Simple models in Mathematical Physics: Random matrices and NLS

这个项目的目标是进一步的数学理解两个简单的物理模型：非线性薛定谔方程和经典的库仑气体与随机矩阵相关。我们研究库仑气体的中心主题是发展逆温度任意值的理论，而不是三个特殊值，即1、2和4。特别是，我们希望证明没有相变。第二个目标是通过首先考虑具有很少随机性的模型，更好地理解经验观察到的随机矩阵统计在明显确定性场景中的出现。项目的第二部分涉及临界正则非线性薛定谔方程的适定性问题。我们的主要目标是在临界规律与强制守恒定律相对应的情况下，特别是能量和质量临界方程的情况下，完成图像。我们将进一步尝试在非保守临界正则性的情况下，在一个特殊的情况下，这似乎是最可行的初步进展。随机矩阵理论和任何物理科学一样，都是由经验数据驱动的：它的根源在于统计分析（特别是方差分析）和实验能级数据的分析。最近，随机矩阵统计在鼓的振动和素数的行为中被观察到。该项目的研究旨在帮助阐明和解释这些真正的数学实验的结果。虽然非线性薛定谔方程被用作几种物理现象的简单有效模型，但本项目的目标是进一步理解该方程解的长期行为，主要是作为一般色散演化方程的模型。通过选择（缩放）临界初始数据，我们将自己精确地置于研究充分的亚临界状态和超临界方程的未知领域之间的边界。作为这种简化方程的发展如何被用于更多的物理模型的一个例子，我们注意到在临界规则下色散方程理论的一个过去的分支：我们现在知道流体流动方程的适定性（与极端湍流行为的自发产生相反）是可以用计算机严格验证任何规定大小的初始数据的。

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.