Time：14:00-16:00, Wednesday, July 8 2026
Venue：E14-116
Speaker：Yusheng Luo, Cornell University
Title：Rigidity and universality in the geometry of limit sets
Abstract: Limit sets are fractal objects that arise naturally in geometry and dynamics, for example from Kleinian groups and from iterating rational maps on the Riemann sphere. A basic question is: how much of the geometry of such a fractal is determined by its topology?
In this talk, I will discuss this question through quasisymmetric geometry, a flexible notion of equivalence well suited to comparing fractals. Two sharply different phenomena appear. In the rigid regime, such as Sierpinski carpet limit sets, the geometry of a limit set remembers the dynamics that produced it. In the universality regime, such as topological circles, topology alone determines the quasisymmetric geometry, and many apparently different dynamical systems give rise to the same geometric object.
I will explain joint work with Haissinsky giving a complete classification for geometrically finite Kleinian groups. I will then place this in a broader context, including recent joint work with Mj and Mukherjee on the universality of the Basilica and McMullen’s work connecting the question mark function, conformal welding, and cusped Julia sets. These examples suggest a broader program for classifying limit sets across conformal dynamics.
