Combinatorial geometry in discrete math and harmonic analysis

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Over the past 25 years, harmonic analysis and combinatorics have become increasingly entwined. Jean Bourgain, Nets Katz, Wilhelm Schlag, Terence Tao, Tom Wolff, and many others have discretized harmonic analysis problems and recast them as questions in combinatorial geometry. In the opposite direction, the recent and growing field of discrete analysis uses tools and ideas from analysis to attack problems that have traditionally been the purview of combinatorics. Broadly speaking, my work lies at the intersection of geometric combinatorics and classical harmonic analysis, and sprawls in several related directions.My research program is guided by two major open problems: the Kakeya problem and the Erdos distinct distance problem. The first of these problems concerns objects called Besicovich sets. A Besicovich set in d dimensions is a subset of d-dimensional space that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that sets of this type must be "large" in a certain technical sense. Today, the Kakeya problem plays a central role in the interactions between classical harmonic analysis, theoretical computer science, and classical algebraic geometry. The Erdos distinct distance problem asks: given a set of n points in Rd, how many different distances can occur between pairs of points? This problem succinctly encapsulates a vast region of combinatorial geometry that is poorly understood. In 2010, Larry Guth and Nets Katz obtained nearly tight bounds for the d=2 version of the problem and in doing so, they developed a new set of tools that are leading to sweeping changes across combinatorial geometry, harmonic analysis, and theoretical computer science. In higher dimensions the problem remains open, and settling it has become a central priority for the combinatorial geometry community.Incidence geometry is a field of extremal combinatorics that analyzes how many intersections (called incidences) can occur within a collections of objects. This has proved to be a powerful and flexible framework for describing many types of phenomena, including the two problems discussed above. A central theme running through my research is the role played by algebraic structure in incidence geometry. If a collection of objects is arranged at random, then there are likely to be few, if any, incidences. This suggests that the most problematic (and thus interesting) arrangements are highly structured. So far, we know of two major sources of interesting structures. The first comes from algebraic geometry, and the second comes from sets that are almost closed under arithmetic operations such as addition and multiplication. If both of these sources of interesting structure are absent, then it seems reasonable to expect a decrease in the number of incidences, which would yield stronger results in the original problem. My work focuses on turning this principle into quantitative results.