CAREER: Analytic and Spectral Methods in Combinatorics

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This CAREER award supports research in combinatorics. The PI aims to develop and apply novel analytic and spectral methods designed for a variety of extremal problems in discrete mathematics. Such investigations ask, for example, how large a set (of points, vectors, integers, etc.) satisfying certain combinatorial properties can be. Results have applications in number theory, computer science, and other fields. Analytic and spectral methods already play central roles in the field; this project will lead to new techniques with further applications in combinatorics and beyond. The project also will have an educational component, including graduate student mentorship and course development.One of the directions of research concerns eigenvalue multiplicities and applications to equiangular lines, spherical two-distance sets, and more generally spherical codes in high dimensions. The PI recently solved a longstanding problem on equiangular lines via new insights in spectral graph theory, paving the way for further developments on high dimensional spherical codes via more generalized spectral graph theoretic problems. A second direction concerns extremal problems in graph theory and additive combinatorics, focusing on techniques connecting the two areas. The PI will study sparse regularity methods, graph homomorphism density inequalities (for example Sidorenko’s conjecture), extremal problems in hypergraphs, and related topics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.