Discrete Problems in Harmonic Analysis, Ergodic Theorems and Singularities

调和分析、遍历定理和奇点中的离散问题

• 批准号: 0202021
• 负责人: Akos Magyar
• 金额: $ 9.33万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202021&HistoricalAwards=false
• 关键词: Discrete; Problems; Harmonic; Analysis; Ergodic

Proposal Number: DMS-0202021PI: Akos MagyarABSTRACTThe proposed research will deal with several problems inharmonic analysis some of which are also of interest inergodic and number theory. The emphasis is on a combinationof techniques from analysis and of other fields such asanalytic number theory and singularity theory. The objectsof study are related to diophantine equations, maximalaverages over subvarieties and their discrete analogues,polynomial type ergodic theorems as well as oscillatoryintegrals and integral operators with degeneracies. A setof problems are discrete in nature, where mapping propertiesof operators related to classical exponential sums play acrucial role. Tough these appear as discrete analogues ofwell known constructs in analysis, they in turn can yieldto new insights both in ergodic and number theory. Anotherdirection is to study problems related to oscillatoryintegrals with degeneracies, where the usual "curvature"type conditions are replaced by assumptions on thesingularities or only assuming genericity.During roughly the past decade the scope of harmonicanalysis has been vastly extended by incorporating methodsfrom seemingly different fields of mathematics. This hasled to an essentially deeper understanding of basic partialdifferential equations under periodic conditions (such asthe nonlinear Schrodinger equation of quantum mechanics andthe Wave equation), new phenomena in ergodic theory meaningthat measurements do not have to be taken regularly butonly very rarely to understand physical processes. Theproposed research will concentrate on these types ofproblems Where a combination of methods of different fieldsof mathematics seems to be needed.

提案编号：DMS-0202021PI：Akos Magyar 摘要所提出的研究将处理调和分析中的几个问题，其中一些问题也与惯性和数论有关。重点是分析技术与其他领域（例如解析数论和奇点理论）的结合。研究对象涉及丢番图方程、子族最大平均及其离散类比、多项式遍历定理以及振荡积分和简并积分算子。一组问题本质上是离散的，其中与经典指数和相关的算子的映射属性起着至关重要的作用。尽管这些在分析中看起来像是众所周知的结构的离散类似物，但它们反过来可以在遍历和数论中产生新的见解。另一个方向是研究与简并振荡积分相关的问题，其中通常的“曲率”类型条件被奇点假设或仅假设一般性所取代。在过去的十年中，调和分析的范围通过合并来自看似不同领域的方法而得到了极大的扩展。数学。这使得人们对周期性条件下的基本偏微分方程（例如量子力学的非线性薛定谔方程和波动方程）有了更深入的理解，遍历理论中的新现象意味着不必定期进行测量，但只需很少的情况下就可以理解物理过程。拟议的研究将集中于似乎需要结合不同数学领域的方法的这些类型的问题。