Asymptotic Analysis of Variational and Hamiltonian PDEs

图案形成系统的研究是科学研究的一个基本领域，其中现代数学分析工具(动力系统理论、偏微分方程组理论、渐近分析和概率论)可以用于物理系统的建模，特别是在系统行为的关键转变附近。对于本提案中正在研究的项目，PI主要对在临界阈值下将连续的平移对称降低到离散的周期性对称时产生的图案感兴趣，从而产生通常所说的“条纹”图案。例如，这种模式在瑞利-贝纳德对流(RBC)中是常见的，这是许多天气模式形成的关键机制。在RBC中，条纹图案对应于在临界温度下形成具有均匀特征宽度的周期性“对流卷”。这项研究的一个主要目标是不仅研究在临界阈值下形成的模式，而且描述当一个人远离阈值时在这些模式中出现的缺陷的类型。这里研究的图案是静态的，与之一致的是，人们可以通过最小化自由能来研究缺陷的形成，该自由能取决于控制参数，其临界值对应于物理系统的临界阈值。这里所研究的自由能被称为“正则化的克诺威尔能量”，它是用局部定义远离缺陷的条纹图案的位相来表示的。从数学的观点来看，问题是当一个临界参数被逼近时，研究一个变分问题的极小值的极限行为。这种极限的确定是一件微妙的事情，但PI正在开发几何和分析工具，使这一确定变得容易处理。这项建议的第二部分还涉及变分问题的渐近研究。这些问题的一般背景是研究当N变大时N×N厄尔米特矩阵的特征值的统计性质。这些统计量是关于一类概率期望的，这些概率期望在保持本征值的对称下是不变的。这个一般的研究领域通常被称为随机矩阵理论，因为它提供了对包括图论、数论和量子场论在内的广泛领域的见解而变得突出。事实上，PI和他的合作者最近证明了随机矩阵配分函数存在一个大的N展开式，在过去的20年里，它在物理界关于量子引力的讨论中发挥了核心作用。这种分析的关键是研究一个变分问题，该问题刻画了特征值的渐近期望密度和相关的Riemann-Hilbert问题的渐近性。目前的建议将研究这种展开式的精细结构，它可以提供关于计数函数的详细信息，这些计数函数在拓扑非平凡曲面上计数图。这些计数函数将被确定为偏微分方程组的解，偏微分方程组是描述特征值密度如何随着概率矩阵期望变化而变化的常微分方程组的哈密顿(实际上是完全可积)系统的连续极限。PI还将探索随机矩阵划分函数的行为，作为这些期望变化的临界阈值。物理学家推测，这种相变可以用来确定一个可行的候选者，用于计算二维量子引力中的场论期望。这项提案的两个部分都可能与有趣的应用程序相关。图案形成方面的工作将对缺陷的形成做出明确的预测，并在实验室中进行测试。事实上，这项工作中出现的一些预测目前正在由实验者进行测试。PI将应用于波型的稳定性分析和波型动力学的一些初步研究。在未来，人们可能会期望这项工作对动物皮毛图案(包括指纹)中的缺陷结构和植物图案的进化进行建模。随机矩阵理论的方法和思想正在产生广泛的影响；在数论、组合学、随机图理论(或网络)、生长过程和多元统计中，仅列出了几个应用领域。重要的是，随机矩阵理论的基本方法和结果必须以严格的最高标准建立，以便它们能够在这些不同的应用中广泛使用。这是研究提案这一部分的首要和根本目标。

The study of pattern forming systems is a fundamental area of scientific investigation in which the tools of modern mathematical analysis (dynamical systems theory, the theory of partial differential equations, asymptotic analysis and probability theory) can be brought to bear on the modelling of physical systems especially near a critical transition in the behavior of the system. For the projects being studied in this proposal the PI is principally interested in patterns that arise when a continuous translational symmetry is reduced, at a critical threshold, to a discrete periodic symmetry resulting in what is often referred to as a "striped" pattern. Such patterns are for instance generic in Rayleigh-Benard convection (RBC) which is a key mechanism in the formation of many weather patterns. In RBC the striped pattern corresponds to the formation, at a criticaltemperature, of periodic "convection rolls" of a uniform characteristic width. A major goal of this research is to study not just the patterns that form at a critical threshold but to characterize the types of defects that arise in these patterns when one is far from threshold. The patterns studied here are stationary and, consistent with that, one may investigate the formation of defects through the minimization of a free energy which depends on a control parameter with a critical value that corresponds to the critical threshold of the physical system. The free energy studied here is known as the "regularized Cross-Newell energy" and is expressedin terms of a phase that locally defines the striped pattern away from defects. From the mathematical point of view the problem is to study the limiting behavior of minimizers to a variational problem as a critical parameter is approached. The determination of such limits is a delicate matter but the PI is developing geometric and analytical tools that should make this determination tractable.A second part of this proposal also involves the asymptotic study of variational problems. The general context of these problems is the study of the statistics of eigenvalues of N x N Hermitean matrices as N becomes large. These statistics are taken with respect to a class of probabilistic expectations which are invariant under the symmetries that preserve the eigenvalues. This general areaof study is generally referred to as random matrix theorywhich has become prominent because of the insights it provides into a wide range of fields including graph theory, number theory and quantum field theory. Indeed the PIand his collaborators recently extablished the existence of a large N expansion for the random matrix partition function which has played a central role over the last twenty years in discussions within the physics community concerning quantum gravity. The key to this analysis is the study of a variational problem that characterizes the asymptotic expected density of eigenvalues and the asymptotics of an associated Riemann-Hilbert problem. The current proposal will study the fine sturcture of this expansionwhich can provide detailed information about counting functions which enumerate graphs on topologically non-trivial surfaces. These counting functions will be determined as solutions to partial differential equations which are continuum limits of Hamiltonian (in fact completely integrable) sytems of ordinary differential equations which describe how the eigenvalue density changes as the probabilistic matrix expectations are varied. The PI will also explorethe behavior of the random matrix partition function as a critical threshold in the variation of these expectations is approached. Physicists have conjectured that this phase transition can be used to determine a viable candidate for calculating field theoretic expectations in two-dimensional quantum gravity. Both parts of this proposal have potential relevance for interestingapplications. The work on pattern formation will make definite predictions on defect formation that can be tested in the laboratory. Indeed some of the predictions emerging from this work are currently being tested by experimentalists. The PI envisions applications to the stability analysis of patterns and some initial studies of the dynamics of wave patterns. In the future one may expect this work to have relevance for modelling defect structure in animal coat patterns (including fingerprints) and the evolution of plantpatterns.The methods and ideas of random matrix theory are having a broad impact; in number theory, combinatorics, random graph theory (or networks), growth processes and multivariate statistics, to name just a few areas of application. It is important that the underlying methods and results of random matrix theory be established with the highest standards ofrigor so that they can be widelyused in these diverse applications. That is an overarching and fundamental goal of this part of the research proposal.