Wave turbulence, the longtime statistical behavior of a sea of weakly nonlinear dispersive waves, unlike the case of hydrodynamic turbulence, has, in many circumstances, a natural asymptotic closure. The closure gives a closed (kinetic) equation for the energy (or number) density clearly revealing the resonance mechanism by which waves of one wavelength and direction transfer energy to others throughout the spectrum. Equations for the higher order cumulants are all linear and satisfied by a frequency renormalization. The kinetic equation has solutions corresponding the equipartition and finite fluxes of all conserved densities. So, the problem is solved, right? The goals of the lecture are twofold: To explain why and how the closure obtains: To discuss the many open challenges that remain and suggest possible solutions.­­

Alan C. Newell, Regents' Professor of Mathematics at the University of Arizona, received degrees from Trinity College, Dublin in 1962 and from MIT in 1966. He has chaired departments of Mathematics at Clarkson University, at the University of Arizona and at the University of Warwick, England, from 1971 until 2000, a period of almost thirty years. His research areas of interest are nonlinear waves and solutions, optics, wave turbulence, plasmas and fluids, coherent structures, cooperative phenomena, and numerical analysis.