[Binghamton Geometry Topology Seminar] Geometry and Topology seminar: Paul Apisa

[Cary Malkiewich]

Hi everyone,

This week we are pleased to have Paul Apisa (University of Wisconsin)
speaking about orbit closures in moduli space, title and abstract below.
This will be an in person talk, on Thursday at 2:50pm in WH 100E.

We will also have a lunch social, meet at 12pm just outside WH100E. See you
there!

Best,
Cary

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*Title:* Hurwitz Spaces, Hecke Actions, and Orbit Closures in Moduli Space
*Abstract:* The moduli space of Riemann surfaces is a space whose points
correspond to the ways to endow a surface with a hyperbolic metric or,
equivalently, complex structure. Geodesic flow on moduli space can be used
to generate an action of GL(2, R) on its cotangent bundle. While work of
Eskin, Mirzakhani, Mohammadi, and Filip implies that GL(2, R) orbit
closures are varieties, the question of which ones occur is wide open.
Aside from two well-understood constructions (taking loci of branched
covers and subloci of rank two orbit closures) there are only 3 known
families of orbit closures: the Bouw-Moller curves, the
Eskin-McMullen-Mukamel-Wright (EMMW) examples, and 2 sporadic examples.
Building on ideas of Delecroix-Rueth-Wright, I will describe work showing
that the Bouw-Moller and EMMW examples can be constructed using just the
representation theory of finite groups. The main idea is to connect these
examples to Hurwitz spaces of G-regular covers of the sphere for an
appropriate finite group G. In the end, I will describe a construction that
inputs a finite group G and a set of generators satisfying a combinatorial
condition and outputs a GL(2, R) orbit closure in moduli space.
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