About fifteen years ago, Patrick Gérard and I introduced the cubic Szegö equation$$\begin{aligned}i \partial_{t} u & =\Pi\left(|u|^{2} u\right), \quad u=u(x, t), \quad x \in \mathbb{T}, t \in \mathbb{R} \\u(x, 0) & =u_{0}(x) .\end{aligned}$$Here $\Pi$ denotes the Szegö projector which maps $L^{2}(\mathbb{T})$-functions into the Hardy space of $L^{2}(\mathbb{T})$-traces of holomorphic functions in the unit disc. It turned out that the dynamics of this equation were unexpected. This motivated us to try to understand whether the cubic Szegö equation is an isolated phenomenon or not. This talk is part of this project.
We consider a family of perturbations of the cubic Szegö equation and look for their traveling waves. Let us recall that traveling waves are particular solutions of the form$$u(x, t)=\mathrm{e}^{-i \omega t} u_{0}\left(\mathrm{e}^{-i c t} x\right), \quad \omega, c \in \mathbb{R}$$We will explain how the spectral analysis of some operators allows to characterize them.
From joint works with Patrick Gérard.[-]

The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.[-]

Liouville conformal field theory (LCFT) was introduced by Polyakov in 1981 as an essential ingredient in his path integral construction of string theory. Since then Liouville theory has appeared in a wide variety of contexts ranging from random conformal geometry to 4d Yang-Mills theory with supersymmetry.
Recently, a probabilistic construction of LCFT on general Riemann surfaces was provided using the 2d Gaussian Free Field. This construction can be seen as a rigorous construction of the 2d path integral introduced in Polyakov's 1981 work. In contrast to this construction, modern conformal field theory is based on representation theory and the so-called bootstrap procedure (based on recursive techniques) introduced in 1984 by Belavin-Polyakov-Zamolodchikov. In particular, a bootstrap construction for LCFT has been proposed in the mid 90's by Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ) on the sphere. The aim of this talk is to review a recent series of work which shows the equivalence between the probabilistic construction and the bootstrap construction of LCFT on general Riemann surfaces. In particular, the equivalence is based on showing that LCFT satisfies a set of natural geometric axioms known as Segal's axioms.
Based on joint works with F. David, C. Guillarmou, A. Kupiainen, R. Rhodes.[-]

We consider the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We prove global wellposedness and give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. Our analysis provides explicit formulae for the multi-soliton component as well as the correction dispersive term. We use the inverse scattering approach and the nonlinear steepest descent method of Deift and Zhou (1993) revisited by the $\bar{\partial}$-analysis of Dieng-McLaughlin (2008) and complemented by the recent work of Borghese-Jenkins-McLaughlin (2016) on soliton resolution for the focusing nonlinear Schrödinger equation. This is a joint work with R. Jenkins, J. Liu and P. Perry.[-]