Simple models in Mathematical Physics: Random matrices and NLS

这个项目的目标是加深对两个简单物理模型的数学理解：非线性薛定谔方程和与随机矩阵相关的经典库仑气体。我们研究库仑气体的中心主题是发展逆温度的任意值的理论，而不是三个特殊的值，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随机性很小的模型，更好地理解在明显确定的情况下根据经验观察到的随机矩阵统计的发生。项目的第二部分讨论了非线性薛定谔方程在临界正则性下的适定性问题。我们的主要目标是在临界规律性对应于强制守恒定律，特别是能量和质量临界方程的情况下完成这幅图。我们将进一步尝试在非守恒临界规律性的情况下，在看起来最可行的特定情况下取得一些初步进展。与任何物理科学一样，随机矩阵理论是由经验数据驱动的：它的根源在于统计分析(具体地说，方差分析)和对实验能级数据的分析。最近，在鼓的振动和质数的行为中观察到了随机矩阵统计。这个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个物理现象的简单有效的模型，但本项目的目的是加深我们对该方程解的长期行为的理解，主要是作为一般色散发展方程的模型。通过选择使用(标度)关键的初始数据，我们精确地将自己置于研究得很好的亚临界区域和隐蔽的超临界方程之间。作为这种简化方程如何发展到更多物理模型中的一个例子，我们注意到过去色散方程理论在临界规则性下的一个分支：我们现在知道，流体流动方程的适定性(相对于极端湍流行为的自发产生)是可以由计算机对任何规定的初始数据大小进行严格验证的。

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.