My research studies how simple “local” rules can force the presence of strong “global” structure in large mathematical objects. I’ve worked on a variety of problems that all share this theme, and the new techniques I develop typically combine ideas from different areas of mathematics. For example, I have shown that if one has a large number of points in the plane and specifies that the distance between each pair of points is a whole number, then most of the points must cluster on a single line or circle. This research made progress on a classic problem in discrete geometry, and required tools from analytic number theory and algebraic geometry.