Mathematical Sciences: Partial Differential Equations and Analysis

主要研究人员将继续他们在一些相关分析领域的工作：调和分析、非线性傅立叶分析和偏微分方程，以及在复杂分析、物理问题和数值工作中的应用。比尔和合作者将更多地使用哈密顿力学中的方法来研究亚椭圆拉普拉斯方程的基本解。Beals还将继续就量子化的三波相互作用和地球物理反问题进行联合工作。科夫曼和迈耶将继续研究非线性傅立叶分析及其在齐次化和非线性相关性方面的应用。科夫曼还将继续开发适应的波形分析，以获得对振荡算子的几何和数值理解。琼斯将继续研究格林函数的调和测度和水平线的精细性质，以及与日冕问题和分析能力有关的奇异积分和d-bar问题。Jones还计划进一步研究拟共形映射以及与Sobolev空间和集合几何有关的问题。分析是从微积分和微分方程式的发明直接发展而来的数学领域。它为解决从纯粹的数学和几何问题到最具体的科学和技术问题提供了一个框架。反过来，这些问题提供了继续完善这一主题的动力，而不是附带地，提供了它继续有用的动力。因此，在纯粹的理论和潜在的或实际的应用之间没有精确的分界线。例如，在短短十年多一点的时间里，纯谐波分析的复杂技术以小波分析的形式出现在实践中，并极大地扩展了分析和存储数字数据的可能性。本提案中要调查的主题反映了全面的分析以及“纯粹的”和“应用的”之间的实际和潜在的相互作用。其中一些有一个世纪或更多的历史，来自经典分析和几何，另一些有几十年的历史，受到物理和地球物理的推动，另一些是在最近十年兴起的，与信号处理和计算机在科学和技术中的具体应用有关。

DMS-9423746 PI: R. Beals, R. R. Coifman, P. W. Jones Abstract The principal investigators will continuing their work in a number of related areas of analysis: harmonic analysis, nonlinear Fourier analysis, and partial differential equations, with applications to complex analysis, physical problems, and numerical work. Beals and collaborators will make more use their methods from Hamiltonian mechanics to investigate fundamental solutions for subelliptic laplacians. Beals will also continue joint work on the quantized three-wave interaction and on geophysical inverse problems. Coifman and Meyer will continue to investigate nonlinear Fourier analysis and applications to homogenization and nonlinear dependence. Coifman will also continue to develop adapted wave form analysis to obtain geometric and numerical understanding of oscillatory operators. Jones will continue his work on the fine properties of harmonic measure and level lines of Green's function, as well as work on singular integrals and d-bar problems related to corona problems and analytic capacity. Jones also plans further investigations in quasiconformal mappings and problems relating Sobolev spaces and geometry of sets. Analysis is the area of mathematics that has grown directly from the invention of calculus and differential equations. It provides a framework for attacking problems that range from the purely mathematical and geometrical to the most concrete scientific and technical questions. In turn, these problems have provided the motivation for the continued refinement of the subject and, not incidentally, its continued utility. Thus there is no precise dividing line between pure theory and potential or actual application. For example, in little more than a decade sophisticated techniques of pure harmonic analysis have emerged in practice as wavelet analysis and have enormously extended the possibilities for analysis and storage of digital data. The sub jects to be investigated in this proposal reflect the full range of analysis as well as the actual and potential interplay between "pure" and "applied." Some have a history of a century or more and come from classical analysis and geometry, others have a history of decades and are motivated by physics and geophysics, others have arisen in the last decade and are connected with signal processing and the concrete use of computers in science and technology.