Renormalization on circle packings

Group Actions, Geometry and Dynamics
Event time: 
Monday, April 1, 2024 - 4:00pm
Yusheng Luo
Speaker affiliation: 
Cornell University
Event description: 
Circle packings have many applications in geometry, analysis and dynamics. The combinatorics of a circle packing is captured by the contact graph, called the nerve of the circle packing. It is natural and important to understand
1. Given a graph G, when is it isomorphic to the nerve of a circle packing?
2. Is the circle packing rigid? Or more generally, what is the moduli space of circle packings with nerve isomorphic to G?
3, How are different circle packings with isomorphic nerves related?
For finite graphs, Kobe-Andreev-Thurston’s circle packing theorem give a complete answer to the above questions. The situation is more complicated for infinite graphs, and has been extensively studied for locally finite triangulations.
In this talk, I will describe how to use renormalization theory to study these questions for infinite graphs. In particular, I will explain how it gives complete answers to the above questions for graphs with subdivision rules.
I will also discuss some applications on quasiconformal geometries for dynamical gasket sets.
This is based on some joint works with Y. Zhang, D. Ntalampekos.