Vinogradov systems and Incidence geometry

In this talk, I will talk about two seemingly disjoint topics - Vinogradov’s mean value theorem, a classically important topic of study in additive number theory concerning solutions to a specific system of diophantine equations, and Incidence geometry, a collection of combinatorial results which focus on estimating the number of incidences between an arbitrary set of points and curves. I will give a brief overview of these two topics along with some basic proofs and applications, and then point out how these subjects connect together.