Discrete Problems in Harmonic Analysis, Ergodic Theorems and Singularities

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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.