Problems in Analysis Related to Lattice Points and Singularities

与格点和奇点相关的分析问题

• 批准号: 9970899
• 负责人: Akos Magyar
• 金额: $ 4.7万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-05-15 至 2001-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970899&HistoricalAwards=false
• 关键词: Problems; Analysis; Related; Lattice; Points

Proposal: DMS-9970899Principal Investigator: Akos MagyarAbstract: The proposed research will deal with several problems in harmonic analysis emphasizing techniques from analytic number theory and singularity theory. The objects of study are related to wave equations, to Fourier integral, restriction and maximal operators and to eigenfunctions of pseudo-differential operators on compact manifolds. The first set of problems are of a discrete nature in the sense that properties of integer lattice points are crucial to them and are exploited by methods for exponential sums such as the Hardy-Littlewood method. The underlying space is the flat torus or its dual, the set of lattice points itself. Similar problems are considered in the settings of a smooth manifold, the usual "curvature" conditions replaced by assuming only that the objects are typical (generic) elements of the family. Techniques of singularity theory are brought into play to establish decay properties of the corresponding oscillatory integrals.In more general terms, this project focuses on problems of analysis that stem from partial differential equations of fundamental importance in nature; namely, the Schrodinger equation governing the evolution of states of physical systems at microscopic scales and the wave equation describing the propagation of light and acoustic waves. The project aims especially at those situations where purely analytic techniques seem to be inadequate and deeper understanding can be achieved only by incorporating methods and results from different fields of mathematics, analytic number theory and singularity theory to name just two. Classical harmonic analysis has been a very succesful mathematical tool for describing the aforementioned physical phenomena but is completely effective in rather restricted circumstances. The proposed project is part of a vast program that has been implemented over the past three decades to enrich the classical theory and to extend its range of applicability

提案：DMS-9970899 首席研究员：Akos Magyar 摘要：所提出的研究将处理调和分析中的几个问题，重点是解析数论和奇点理论的技术。研究对象涉及波动方程、傅里叶积分、限制和极大算子以及紧流形上伪微分算子的本征函数。第一组问题具有离散性质，因为整数格点的属性对它们至关重要，并且可以通过指数和方法（例如 Hardy-Littlewood 方法）来利用。底层空间是平坦的环面或其对偶，即格点集本身。在光滑流形的设置中考虑类似的问题，通常的“曲率”条件被仅假设对象是族的典型（通用）元素所取代。奇点理论的技术被用来建立相应振荡积分的衰减特性。更一般地说，该项目重点关注源于自然界中具有根本重要性的偏微分方程的分析问题；即控制微观尺度物理系统状态演化的薛定谔方程和描述光波和声波传播的波动方程。该项目特别针对那些纯粹的分析技术似乎不够充分的情况，只有通过结合数学、解析数论和奇点理论等不同领域的方法和结果才能实现更深入的理解。经典调和分析是描述上述物理现象的非常成功的数学工具，但在相当有限的情况下是完全有效的。拟议的项目是过去三十年来实施的庞大计划的一部分，旨在丰富经典理论并扩大其适用范围