Haitian Yue (he/him)
Brief introduction to the history of local wellposedness theory of nonlinear Schrödinger equation (NLS).
Background and motivation of the random initial data theory for NLS and probabilistic scaling heuristic for NLS.
Probabilistic tools (large deviation properties) and number theory tools (integer lattice counting).
Bourgain's recentering method for 2D cubic NLS [A2] and some furthur discussions on the following developments [A4, A5].
Project A.1: Prove (even improve) the almost surely local wellposedness for the toy model for 2D cubic NLS in [A2].
Project A.2: Recover RAO ansatz for toy model (e.g. resonant system) and prove some key estimates in [4].
[A1] J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), 1–26.
[A2] J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), 421–445.
[A3] N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173(3):449–475, 2008.
[A4] Y. Deng, A. R. Nahmod, and H. Yue, Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two, Ann. of Math. (2), 200(2):399–486, 2024.
[A5] Y. Deng, A. R. Nahmod, and H. Yue, Random tensors, propagation of randomness, and nonlinear dispersive equations, Invent. Math., 228(2):539–686, 2022.
Nicolas Camps (he/him)
The statistical approach in the study of nonlinear waves: motivations.
Invariant Gibbs measure.
Bourgain's globalization argument.
Some words about quasi-invariance properties of out-of-equilibrium Gaussian measures.
Project B.1: First applications to the existence of a Gibbs measure in the context of the cubic Schrödinger equation. [B1]
Project B.2: Transport of Gaussian measures by the flow of a nonlinear Schrödinger equation with enhanced dispersion. [B2, B3]
[B1] N. Burq, L. Thomann, N. Tzvetkov, Remarks on the Gibbs Measures for Nonlinear Dispersive Equations, Ann. Fac. Sci. Toulouse, Math. (6), (2018), 527-597.
[B2] T. Oh, P. Sosoe, N. Tzvetkov, An Optimal Regularity Result on the Quasi-Invariant Gaussian Measures for the Cubic Fourth-Order Nonlinear Schrödinger Equation, J. Ec. Polytech., Math. 5 (2018), 793-841.
[B3] T. Oh, N. Tzvetkov, Quasi-Invariant Gaussian Measures for the Cubic Fourth-Order Nonlinear Schrödinger Equation, Probab. Theory Relat. Fields 169 (2017), 1121-1168.
Ricardo Grande
Overview of statistical mechanics for nonlinear waves. Formal derivation of Wave Kinetic Equations from dispersive PDEs.
Scaling laws, resonance vs quasi-resonances and counting lemmas.
Probability toolbox: Gaussian Hilbert Spaces and Gaussian Hypercontractivity.
Picard iteration scheme. Sharp estimates and combinatorics for the first Picard iterate.
Project C.1: Convergence to the WKE: rigorous justification of the limit using tools from harmonic analysis and exponential sums.
Project C.2: High-probability estimates on Picard iterates: trees, decorations and counting algorithms.
[C1] T. Buckmaster, P. Germain, Z. Hani and J. Shatah, Onset of the wave turbulence description of the longtime behavior of the nonlinear Schroedinger equation, Invent. Math., 225:787–855, 2021.
[C2] Y. Deng, Z. Hani, On the derivation of the wave kinetic equation for NLS, Forum Math. Pi 9 (2021), e6, 1–37.
[C3] Y. Deng, Z. Hani, Full derivation of the wave kinetic equation, Invent. Math., 233:543–724, 2023.