Discrete Problems in Harmonic Analysis, Ergodic Theorems and Singularities

建议编号：DMS-0202021PI：Akos Magya ABSTRACT建议的研究将涉及调和分析中的几个问题，其中一些问题也是令人感兴趣的惯性和数论。重点是分析技术和其他领域的技术的结合，如解析数论和奇点理论。研究的对象涉及丢番图方程、子簇及其离散类似物上的最大平均、多项式型遍历定理以及具有退化的振荡积分和积分算子。一组问题本质上是离散的，其中与经典指数和有关的算子的映射性质起着至关重要的作用。尽管它们在分析中看起来像是众所周知的概念的离散类似物，但它们反过来又可以在遍历和数论中产生新的见解。另一个方向是研究与具有简并性的振荡积分有关的问题，其中通常的“曲率”型条件被关于奇异性的假设或只是假设的广泛性所取代。在大约过去的十年里，通过结合来自似乎不同的数学领域的方法，调和分析的范围得到了极大的扩展。这本质上加深了对周期条件下的基本偏微分方程式(如量子力学的非线性薛定谔方程和波动方程)的理解，这是遍历理论中的新现象，意味着不必定期进行测量，而只需极少地进行测量即可了解物理过程。拟议的研究将集中在这些类型的问题上，这些问题似乎需要结合不同数学领域的方法。

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.