**Introduction to Wave Turbulence theory: basic concepts and main results.**

Sergey Nazarenko

*Université de Cote d'Azur*

Abstract: This course is an introduction to the Wave Turbulence theory which will consist of the following themes.

1. General concepts in Turbulence and their manifestations in Wave Turbulence. Richardson cascade. Kolmogorov spectrum. Fjortoft's dual cascade argument.

2. Basic results via a dimensional analysis. Resonant wave interactions. Direct and inverse cascade Kolmogorov-Zakharov spectra.

3. Formulation of the statistical setup: Random Phase and Amplitude fields.

4. Weak nonlinearity (Duhamel) expansion. Statistical closure: Wave-Kinetic equation. Conservation laws.

5. Derivation of Kolmogorov-Zakharov spectra via Zakharov-Kraichnan transformation. Energy and wave action fluxes. Locality of interaction.

6. Evolving Wave Turbulence, self-similar solutions of the first and the second kinds. Spectrum blowup and Bose-Einstein condensation.

7. Time permitting: differential appoximation models, discussion of physical examples.

**Mathematical Properties of the the Wave Turbulence Equation**

Juan Velázquez

*HCM, Bonn*

Abstract: In this course, I will describe some rigorous mathematical results available for the dynamics of the Wave Turbulence Equations. Some particular issues that will be discussed are the following ones. Power law solutions describing fluxes of mass and energy between different regions of the phase space. The onset of singularities in finite time. Solutions beyond the blow-up time: Weak solutions. The analogies between the Wave Turbulence kinetic equation and the classical Smoluchowski coagulation equation will be discussed. Some results concerning the dynamic behaviour of the coagulation equations, including the onset of time dependent oscillatory solutions, the properties of some multicomponent coagulation equations and the existence of flux solutions which are not power laws.

**The mathematical theory of wave turbulence**

Zaher Hani

*University of Michigan*

Abstract: In this mini course, we will survey some recent advances that were made in the past couple of years on building a mathematically rigorous theory of wave turbulence. More precisely, we will be discussing the rigorous derivation of the fundamental equations of wave kinetic theory starting from the microscopic dynamics given by a nonlinear dispersive PDE. This is Hilbert’s sixth problem in the context of nonlinear waves. The bulk of the discussion will focus on a series of recent works with Yu Deng (USC) in which we give the full derivation of the wave kinetic equation starting from the nonlinear Schrodinger equation. The key ingredient is a thorough analysis of the diagrammatic expansion that allows to: a) uncover elaborate cancellations within it, and b) overcome difficulties coming from factorial divergences in the expansion and the criticality of the problem. We shall discuss some of the combinatorial and analytic techniques involved in such an endeavor.