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.