In the 60s and 70s, there was a flurry of activity concerning the question of whether or not various subgroups of homeomorphism groups of manifolds are simple, with beautiful contributions by Fathi, Kirby, Mather, Thurston, and many others. A funnily stubborn case that remained open was the case of area-preserving homeomorphisms of surfaces. For example, for balls of dimension at least 3, the relevant group was shown to be simple by work of Fathi from the 1970s, but the answer in the two-dimensional case was not known. I will explain recent joint work proving that the group of compactly supported area preserving homeomorphisms of the two-disc is in fact not a simple group, which answers the ``Simplicity Conjecture” in the affirmative. Our proof uses a new tool for studying area-preserving surface homeomorphisms, called periodic Floer homology (PFH) spectral invariants; these recover the classical Calabi invariant via a kind of Weyl law. Time permitting, I will also briefly mention a generalization of our result to compact surfaces of any genus.