UVictoria Discrete Math Seminar: Sammy Luo

A central tool in additive combinatorics is the polynomial method, a family of powerful techniques for proving results about combinatorial objects by encoding them in algebraic terms. In this talk, we give a brief overview of some iconic modern developments in the use of polynomial methods, including the Combinatorial Nullstellensatz, the slice rank method, and Dvir’s approach to the finite field Kakeya problem. Then, we introduce a new method, based on what we call “shift operators,” that unifies these tools by providing a natural common framework for them while also yielding new results in the study of sumsets.