrt{q}.$$

Curves attaining this bound are called $\mathbb{F}_q$-maximal. The Hermitian curve is a key example of an $\mathbb{F}_q$-maximal curve, as it is the unique curve, up to isomorphism, attaining the maximum possible genus of an $\mathbb{F}_q$-maximal curve. 

It is a result commonly attributed to Serre that any curve which is $\mathbb{F}_q$-covered by an $\mathbb{F}_q$-maximal curve is still $\mathbb{F}_q$-maximal. In particular, quotient curves of $\mathbb{F}_q$-maximal curves are $\mathbb{F}_q$-maximal. Many examples of $\mathbb{F}_q$-maximal curves have been constructed as quotient curves of the Hermitian curve by choosing a subgroup of its very large automorphism group.

It is a challenging problem to construct maximal curves that cannot be obtained in this way, as well as to construct maximal curves with many automorphisms (in order to use the machinery described above). A natural question arises also: given two maximal curves over the same finite field, how can one decide whether they are isomorphic or not? A way to try to give an answer to this question is to look at the birational invariants of the two curves, that is, their properties that are invariant under isomorphism. 

In this talk, we will describe our main contributions to the theory of maximal curves over finite fields and their applications to coding theory. In relation with the question described before, during the talk, the behaviour of the birational invariant of maximal curves will also be discussed.