Department Mathematik
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Oberseminar Geometrie und Topologie

(run by Sebastian Hensel, Christian Lange, and Bernhard Leeb)

The seminar takes place on Thursdays, 16:15, virtually via zoom.
If you want to participate, please send a quick email to Sebastian Hensel to be added to the mailing list, and you will receive invitations and zoom links.

  • May 6th, 2021: Richard Webb (Manchester): How non-positively curved is the mapping class group?
    Abstract: In general, mapping class groups are not CAT(0) but one is still interested in finding non-proper yet cocompact actions on CAT(0) spaces. We will show that, in general, the arc complex admits no CAT(0) metric with finitely many shapes. In particular there is no finite-index subgroup of the mapping class group that preserves a CAT(0) metric on the arc complex. The hex theorem from combinatorics plays a role in the proof. The analogous statements are true for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. I will give background and motivation, and some connections of the work with other areas of algebra and topology.
  • May 27th, 2021: Alessandro Sisto (Heriot-Watt): What does a generic 3-manifold look like?
    Abstract: I will discuss two constructions of "random" 3-manifolds, namely Heegaard splittings and mapping tori where the gluing map is chosen using a random walk on a mapping class group. As it turns out, in both cases one obtains hyperbolic manifolds with asymptotic probability 1. I will give a brief overview of further geometric properties of these manifolds, and then focus on a deterministic result on Heegaard splittings, joint with Peter Feller and Gabriele Viaggi, that applies to random Heegaard splittings.
  • June 10th, 2021: Jose Andres Rodriguez Migueles (LMU) Volumes associated to periodic orbits of the geodesic flow on the Modular surface
    Abstract: Closed geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle. The isotopy class of any periodic orbit can be considered as a knot in a 3-manifold. The complement of those knots is always a hyperbolic 3-manifold, and hence has a well-defined volume. We show that there exist sequences of closed geodesics for which this volume is bounded linearly in terms of the period of the geodesic’s continued fraction expansion.
  • June 17th, 2021: Christian Lange (LMU)