Applied Mathematics Seminar

Vincent Lepetit (Imagine)

Thursday, October 3rd, 10h — Salle de séminaire du CERMICS

3D Scene Understanding from a Single Image

I will present several works we very recently developed for 3D scene understanding. The first work is a method for 3D object recognition and pose estimation based on a feedback loop inspired by biological mechanisms, and providing very accurate and reliable results. The second work is a method for understanding the 3D layout (walls, floor, ceiling, ..) of an indoor environment from a single image despite possible occlusions by furnitures. I will then discuss the challenges in creating training and evaluation data for 3D registration problems, and present the direction we are currently exploring.

Xavier Blanc (LJLL et Matherials)

Thursday, October 17th, 10h — Salle de séminaire du CERMICS


Horia Cornean (Aalborg)

Tuesday, October 29, 10h — Salle de séminaire du CERMICS

Parseval frames of exponentially localized magnetic Wannier functions

Abstract: Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension d \le 3, we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model m occupied energy bands by a real-analytic and {{\mathbb {Z}}}^{d}-periodic family \left\{ P(\mathbf{k}) \right\} _{\mathbf{k}\in {{\mathbb {R}}}^{d}} of orthogonal projections of rank m. A moving orthonormal basis of {{\,\mathrm{Ran}\,}}P(\mathbf{k}) consisting of real-analytic and {{\mathbb {Z}}}^d-periodic Bloch vectors can be constructed if and only if the first Chern number(s) of P vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we show how to algorithmically construct a collection of m-1orthonormal, real-analytic, and {{\mathbb {Z}}}^d-periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of m+1 real-analytic and {{\mathbb {Z}}}^d-periodic Bloch vectors which generate {{\,\mathrm{Ran}\,}}P(\mathbf{k}). Both algorithms are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. A moving Parseval frame of analytic, {{\mathbb {Z}}}^d-periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting. Our results are illustrated in crystalline insulators modelled by 2d discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schrödinger operators as well.
This is joint work with Domenico Monaco and Massimo Moscolari.

Denis Talay (Inria Sophia-Antipolis)

Wednesday, November 27th, 14h30 — Salle F106


Laure Dumaz (CEREMADE)

Thursday, December 12th, 10h — Salle de séminaire du CERMICS

Localization of the continuous Anderson hamiltonian in 1-d and its transition

We consider the continuous Schrödinger operator – d^2/d^x^2 + B’(x) on the interval [0,L] where the potential B’ is a white noise. We study the spectrum of this operator in the large L limit. We show the convergence of the smallest eigenvalues as well as the eigenvalues in the bulk towards a Poisson point process, and the localization of the associated eigenvectors in a precise sense. We also find that the transition towards delocalization holds for large eigenvalues of order L, where the limiting law of the point process corresponds to Sch_tau, a process introduced by Kritchevski, Valko and Virag for discrete Schrodinger operators. In this case, the eigenvectors behave like the exponential Brownian motion plus a drift, which proves a conjecture of Rifkind Virag. Joint works with Cyril Labbé.

Archive of past seminars: here

Organizers: Antoine Levitt, Julien Reygner.