2025

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), 2 pm

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), 10:30 am

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

Miri Son (Rice)

Classification of SL(n,R)-actions on closed manifolds and linearization of smooth SL(n,R)-actions

Recently, Fisher and Melnick classified SL(n,R)-actions on n-dimensional manifolds for n≥3. We generalize this result by classifying smooth or real-analytic SL(n,R)-actions on m-dimensional manifolds for 3≤n≤m≤2n-3. This work is motivated by the Zimmer program and is central to it, as Lie group actions restrict to their lattice actions. This classification relies on the linearization of SL(n,R)-actions when there is a global fixed point. The analytic case was proved by Guillemin--Sternberg and Kushinirenko, and we prove the smooth case jointly with Insung Park. Additionally, we discuss ongoing joint work with Insung Park studying linearizations of smooth SL(n,R)-actions with respect to their representations.