Abstract: Wave Turbulence Theory is widely used across several areas of physics to describe the statistical behavior of dispersive waves. It was formulated throughout the twentieth century by Peierls, Hasselman and Zakharov. Central to this theory is the derivation of the Wave Kinetic Equation which describes energy cascades between waves. In recent years a rigorous mathematical proof of the derivation of the Wave Kinetic Equation and its range of applicability started to emerge, a feat that has taken a long time since the theory was first proposed. In this expository talk I will be discussing rigorous results and mathematical techniques that lead to proving the validity of the kinetic description of waves. I will also discuss the range of parameters where the kinetic description is valid.

Jalal M. I. Shatah is a Silver Professor of Mathematics at Courant, New York University. His research focuses on the problem of global existence and asymptotic behavior of small solutions to nonlinear hyperbolic and dispersive equations.

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This lecture was part of the bi-annual Abel Symposium.

This year the title of the symposium was Partial Differential Equations waves, Nonlinearities and Nonlocalities.

The symposium was funded by
- The Norwegian Academy of Sciences and Letters via the Abel board and The Norwegian Mathematical Society
- NTNU Norwegian University of Science and Technology
- Research Council of Norway via the grant Waves and Nonlinear Phenomena
- Trond Mohn Foundation