Jason Behrstock: Temple geometry seminar (5 pm,Tuesday, Jan 22)

Jason Behrstock of CUNY will be speaking in the Temple Geometry and Topology Seminar.

Title: Divergence, thick groups, and morse geodesics

Abstract: In a metric space the divergence of a pair of rays is a way
to measure how quickly they separate from each other.  Understanding
what divergence rates are possible in the presence of non-positive
curvature was raised as a question by Gromov and then refined by
Gersten.  We will describe a construction of groups with several
interesting properties, some of which shed light on the above
question.  (Joint work with Cornelia Drutu.)

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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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Joseph Maher’s talk at the Geometry/Topology seminar (4 pm Tuesday, November 20)

Joseph Maher will be speaking in the Temple Geometry and Topology Seminar.

Title: Statistics for Teichmuller geodesics.

Abstract: We describe two ways of picking a geodesic “at random” in a
space, one coming from the standard Lebesgue measure on the visual sphere, and the other coming from random walks. The spaces we’re interested in are hyperbolic space and Teichmuller space, together with some discrete group action on the space. We investigate the growth rate of word length as you move along the geodesic, and we show these growth rates are different depending on how you choose the geodesic. This is joint work with Vaibhav Gadre and Giulio Tiozzi.

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Stefan Friedl’s talk at the Geometry/Topology seminar (4pm Tuesday, October 16th)

In 2007, Agol showed that any irreducible 3-manifold such that its fundamental groups is ‘virtually RFRS’ is virtually fibered. I will give a somewhat different proof using complexities of sutured manifolds. This is joint work with Takahiro Kitayama.

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Igor Rivin speaking at the Geometry and Dynamics seminar at Penn State

Igor will be speaking in the Geometry and Dynamics Seminar at Penn State on November 28th, on the topic of “counting curves on usually hyperbolic) surfaces.”

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Igor Rivin speaking at Penn State MASS Colloquium

Igor will be speaking at the MASS Colloquium on November 29th. Watch this space for more details

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Patricia Cahn’s talk at the Geometry/Topology seminar (October 9th, 4pm)

Title: Algebras counting intersections and self-intersections of curves

Abstract: Goldman and Turaev discovered a Lie bialgebra structure on the vector space generated by free homotopy classes of loops on an oriented surface. Goldman’s Lie bracket gives a lower bound on the minimum number of intersection points of two loops in two given free homotopy classes. Turaev’s Lie cobracket gives a lower bound on the minimum number of self-intersection points of a loop in a given free homotopy class. Chas showed that these bounds are not equalities in general. We show that for other operations, namely, the Andersen-Mattes-Reshetikhin Poisson bracket and a new operation μ, the corresponding bounds are always equalities. Some of this is joint work with Vladimir Chernov.

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