Problems in Analysis Related to Lattice Points and Singularities

建议：DMS-9970899首席研究员：Akos Magyar摘要：拟议的研究将处理谐波分析中的几个问题，强调解析数论和奇异性理论的技术。研究的对象涉及波动方程，傅立叶积分，限制和最大运营商和本征函数的伪微分算子的紧流形。第一组问题是一个离散的性质，在这个意义上说，整数格点的属性是至关重要的，并利用指数和方法，如哈迪-利特尔伍德方法。底层空间是平坦环面或其对偶，即格点本身的集合。类似的问题被认为是在设置一个光滑的歧管，通常的“曲率”的条件下，只假设对象是典型的（通用）元素的家庭。利用奇异性理论的技巧来建立相应的振荡积分的衰减特性。更一般地说，本项目侧重于分析源于自然界中具有根本重要性的偏微分方程的问题，即控制微观尺度下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是不够的情况，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。这个被提议的项目是一个庞大计划的一部分，这个计划在过去的三十年里一直在实施，以丰富经典理论并扩大其适用范围

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