Simple models in Mathematical Physics: Random matrices and NLS

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The goal of this project is to further the mathematical understanding of two simple physical models: the nonlinear Schrodinger equation and the classical Coulomb gas associated with random matrices. The central theme in our investigations of the Coulomb gas is to develop the theory for arbitrary values of the inverse temperature, as opposed to the three special values, namely, 1, 2, and 4. In particular, we wish to show that there is no phase transition. A second goal is to better understand the empirically observed occurrence of random matrix statistics in manifestly deterministic scenarios, by first considering models with very little randomness. The second part of the project concerns well-posedness questions for the nonlinear Schrodinger equations at critical regularity. Our primary goal is to complete the picture in those cases where the critical regularity corresponds to a coercive conservation law, specifically, the energy- and mass-critical equations. We will further attempt to make some initial inroads into the case of non-conserved critical regularity in a particular case where this seems most feasible. The theory of random matrices is driven by empirical data as surely as any physical science: Its roots lie in statistical analysis (specifically, ANOVA) and the analysis of experimental energy-level data. More recently, random matrix statistics have been observed in the vibrations of drums and the behaviour of the prime numbers. The researches of this project are aimed at helping to elucidate and explain the results of these truly mathematical experiments. Although the nonlinear Schrodinger equation is used as a simple effective model for several physical phenomena, the goal of this project is to further our understanding of the long-time behaviour of solutions to this equation principally as a model for general dispersive evolution equations. By choosing to work with (scaling-)critical initial data, we place ourselves precisely at the border between the well-studied subcritical regime and the terra incognita of supercritical equations. As an example of how developments of such simplified equations has fed into more physical models, we note one past off-shoot of the theory of dispersive equations at critical regularity: we now know that the well-posedness of the equations of fluid flow (as opposed to the spontaneous generation of extremely turbulent behaviour) is something that can be rigorously verified by computer for any prescribed size of initial data.