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.
As a result of this project, my collaborators and I gained a much deeper understanding of the distribution of finite point configurations in Euclidean space. We developed a multi-linear and group action machinery in order to opbtain near sharp exponents for the natural generalization of the Erdos-Falconer distance problem in Euclidean space and Riemannian manifolds. The PI and his co-authors also developed Fourier analysis and combinatorial techniques in the context of Erdos type problems in vector spaces over finite fields. As a result, they took an important step towards unifying the discrete and continuous theories in the context of modules over locally compact rings. There is an excellent chance this program will result in a thorough general understanding of the matter within the next couple of years. The problems under considerations have interesting connections with the study of large data sets and in the process of working on this project, the PI developed a number of concrete ideas about further explorations designed to enhance our understanding of these connections. Several undergraduate and graduate students participated in this project, including three female students: Esen Aksoy (graduating with a Ph.D. this year), Krystal Taylor (2012 Ph.D. and currently a posdoctoral resercher in Minnesota) and Yujia Zhai (currently a Ph.D. student at Cornell). The PI also ran several summer research programs for undergraduate students at the Unviersity of Rochester.
