Algebraic and Probabilistic Methods in Extremal Combinatorics

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Extremal combinatorics is an area of mathematics that investigates how large or small configurations of mathematical objects can be under certain constraints. This area is rapidly developing and has close connections to many other areas of mathematics, as well as to theoretical computer science. This research project aims to make progress on questions in extremal combinatorics using methods from algebra and probability theory. The research focuses on some longstanding open questions and conjectures, as well as several related problems. The work will lead to the development of new mathematical tools and techniques and push the limits of known methods. Moreover, through her teaching and mentoring, the investigator strives to encourage students to learn about mathematics and to pursue careers in STEM fields.The questions studied in this project fall into two rough topic areas. The first of these areas is centered around the slice rank polynomial method that was introduced in 2016. This method has led to several spectacular results in additive combinatorics, but many related questions remain open. The investigator intends to study specific problems exemplifying the current limitations of the slice rank polynomial method. One aim of this project is to find ways to make the method more flexible and more widely applicable. The second topic area of the project concerns the inducibility problem, which was posed over forty years ago and is still wide open. Given a fixed graph H, and a large integer n, this problem asks about the maximum number of induced copies of H that an n-vertex graph can contain. A major open question in this area is the case where the graph H is a path or a cycle. Using probabilistic techniques, the PI plans to investigate this question as well as other inducibility-type problems.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.