2025

July 31 (Thu), 11 am

KIAS 1423 & Zoom https://kimsh.kr/vz

George Domat (University of Michigan)

Surface Houghton Groups

I will introduce a type of "medium" mapping class groups: Surface Houghton groups. As the name suggests, these were first introduced as a surface analog of the classical Houghton groups and were shown to exhibit certain finiteness properties. They arise as asymptotically rigid subgroups of mapping class groups of infinite-type surfaces. They are finitely generated, but also still dense inside of big mapping class groups, hence "medium." This talk will mostly be a general introduction to these groups and motivations for their study. I will also talk on some past work relating to isomorphism and commensurability questions, and current work in progress relating these groups to square-integrable quasiconformal deformation spaces of Riemann surfaces, Weil-Petersson metrics, and three manifolds. All of this is joint work with Javier Aramayona and Chris Leininger.

August 5 (Tue), 11 am

KIAS 1423 & Zoom https://kimsh.kr/vz

Ryoo, Seung-Yeon (Caltech)

Sharpening the Assouad embedding theorem

The Assouad embedding theorem is a fundamental embedding result for a broad class of metric spaces, namely doubling metric spaces. Obtaining a sharp version of this embedding, with optimal distortion and target dimension, is a challenging open problem whose resolution is likely to involve new techniques applicable to metric dimension reduction. We sketch some partial results in this direction, along with a structural conjecture which would imply a sharp form of the Assouad embedding theorem. This is based on forthcoming joint works with Alan Chang and Hyun Chul Jang.

August 8 (Fri), 11 am

KIAS 1423 & Zoom https://kimsh.kr/vz

Jang, Seung-Uk (Rennes)

Dynamics of the Sturmian Trace Skew Product

In this talk, we focus on the spectrum of the discrete Schrödinger equation with a quasi-periodic potential called Sturmian potential. Eigenvector problem with a Sturmian potential is associated to a dynamics of the Markov surface, together with a control variable of "rotation angle," leading us to a study of a skew product system.

Our understanding is that this skew product system exhibits a sort of hyperbolicity. As a first step to establish it, we have shown that there exists a cone field on the Markov surface that contracts by the dynamics, which is independently defined by the angle variable. The discovery is more or less elementary, initiated by some geometric observations of the Markov dynamics. After sketching the tricks, we will announce some prospective after having a cone field, including the "holonomy" between Sturmian spectra.

This talk is based on a joint work with Anton Gorodetski and Victor Kleptsyn.

August 11 (Mon), 11 am

KIAS ??? & Zoom https://kimsh.kr/vz

Miri Son (Rice)