Discrete problems in harmonic analysis with applications to ergodic theory and additive number theory

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The proposed research will deal with problems that are on the boundary between different fields of mathematics; harmonic analysis, ergodic theory, and number theory. The problems occur in different contexts however a central role is played by properties certain exponential sums and related methods of analytic number theory, such as the Hardy-Littlewood method of exponential sums. One set of problems concern maximal and singular integral operators associated with polynomial surfaces on discrete nilpotent groups. These are discrete analogues of problems harmonic analysis on Euclidean spaces, and also are naturally connected to pointwise ergodic theory of non-commuting transformations. Technically, a crucial role is played by an extension of the "circle method" to an operator valued settings, which arises because of non-commutativity of the underlying group. Another set of the proposed problems are in the settings of additive number theory/combinatorics, as they are to show the existence of certain structures in subsets of positive density, or in cells obtained after a finite partitioning of integer lattices. The emphasis is on the Fourier analytic approach to address certain problems in the multidimensional or nonlinear settings.Because of its interdisciplinary nature, the project is expected to have an impact on the various fields involved. It aims utilize and extend several techniques, some of which is recent and is still being developed. The emphasis is on the interplay of techniques of the above fields, to attack open problems of interest. The project would enable the PI to continue to support graduate students and post-docs, and in general, to introduce young researchers to this rapidly developing area of mathematics.