Abstract

An Apollonian circle packing is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices of four mutually tangent circles, via an old theorem of Apollonius of Perga (262-190 BC). They give rise to one of first examples of a fractal in the plane. In the first lecture, we will discuss recent results on counting and distribution of circles in Apollonian packings in fractal geometric terms and explain how the dynamics of flows on infinite volume hyperbolic manifolds are related.

A beautiful theorem of Descartes observed in 1643 implies that if the initial four circles have integral curvatures, then all the circles in the packing have integral curvatures, as observed by Soddy, a Nobel laureate in Chemistry. This remarkable integrality feature gives rise to several natural Diophantine questions about integral Apollonian packings such as ``how many circles have prime curvatures''. In the second lecture, we will discuss progress on these questions while introducing recent developments on expanders and the affine sieve by Bourgain-Gamburd-Sarnak.