Algebraic Methods in Extremal Combinatorics

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Questions in extremal combinatorics ask to optimize a combinatorial invariant subject to a set of constraints. This deliberately broad question has ubiquitous applications both in industry and throughout mathematics, but also leads to questions that are interesting in their own right. This project will attempt to quantify statements of the form: if an object is "large" in a suitable sense, then it contains ordered substructure. The project will specifically seek to use methods from algebra and geometry to answer problems in extremal graph theory and combinatorial number theory. The main benefits of the project are two-fold: first it will answer important open problems in the intersection of several mathematical areas, and second it will involve undergraduate students in research.The PI will focus specifically on Turan and Ramsey type problems, on additive combinatorics, and on spectral graph theory. Algebraic and geometric constructions have long been useful for extremal graph theory problems, and this project will continue to use these methods as well as explore their limitations and connections to each other. The project will also seek to extend known methods in both areas to hypergraphs, where our knowledge is limited. The project goals sit in the intersection of combinatorics, algebra, combinatorial number theory, and geometry, and as such many techniques are applicable and problems from several different areas will be worked on.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.