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# La semaine de l’IMB

Le 2 mars 2021
à 10:00
Séminaire de Théorie Algorithmique des Nombres
*Online*
Jade Nardi (Inria Saclay, LIX)
**Explicit construction and parameters of projective toric codes**

Le 2 mars 2021
à 11:15
Séminaire de Physique Mathématique - EDP
*Visio*
Badreddine Benhellal
**Quantum Confinement induced by Dirac operators with anomalous magnetic $delta$-shell interactions.**
Abstract: Let $\Omega$ be a bounded domain and $\upsilon\in\mathbb{R}$. I will consider the coupling $\mathcal{H}_{\upsilon}=\mathcal{H}+ V_\upsilon$, where $\mathcal{H}$ is the free Dirac operator in $\mathbb{R}^3$ and $V_\upsilon= i\upsilon\beta(\alpha\cdot \mathit{N})\delta_{\partial\Omega}$ is the anomalous magnetic $\delta$-interactions potential. In the first instance, assuming that $\upsilon^2 eq 4$ and under some regularity assumption on the domain $\Omega$, we prove that $\mathcal{H}_{\upsilon}$ is self-adjoint and its domain is included in the Sobolev space $\mathit{H}^{1}(\mathbb{R}^3\setminus \partial\Omega)^4$. Moreover, a Krein-type resolvent formula and a Birman-Schwinger principle are obtained, and several qualitative spectral properties of $\mathcal{H}_{\upsilon}$ are given. Finally, we study the self-adjoint realization of $\mathcal{H}_{\upsilon}$ in the case $\upsilon^2=4$. In particular, if $\Omega$ is $\mathit{C}^1$-smooth, we then show that $\mathcal{H}_{\upsilon}$ is essentially self-adjoint and the domain of the closure is not included in any Sobolev space $\mathit{H}^{s}(\mathbb{R}^3\setminus \partial\Omega)^4$, for all $s>0$. In addition, we show that $\overline{\mathcal{H}_{\pm2}}$ generates confinement and prove the existence of embedded eigenvalues on the essential spectrum of $\overline{\mathcal{H}_{\pm2}}$.

Le 4 mars 2021
à 11:00
Séminaire Images Optimisation et Probabilités
*Salle de Conférences*
François-Pierre Paty (CREST, ENSAE Paris)
**Regularizing Optimal Transport through Regularity Constraints**
Optimal transport (OT) suffers from the curse of dimensionality. Therefore, OT can only be used in machine learning if it is substantially regularized. In this talk, I will present a new regularization of OT which leverages the regularity of the Brenier map. Instead of considering regularity as a property that can be proved under suitable assumptions, we consider regularity as a condition that must be enforced when estimating OT. From a statistical point of view, this defines new estimators of the OT map and 2-Wasserstein distance between arbitrary measures. From an algorithmic point of view, this leads to an infinite-dimensional optimization problem, which, when dealing with discrete measures, can be rewritten as a finite-dimensional separately-convex problem. I will finish by sharing some recent ideas on how to speed up the algorithms. The talk is based on some joint work with Marco Cuturi and Alexandre d'Aspremont.

Le 5 mars 2021
à 10:00
Soutenance de thèse
*Salle de Conférences*
Gaël GUILLOT
**Sujet :'Méthodes d'agrégation et désagrégation de programmes linéaires en nombres entiers'. Directeur de thèse : François Clautiaux, co-directeur : Boris Detienne.**

Le 5 mars 2021
à 16:00
Séminaire de Théorie des Nombres
*Visio*
Türkü Özlüm Çelik (Simon Fraser University, Vancouver)
**KP equation in Symbolic, Numerical and Combinatorial Algebraic Geometry**
The Kadomtsev-Petviashvili (KP) equation is a partial differential equation that describes nonlinear wave moves. It is known that algebro-geometric approaches to the KP equation provide solutions coming from a complex algebraic curve, in terms of the Riemann theta function associated with the curve. Reviewing this relation, I will introduce an algebraic object and discuss its geometric features: the so-called Dubrovin threefold of a complex algebraic curve, which parametrizes the solutions. Mentioning the relation of this threefold with the classical algebraic geometry problem, namely the Schottky problem, I will report a procedure that isvia the threefold and based on numerical algebraic geometric tools, which can be used to deal with the Schottky problem from the lens of computations. I will finally focus on the geometric behaviour of the threefold when the underlying curve degenerates. This is joint work with Daniele Agostini and Bernd Sturmfels.

**La semaine de l’IMB recense l’ensemble des événements de la semaine en cours**

Toric codes, introduced by Hansen in 2002, generalize (weighted) Reed-Muller codes on other toric varieties than projective spaces. They consist of evaluation codes of monomials at tuples of non-zero coordinates, which correspond to the points on the dense torus contained in the associated toric variety. Our aim is to ‘projectivise’ these codes, in the same spirit that turns a Reed-Muller codes into a projective one: we consider codes obtained by evaluating global sections on the whole set of the rational points of a toric variety. We focus on simplicial toric varieties, which come with a nice quotient description, and we give an explicit construction of projective codes on them, as well as a combinatorial way to determine their parameters. ‘Projectivizing’ toric codes opens new possibilities of getting codes with excellent parameters, by extending some champion classical toric codes geometrically.