Train Einstein

Alexis Drouot
alexis dot drouot at gmail dot com

Welcome on my website!
I am now a postdoctoral fellow at Columbia University. CV.


Resonances for random highly oscillatory potentials, preprint.
I study the behavior of resonances of  a random potential obtained as a sum of a deterministic part with a hihgly oscillatory, random part, whose typical scale of variation if 1/N, N large. Using pertubation analysis, I show almost sure convergence of the resonances as N goes to infinity. I identify a stochastic and a deterministic regime for the speed of convergence, depending whether the large deviations effects take over the constructive interference

Bound states for rapidly oscillatory Schrödinger operators in dimension 2, preprint.
I show that the Schrödinger operator Δ+VΔ+V-∆+V on the plane, with V oscillating at frequency ε, admits a single eigenvalue that is expoentially close to the edge of the continuous spectrum.
Existence and non-existence of extremizers for k-plane transform inequalities, preprint.
I study certain sharp k-plane transform inequalities on the sphere and on the hyperbolic space. The best constant is computed. For the case of the sphere I characterize all extremizers, while for the hyperbolic space I prove that no extremizers exist.
Pollicott-Ruelle resonances via kinetic Brownian motion, to appear in Communications in Mathematical Physics.
The kinetic Brownian motion on the cosphere bundle of a Riemannian manifold M is a stochastic process that models the geodesic equation perturbed by a random white force of size ε. When M is compact negatively I show that the spectrum of the infinitesimal generator of this process converges to the Pollicott-Ruelle resonances of M as ε→0.
Scattering resonances for highly oscillatory potentials, to appear in Annales Scentifiques de l'Ecole Normale Supérieure.
I study potentials that are perturbations of a slowly varying term by a term oscillating at frequency ε, in odd dimension.  If the slowly varying part vanishes I prove that the scattering resonances escape every compact sets as ε tends to 0 - apart from a special case in dimension 1. If the slowly varying part does not vanish I prove that the resonances of the perturbed potential converge to the resonances of the unperturbed one, and give an expansion of these resonances in powers of ε. I also derive a simple effective potential whose scattering properties are similar to the oscillatory one.
A quantitative version of Hawking radiation, Annales Henri Poincaré 18(2017), No. 3, 757–806.
I show that the gravitational collapse of stars generates the emergence of a quantum thermal state from an initial vacuum state, whose temperature is given by the Hawking temperature. The convergence to the equilibrium is exponential. The proof uses conformal scattering theory; a semiclassical descrption of the blueshift effect; and the construction of a WKB parametrix near the surface of the star.
Quantitative form of certain k-plane transform inequalities, Journal of Functional Analysis 268(2015), No. 5, 1241–1276.
I extend the restricted precompactness result stated in a previous work to the radial case. Mixing it with a spectral problem I show the following inverse problem: if a radial function has a large k-plane transform then it must be close to an extremizer.
Best constant for a k-plane transform inequality, Analysis and PDE 7(2014), No. 6, 1237–1252.
I study the sharp form of certain inequalities satisfied by the k-plane transform. I prove a restricted precompactness result, that implies the existence of extremizers. I identify an extremizer and derive the value of the best constant.
A quantitative version of the Catlin-D'Angelo-Quillen theorem (with Maciej Zworski), Analysis and Mathematical Physics 3(2013), No. 1, 1–19.
We give a quantitative proof of the Catlin-D'Angelo-Quillen theorem, that states that a bihomogeneous complex form can be written as a sum of square of polynomials, when multiplied by a certain power of the complex norm. Our semiclassical analysis-based proof leads to quantitative upper bounds on this power. 

Selected talks

Resonances for stochastic, highly oscilltory potentials. AMS meeting on Spectral theory and Microlocal Analysis, WSU - Pullman, April 2017.
Pollicott-Ruelle resonances via kinetic Brownian motion. Analysis and Geometry Seminar (Nice), June 2016; PDE/Analysis seminar (MIT), September 2016; Workshop on Mathematical Physics (Tokyo), January 2017; Analysis and PDE seminar (UCLA), January 2017; Conference on geometrical aspects of resonances (Marseille), March 2017.
Scattering resonances for highly oscillatory potentials.Bay Area Microlocal Analysis Seminar, (Stanford University), November 2015; Analysis and Geometry Seminar (Columbia University), November 2016. A more technical version is available here.
Existence and non-existence of extremizers for a k-plane transform inequality. Conference in honor of Michael Christ, UW - Madison, May 2016.
The principle of concentration-compactness and an application. Bonn Summer School on Sharp Inequalities in Analysis, September 2015.