The Polynomial Method: Algebraic Applications in Combinatorics
Abstract: A common technique in extremal combinatorics is the embedding of discrete structures into well studied objects. We will be discussing the polynomial method, which attempts to correspond a set of objects to the zeros of some polynomial. By bounding the degree of this polynomial, we can restrict the size of our original set. We will be introducing the solution to the Kakeya conjecture over finite fields, the Combinatorial Nullstellensatz theorem, and finish with discussing applications in discrete geometry.