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.


 
Analysis and Optimization (Math V2500)




Papers

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

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.
 
Bound states for rapidly oscillatory Schrödinger operators in dimension 2, to appear in SIAM Journal on Mathematical Analysis.
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.
 
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.
 
Pollicott-Ruelle resonances via kinetic Brownian motion, Communications in Mathematical Physics 356(2017), No. 2, 357–396.
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.
 
 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

Topologically protected edge states via highly oscillatory potentials. Conference on spectral geometry and semiclassical analysis, Aussois, December 2017.
 
Resonances for stochastic, highly oscilltory potentials. AMS meeting on Spectral theory and Microlocal Analysis, WSU - Pullman, April 2017; PDE seminar;  PDE seminar, CMU.
 
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; PDE seminar, Orsay, December 2017.
 
Scattering resonances for highly oscillatory potentials.Bay Area Microlocal Analysis Seminar, (Stanford University), November 2015; Analysis and Geometry Seminar (Columbia University), November 2016; Courant institute, October 2017; UNC PDE minischool, November 2017. 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.