Mathematical Sciences: Partial Differential Equations and Analysis

DMS-9423746 PI：R。比尔斯河R. Coifman，P. W.琼斯 摘要 主要调查人员将继续他们的工作， 相关领域的分析：谐波分析，非线性傅立叶 偏微分方程及其应用 到复杂的分析、物理问题和数值工作。 比尔斯 合作者将更多地利用他们的方法从哈密尔顿 研究次椭圆方程基本解的力学 拉普拉斯算子。 比尔斯还将继续联合工作的量化 三波相互作用和地球物理逆问题。 Coifman和Meyer将继续研究非线性傅立叶 分析和应用均匀化和非线性 依赖 Coifman还将继续开发适应wave 形式分析，以获得几何和数字的理解， 振动算子 琼斯将继续他的工作， 调和测度的性质和绿色函数的水平线， 以及奇异积分和d-杆问题的工作， 电晕问题和分析能力。琼斯还计划进一步 拟共形映射及其相关问题的研究 索伯列夫空间与集合几何。 分析是数学的一个领域，它直接从 微积分和微分方程的发明。 它提供 一个解决问题的框架，从纯粹的 从数学和几何到最具体的科学和 技术问题。 反过来，这些问题又提供了 继续改进主题的动机，而不是 顺便说一句，它的持续效用。 因此，没有精确的 纯理论和潜在或实际的分界线 应用程序. 例如，在十多年的时间里， 出现了复杂的纯谐波分析技术 在实践中，小波分析，并极大地扩展了 分析和存储数字数据的可能性。 的 本建议书中拟调查的子项目反映了 分析范围以及实际和潜在的相互作用 在“纯粹”和“应用”之间。“有些有世纪的历史 或更多，来自经典分析和几何，其他 有几十年的历史，受物理学的启发， 物理学，其他人在过去十年中出现， 与信号处理和具体使用有关， 计算机在科学和技术中的应用

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.