The Simplicity Conjecture
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 in 1980; but, the answer in the two-dimensional case, asked by Mather in the 70s, was not known. We recently answered Mather's question by showing that the group of compactly supported area-preserving homeomorphisms of the two-disc is in fact not simple. After surveying the history described above, I will give a very gentle introduction to some of the key ideas in our proof; what is crucial is the fact that the 2-ball with its volume form is a symplectic manifold. Our work underscores that it is natural to study continuous symplectic geometry, and I will briefly explain what this means.