The main thrust of this proposal is the study of k-point configurations in discrete, continuous and arithmetic settings. The basic question is to determine how large a subset of a given vector space needs to be to ensure that it determines a substantial proportion of all possible finite point configurations up to congruence. The research involves a tight interaction between analytic, number theoretic and combinatorial methods in a seamless symbiosis that allows for a deeper understanding of the associated techniques and ideas.
The techniques and ideas of the proposal do not only involve interaction between different areas of mathematics. They also involve ideas from theoretical computer science with potential applications to data mining, coding, signal processing, bioinformatics and many other areas of modern science. Modern harmonic analysis, which is PI?s main area of expertise, is a treasure trove of techniques and ideas that have found relevance in virtually every scientific disciplines and this influence will continue to grow in the years and decades to come.
