Title: Bounds on image sets of some functions

A planar function $f$ is a function defined on a finite field with the property that for every $a\in\mathbb F_q^*,$ the function $x\mapsto f(x+a)-f(x)$ is a bijection. One long standing problem in the intersection of finite geometry, cryptography, and finite fields is obtaining an upper bound for the size of the image set of a planar function. Over a decade ago, Steve Senger and I gave the best known (also the only non-trivial!) upper bound via an almost completely combinatorial argument. However, we've always believed that our bound could be improved. In this talk I'll explain how we managed to show we were right, except possibly for $q=343$, where we instead show that an example meeting the bound would imply the existence of a projective plane of order 18.