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Mathematical Conversations

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.


Daniel Cristofaro-Gardiner

Speaker Affiliation

University of California, Santa Cruz; von Neumann Fellow, School of Mathematics



Event Series

Date & Time
May 13, 2020 | 5:307:00pm


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