Problems in Analysis Related to Lattice Points and Singularities

建议：DMS-9970899首席研究员：Akos Magya摘要：建议的研究将处理调和分析中的几个问题，强调从解析数理论和奇点理论的技巧。研究对象涉及波动方程、傅立叶积分、限制和极大算子以及紧流形上拟微分算子的特征函数。第一组问题是离散的，因为整数格点的性质对它们至关重要，并被诸如Hardy-Littlewood方法等求指数和的方法所利用。下面的空间是平面环面或其对偶，即晶格点本身。在光滑流形的设置中也考虑了类似的问题，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。利用奇点理论技术建立相应的振荡积分的衰变性质。更广泛地说，这个项目集中于源于自然界中基本的偏微分方程组的分析问题，即在微观尺度上控制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，只有将数学、解析数论和奇点理论等不同领域的方法和结果结合在一起，才能实现更深层次的理解。经典调和分析是描述上述物理现象的一种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。拟议中的项目是过去30年来为丰富经典理论和扩大其适用范围而实施的庞大计划的一部分。

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