The proposed research will deal with several problems in harmonic analysis some of which are also of interest in ergodic and number theory. The emphasis is on a combination of techniques from analysis and of other fields such as analytic number theory and singularity theory. The objects of study are related to diophantine equations, maximal averages over subvarieties and their discrete analogues, polynomial type ergodic theorems as well as oscillatory integrals and integral operators with degeneracies. A set of problems are discrete in nature, where mapping properties of operators related to classical exponential sums play a crucial role. Tough these appear as discrete analogues of well known constructs in analysis, they in turn can yield to new insights both in ergodic and number theory. Another direction is to study problems related to oscillatory integrals with degeneracies, where the usual "curvature" type conditions are replaced by assumptions on the singularities or only assuming genericity.
During roughly the past decade the scope of harmonic analysis has been vastly extended by incorporating methods from seemingly different fields of mathematics. This has led to an essentially deeper understanding of basic partial differential equations under periodic conditions (such as the nonlinear Schrodinger equation of quantum mechanics and the Wave equation), new phenomena in ergodic theory meaning that measurements do not have to be taken regularly but only very rarely to understand physical processes. The proposed research will concentrate on these types of problems Where a combination of methods of different fields of mathematics seems to be needed.
