**See the Google Agenda of the seminar, and add the iCal url to your calendar.**

## Xavier Blanc (LJLL et Matherials)

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

Schémas préservant l’asymptotique sur maillages coniques

On considère un système hyperbolique linéaire appelé équation

de la chaleur hyperbolique. Ce système est une approxlmation très

utilisée de l’équation du transfert radiatif, en particulier dans les

expériences de fusion par confinement inertiel (laser méga-joule). Dans

la limite d’une forte interaction du rayonnement avec la matière, cette

équation dégénère vers une équation de diffusion. Nous présenterons des

schémas volumes finis qui respectent cette limite, d’abord sur des

maillages polygonaux, puis sur des maillages dont les arêtes sont

définies par des coniques. Il s’agit d’un travail en commun avec V.

Delmas et P. Hoch (CEA).

## Pierre Bellec (Rugers University)

*Thursday, October 17th, 15h — Salle de séminaire du CERMICS
*

First order expansion of convex regularized estimators

We consider first order expansions of convex penalized estimators in

high-dimensional regression problems with random designs. Our setting includes

linear regression and logistic regression as special cases. For a given

penalty function $h$ and the corresponding penalized estimator $\hat\beta$, we

construct a quantity $\eta$, the first order expansion of $\hat\beta$, such that

the distance between $\hbeta$ and $\eta$ is an order of magnitude smaller than

the estimation error $\|\hat{\beta} – \beta^*\|$. In this sense, the first

order expansion $\eta$ can be thought of as a generalization of influence

functions from the mathematical statistics literature to regularized estimators

in high-dimensions. Such first order expansion implies that the risk of

$\hat{\beta}$ is asymptotically the same as the risk of $\eta$ which leads to a

precise characterization of the MSE of $\hat\beta$; this characterization takes a

particularly simple form for isotropic design. Such first order expansion also

leads to confidence intervals based on $\hat{\beta}$. We provide sufficient

conditions for the existence of such first order expansion for three

regularizers: the Lasso in its constrained form, the lasso in its penalized

form, and the Group-Lasso. The results apply to general loss functions under

some conditions and those conditions are satisfied for the squared loss in

linear regression and for the logistic loss in the logistic model.

Joint work with Arun K Kuchibhotla (UPenn)

## 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.

## Sonia Fliss (ENSTA)

*Thursday, November 7th, 10h — Salle de séminaire du CERMICS
*

TBA

## David Herzog (Iowa State University)

*Thursday, November 21st, 10h — Salle de séminaire du CERMICS
*

TBA

## Denis Talay (Inria Sophia-Antipolis)

*Wednesday, November 27th, 14h30 — Salle F106
*

TBA

## 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.