Jesse Wolfson University of California, Irvine (UCI) When nature hands you an algebraic function, you have two tasks: 1) find a simple solution and 2) prove no simpler solution is possible. In this talk I'll focus on 2), starting with Hilbert's 13th Problem, where he conjectured that the Hamilton/Klein solutions of the degree 7 were the simplest possible, and suggested that this was for topological reasons. Amazingly, no progress has been made on the fundamental problem since Hilbert, or even since Abel. In this talk, I’ll review some of the most promising ideas that have been developed to tackle this problem, with a discussion of what they can achieve, and what fundamental obstacles remain.