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 part, we will discuss counting and equidistribution results for
circles in Apollonian packings in fractal geometric terms and explain how
the dynamics of flows on infinite volume hyperbolic manifolds are related
(different parts based on joint works with Kontorovich, Shah and Lee).

A beautiful theorem of Descartes 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?’’ and “what kind of integers arise
as curvatures?”

In the second part, we will discuss progress on these questions which
are based on expanders and the affine sieve.