Randomness, analysis and PDEs : a conference in honour of Nicolas Burq on the occasion of his 60th birthday

9:30 - 10:10 (40min)

Leonardo Tolomeo

Title: A statistical consequence of the soliton resolution conjecture for the focusing nonlinear Schrödinger equation. Abstract: In this talk, I will discuss concentration of measure phenomena for the Gibbs of the focusing, mass-subcritical Schrödinger equation on the real line. I will first discuss how these results connect to the soliton resolution conjecture for NLS, and then explain how to approach this family of problems, combining the variational representation of solitons with more probabilistic techniques. This talk is based on joint works with T. Oh (Edinburgh), M. Okamoto (Osaka), H. Weber (Münster), J. Forlano (Monash), and F. Höfer (Münster).

10:20 - 11:00 (40min)

Fabrice Planchon

Title: 10 ways to prove local smoothing. Abstract: I will revisit our old results (joint with N. Burq) on the 1d Schrödinger equation with a BV coefficient, with emphasis on how to avoid spectral theory (to the extend it is possible to do so!) and why we should. In the process we will greatly simplify the original proofs and discuss possible extensions.

11:00 - 11:30 (30min)

Pause café

11:30 - 12:10 (40min)

Wilhelm Schlag

Title: Radiation Damping and Asymptotic Stability of the Degree-One Vortex in the Abelian Yang-Mills-Higgs Model. Abstract: The abelian Yang-Mills-Higgs equations in two space dimensions admit topological vortex solutions. I will discuss the asymptotic stability of the degree-one vortex at self-dual coupling, for small corotational perturbations in Stuart’s gauge.Joint work with Jose Palacios, Fabio Pusateri, Jonas Lührmann, and Sohrab Shahshahani.

12:10 - 14:00 (1h50)

Déjeuner

15:00 - 15:40 (40min)

Maciej Zworski

Title: WKB structure in a scalar model of flat bands Abstract: The scalar model of flat bands is a simplification of models in condensed matter physics. It allows the study of relevant spectral problems using a 2nd order scalar equation, akin to the Schroedinger equation with the square of dbar on a torus replacing the Laplacian. It displays many features of original models such as the ``quantisation" of the reciprocals of magic angles at which flat bands appear. The space of solutions can be described using a rank 2 holomorphic vector bundle over the torus and its properties as alpha varies are related to the structure of bands leading to a trichotomy: tangential touching (most of alphas), Dirac points (discrete set of alphas) and flat bands (discrete set). (Mengxuan Yang and Bryan Li observed that the same argument works in a more physically realistic setting of twisted two-layered wafers of graphene.) In my talk I will describe the basic properties of the scalar model and of the ge

15:50 - 16:30 (40min)

Chenmin Sun

Title: Solving NLS on the three-dimensional Ball with radial, supercritical random data. Abstract: We consider the cubic nonlinear Schrödinger equation on the three-dimensional ball with radial random initial data in a supercritical regime. We construct probabilistic strong solutions, substantially improving a previous result of Bourgain and Bulut. The criticality arises from a divergent contribution that, unlike for the NLS on the torus, cannot be removed by a global transformation. We overcome this obstruction by introducing a refined gauge transform which leaves the equation unchanged. I will then explain how this gauge transform can be incorporated into the random averaging operator ansatz to solve the problem. This is joint work with Nicolas Burq, Nicolas Camps, and Nikolay Tzvetkov.

16:30 - 17:00 (30min)

Pause café

17:00 - 17:40 (40min)

Massimiliano Berti  

Title: Long time dynamics of space periodic water waves. Abstract: The goal of this talk is to review recent advances regarding the long-time dynamics of space-periodic water waves, focusing on 1) bifurcation of quasi-periodic solutions, both standing and traveling; 2) long-time well-posedness results; 3) modulational instability of Stokes waves. These results rely on unconventional approaches to KAM and Birkhoff normal form theories for Hamiltonian quasi-linear PDEs and symplectic Kato perturbation theory for separated eigenvalues of reversible and Hamiltonian operators.

Programme