Problems in Analysis Related to Lattice Points and Singularities

摘要：本课题将重点研究解析数论和奇点理论中的谐波分析问题。研究了紧流形上的波动方程、傅里叶积分、限制算子和极大算子以及伪微分算子的特征函数。第一组问题具有离散性质，因为整数点阵点的性质对它们至关重要，并且可以通过诸如Hardy-Littlewood方法之类的指数和方法加以利用。底层空间是平面环面或者它的对偶，晶格点的集合本身。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该族的典型（一般）元素所取代。利用奇点理论建立了相应振荡积分的衰减性质。更一般地说，这个项目的重点是分析问题，这些问题源于在自然界中具有基本重要性的偏微分方程；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解析技术似乎不足的情况，只有通过结合不同数学领域的方法和结果才能实现更深入的理解，解析数论和奇点理论仅举两例。经典谐波分析是描述上述物理现象的一个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。在过去的三十年中，为了丰富经典理论并扩展其适用范围，该计划是一个庞大计划的一部分

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