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-0202021 PI：Akos Magyar摘要本研究将探讨调和分析中的几个问题，其中一些问题也与万有引力和数论有关.重点是结合技术分析和其他领域，如解析数论和奇异性理论。研究对象涉及丢番图方程、子簇上的极大平均及其离散类似、多项式型遍历定理以及退化积分和积分算子。一类本质上是离散的问题，其中与经典指数和有关的算子的映射性质起着关键作用。坚韧这些出现作为离散的类似物众所周知的结构在分析中，他们反过来可以yieldto新的见解都遍历和数论。另一个方向是研究与退化积分相关的问题，其中通常的“曲率“型条件被奇异性假设或仅假设一般性所取代。在大约过去的十年中，调和分析的范围通过结合来自看似不同的数学领域的方法而得到了极大的扩展。这使得人们对周期条件下的基本偏微分方程（如量子力学中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，但要理解物理过程，测量次数很少。拟议的研究将集中在这些类型的问题，其中不同领域的数学方法的组合似乎是必要的.