Mathematical Sciences: Partial Differential Equations & Analysis

该奖项支持三位研究人员的工作，他们研究了几个具有广泛数学兴趣和重要性的领域。一些工作将联合进行，而另一些工作将由单个调查人员进行。人们可以将研究分为三个主要主题：非线性偏微分方程、小波分析和复变函数理论。首先，将努力探索如何将取代经典作用角变量的一维量子反散射方法应用于理解相应的高维系统。涉及表面波现象的逆地球物理问题的研究工作也在取得进展。模型标量情况的研究进展表明，可以对整个方程组进行分析。自适应波形分析的研究，类似FFT的自适应变换算法的集合，继续发现和完善新的标准正交基，将函数和算子分解成几乎对角的形式。这些理论进步与数值信号处理应用之间的相互作用导致了基于熵的停止时间搜索的高度成功的分解技术。这些应用反过来又导致了谐波分析中的一系列新问题。其中之一将是创建一个函数的理论维度的新概念，用于测量用给定库中的波形来描述函数所需的参数数量。对谐波测度精细结构的研究也在继续。这里的目标是找出谐波测度在各种情况下的渐近行为。本文还将对有界单连通域上格林函数的水平线长度进行估计。认为水准线长度随常数值的四次方根而变化。偏微分方程是物理科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是说明关系，而是从中提取定性和定量的意义，并验证表达解决方案的方法。调和分析侧重于将函数分解成最能反映其振荡特征的组成部分，而复函数理论则试图描述复变量的可微函数的几何和解析性质及其在高维上的推广。

This award supports the work of three researchers investigating several areas of broad mathematical interest and importance. Some work will be conducted jointly while other will be addressed by single investigators. One can divide the studies into three main themes: nonlinear partial differential equations, wavelet analysis and complex function theory. In the first, efforts will be made to explore how the one-dimensional quantum version of the inverse scattering methods which replace the classical action-angle variables can be applied to understanding the corresponding higher dimensional systems. Work is also progressing on the investigation of inverse geophysical problems involving surface wave phenomena. Progress on model scalar cases suggests that analysis of the full system of equations can be obtained. Studies of adapted wave form analysis, a collection of FFT- like adapted transform algorithms, continues to discover and refine new orthonormal bases which decompose functions and operators into almost diagonal form. The interplay between these theoretical advances and the application to numerical signal processing has led to highly successful decomposition techniques using entropy based stopping time searches. The applications lead, in turn, to a host of new questions in harmonic analysis. One of these will be that of creating a new concept of theoretical dimension of a function designed to measure the number of parameters necessary to describe the function with wave forms taken from a given library. Studies on the fine structure of harmonic measure also continue. Here the goals are to find the asymptotic behavior of harmonic measure in a variety of settings. Work will also be done estimating the length of level lines of the Green's function on bounded simply connected domains. It is believed that level line length varies with the fourth root of the value of constancy. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions. Harmonic analysis focuses on the decomposition of functions into component parts which best reflect their oscillatory characteristics, while complex function theory seeks to describe the geometric and analytic properties of differentiable functions of a complex variable and their generalizations to higher dimensions.