Geometry and Topology Seminar 2019-2020
|September 13, 10:00 AM in 901!||Alex Zupan (Texas)||Totally geodesic subgraphs of the pants graph||Kent|
|October 18||Jayadev Athreya (Illinois)||Gap Distributions and Homogeneous Dynamics||Kent|
|October 25||Joel Robbin (Wisconsin)||TBA||local|
|November 1||Anton Lukyanenko (Illinois)||Uniformly quasi-regular mappings on sub-Riemannian manifolds||Dymarz|
|November 8||Neil Hoffman (Melbourne)||Verified computations for hyperbolic 3-manifolds||Kent|
Alex Zupan (Texas)
Totally geodesic subgraphs of the pants graph
Abstract: For a compact surface S, the associated pants graph P(S) consists of vertices corresponding to pants decompositions of S and edges corresponding to elementary moves between pants decompositions. Motivated by the Weil-Petersson geometry of Teichmüller space, Aramayona, Parlier, and Shackleton conjecture that the full subgraph G of P(S) determined by fixing a multicurve is totally geodesic in P(S). We resolve this conjecture in the case that G is a product of Farey graphs. This is joint work with Sam Taylor.
Jayadev Athreya (Illinois)
Gap Distributions and Homogeneous Dynamics
Abstract: We discuss the notion of gap distributions of various lists of numbers in [0, 1], in particular focusing on those which are associated to certain low-dimensional dynamical systems. We show how to explicitly compute some examples using techniques of homogeneous dynamics, generalizing earlier work on gaps between Farey Fractions. This works gives some possible notions of `randomness' of special trajectories of billiards in polygons, and is based partly on joint works with J. Chaika, J. Chaika and S. Lelievre, and with Y.Cheung. This talk may also be of interest to number theorists.
Joel Robbin (Wisconsin)
Anton Lukyanenko (Illinois)
Uniformly quasi-regular mappings on sub-Riemannian manifolds
Abstract: A quasi-regular (QR) mapping between metric manifolds is a branched cover with bounded dilatation, e.g. f(z)=z^2. In a joint work with K. Fassler and K. Peltonen, we define QR mappings of sub-Riemannian manifolds and show that: 1) Every lens space admits a uniformly QR (UQR) mapping f. 2) Every UQR mapping leaves invariant a measurable conformal structure. The first result uses an explicit "conformal trap" construction, while the second builds on similar results by Sullivan-Tukia and a connection to higher-rank symmetric spaces.
Neil Hoffman (Melbourne)
Verified computations for hyperbolic 3-manifolds
Abstract: Given a triangulated 3-manifold M a natural question is: Does M admit a hyperbolic structure?
While this question can be answered in the negative if M is known to be reducible or toroidal, it is often difficult to establish a certificate of hyperbolicity, and so computer methods have developed for this purpose. In this talk, I will describe a new method to establish such a certificate via verified computation and compare the method to existing techniques.
This is joint work with Kazuhiro Ichihara, Masahide Kashiwagi, Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu.
|April 4||Matthew Kahle (Ohio)||TBA||Dymarz|
Matthew Kahle (Ohio)