In 1980, Fathi showed that the group of compactly supported area-preserving homeomorphisms of the n-ball is a simple group when n > 2, and asked whether or not this group is simple in the n = 2 case.  I will explain recent joint work showing that in the n = 2 case, this group is in fact not simple;  this answers what is known as the "simplicity conjecture" in the affirmative.  The crux of our proof involves trying to recover the Calabi homomorphism, which is defined on the diffeomorphism subgroup, in terms of quantities called "spectral invariants" that are C^0 continuous and so extend to the full group.  These quantities are defined using Floer homology, and I will give an impressionistic sketch of how this works.