Nicolas Camps

I'm a graduate student at Orsay

About

Introduction.

I am a third year graduate student under the supervision of Nicolas Burq and Frédéric Rousset. I'm interested in the asymptotic dynamics of solutions to nonlinear dispersive PDE's, such as Schrödinger equations, with random initial data in supercritical regimes.

Keywords.

Nonlinear Schrödinger equation, Probabilistic Cauchy theory, Distroted Fourier transform, Asymptotic stability, Scattering, Semiclassical analysis, I-method, Harmonic analysis, Schrödinger equation on compact manifolds.

Research Interests.

Schrödinger with a potential

In the presence of a short-range potential, the nonlinear Schrödinger equation may have a center-manifold made of ground states solutions. Following the work of A. Soffer and M.I. Weinstein, one can prove asymptotic stability for small energy data of these coherent states. I investigated the asymptotic dynamics of a probabilistic flow below the energy space, and proved that asymptotic stability holds almost-surely.

Scattering below the energy space

Over the last decades, modified energy strategies and the so-called I-method were widely used and refined in an effort to settle large-time dynamics questions for solutions to dispersive equtions with data below the energy space. Folowing a series of works on the almost sure global well-posedness theory, I make use of these recent developements in a stochastic setting to prove alsmot-sure scattering for a probabilistic flow, in a supercritical regime far below the enregy space.

Schrödinger on the sphere

I am interested in the adaptation to the sphere of the pionner works of Bourgain on the torus, who investigated the Schrödinger equation on the support of the Gibbs measure thanks to the stochastic nonlinear smoothing gain for the Duhamel term. However, the cubic Shrödinger equation is quasilinear at low regularity, and it turns out that the adaptation of the work of Bourgain needs new insights.

Publications and Preprints

Publications

2020
  • Asymptotic stability of small ground states for NLS under random perturbations, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, to appear.

    Arxiv: https://arxiv.org/abs/2012.05878

Preprints

2021
  • Scattering for the cubic Shrödinger equation in 3D with randomized initial data

    (in preparation)

Curriculum Vitae

Education

Ph.D

Laboratoire de Mathématiques d'Orsay

2019 - Present

Dynamical behavior of dispersive PDE’s: from explicit solutions to generic dynamics

Under the supervision of Nicolas Burq & Frédéric Rousset

Magistère of Mathematics

ENS Paris-Saclay & University Paris-Saclay

2015 - 2019
  • 2018-2019: Master "AMS" in Mathematics, Orsay (with high honours),
  • 2017-2018: Agrégation de Mathématiques, (national french teaching competitive exam) Rank: 11th,
  • Second concours de l’ENS (Ranks : 8th at ENS Paris-Saclay, 7th at ENS Rennes),
  • Bachelor in Mathematics, Orsay (with high honours, Laureate of Hadamard grant)

Classes préparatoires

2012 - 2015
  • MPSI/MP*, Lycée Joffre (Montpellier)

Teaching Experience

Assistant at Laboratoire de Mathématiques d'Orsay
  • Differential Calculus and Geometry, 2020-2021,Magistère 1st year,
  • Differential Calculus and Topology, 2020-2022,Bachelor 3rd year,
  • Linear Algebra,2020-2021,Bachelor 2nd year,
Assistant at IUT d’Orsay, Université Paris Saclay
  • Calculus, 2019-2020, Bachelor 1st year.

Other

Responsibilities

July 2021
  • Organization commitee of the Conference in honnor of Patrick Gerard's 60th anniversary , Orsay
2019-2021
  • co-organizer of the seminar for gradute students in Analysis, Orsay

Semesters

September - December 2021
  • Invited to the Semester "Hamiltonian technics in dispersive PDE's", ICERM

Conferences

Given talks

Seminars
  • Séminaire des doctorants en EDP, Orsay, Octobre 2020
  • Séminaire de vulgarisation des doctorants, Orsay, April 2021
  • Séminaire de AN-EDP, Orsay, June 2021
Conferences
  • Conference on Schrödinger equation, Le Croisic, May 2021

Attended conferences and worshops

2021
  • Colloque Analyse Harmonique et EDP, Evry, France, July .
  • Conference in honnor of P. Gérard 60th anniversary, Orsay, France, July .
  • Journées EDP, Obernai, France, June .
2020

January

  • 11th Itinerant Workshop in PDEs, Haussdorff Center for Mathematics, Bonn, Germany.

Contact

Location:

Université Paris-Sud, 91405, Orsay Cedex, France

Office:

Batiment 307, office 2A1.