Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond

We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schottky dances, etc.Comment: To appear in the Proceedings of ICM, 201

Apollonian circle packings: Dynamics and Number theory

We give an overview of various counting problems for Apollonian circle packings, which turn out to be related to problems in dynamics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the fall of 2013.Comment: To appear in Japanese Journal of Mat