Topics in harmonic analysis and additive combinatorics

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The project involves work on two distinct types of problem in harmonic analysis and additive combinatorics. The first collection of problems will be concerned with on the further development of the theory of strongly singular integrals (linear operators whose integral kernels are too singular at the origin to be of Calderon-Zygmund type) and in particular will focus the use of oscillatory integral techniques to investigate the regularity of these classical operators both along varieties and on different Lie groups. The second collection of problems will concern the application of Fourier analytic techniques to problems in additive combinatorics, in particular we will focus on quantitative results relating to the beautiful observation (made independently by Sarkozy and Furstenberg) that subsets of the integers with positive upper density necessarily contain square differences.Broadly speaking the project will focus on two main problems. The first of these is specifically concerned with the generalization (and extension to more general abstract settings) of some well known results from classical harmonic analysis. The second problem relates to proving quantitative versions of results from density Ramsey theory establishing the existence of certain arithmetic patterns in large sets of integer points.