Nathan Kaplan Yale University We will introduce some important properties of polynomials over finite fields, focusing on connections between geometry and algebra that come from identifying a polynomial with the variety defined by its set of zeros. We will study when a low-degree polynomial vanishes on a given set of points and discuss algebraic results that have proven useful in combinatorial applications. We will describe some successes of the polynomial method, focusing on Dvir’s proof of the finite field Kakeya conjecture and subsequent refinements and extensions of these ideas.