What is an Apollonian Packing?

Abstract: Fix four mutually tangent circles in the plane. Fill in the spaces between these circles with additional tangent circles. By repeating this process ad infinitum, on smaller and smaller scales, we obtain an Apollonian circle packing. In this talk I will sketch a proof of Descartes' theorem on circle configurations, and introduce a group which acts on packings in two different ways, with a subtle duality between them. If time allows, I will also talk about my own recent work relating packings to Kac-Moody root systems. This connection is via a four-variable generating function for curvatures that appear in an Apollonian packing, which is essentially a character for a rank 4 indefinite Kac-Moody root system. I will discuss its domain of convergence, the Tits cone of the root system, which inherits the rich geometry of Apollonian packings.

algebraic geometrycombinatoricsdifferential geometrynumber theoryrepresentation theory

Audience: learners

What is ...? Seminar

Series comments: The ``What is ... ? Seminar'' (WiSe) is a weekly Zoom mathematics seminar. The goal is to explain interesting things to each other in a casual manner. The name is inspired by the What is ...? column of the Notices of the American Mathematical Society. See the website for more details. If you want to receive the zoom link for the talks please fill out the form linked on the website or contact Anna at a.puskas@uq.edu.au.