PATCH seminar at UPenn

Speakers will be John Pardon and Larry Guth.

John Pardon (3:30):

Totally disconnected groups (not) acting on three-manifolds.

Hilbert´s Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved inthe affirmative by Gleason and Montgomery–Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert–Smith Conjecture,which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery–Zippin) that it suffices to rule out the case of the additive group of p-adic integers acting faithfully on a manifold. I will present a solution in dimension three. The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group.

Larry Guth (5:00):
Contraction of areas and homotopy type of mappings.

I´m going to talk about connections between the geometry of a map and its homotopy type. Suppose that we have a map from the unit m-sphere to the unit n-sphere. We say that the k-dilation of the map is < L if each k-dimensional surface with k-dim volume V is mapped to an image with k-dim volume at most LV. Informally, if the k-dilation of a map is less than a small epsilon, it means the map strongly shrinks each k-dimensional surface. Our main question is: can a map with very small k-dilation still be homotopically non-trivial?
Here are the main results. If k > (m+1)/2, then there are homotopically non-trivial maps from S^m to S^{m-1} with arbitrarily small k-dilation. But if k is at most (m+1)/2, then every homotopically non-trivial map from S^m to S^{m-1} has k-dilation at least c(m) > 0.

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