Expanders, Fourier analysis and point-curve incidence theory

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The main goal of the proposed project is to study the effect that the ambient field has on the behavior of set expanders. This is explored by studying the Euclidean analogs of two problems which were recently investigated in the context of vector spaces over finite fields. The first problem is a natural generalization of the Falconer distance problem to the study of point configurations. The second problem is drawn from a large class of problems known as the sum-product problems. The continuous nature of the first problem and the discrete nature of the second allows the researcher to incorporate a variety of methods from various fields in order to better illustrate the general principles involved. These include methods drawn from arithmetic and geometric combinatorics, number theory, and classical harmonic analysis. The juxtaposition of the methods from the Euclidean cases with the methods in the context of the finite field geometry could provide a more complete picture of the set expansion phenomenon.This project involves studying the interaction between several areas of mathematics using a problem-solving approach, in which two problems are considered whose analogs have known solutions in one subfield of mathematics known as finite field geometry. The problems are examined in two additional related subfields known as discrete and continuous Euclidean geometry. The goal of the project is to not only find solutions to the two problems in these alternate contexts, but to explore the interplay between a variety of known methods and techniques in order to obtain a more complete understanding of the scope of the general principles involved. The major areas of mathematics these problems are taken from include additive and geometric combinatorics, number theory, and classical harmonic analysis. These areas have a variety of applications in encryption, data mining, signal processing, and bioinformatics.