# Colloquium

Apollonian circle packings

Hee Oh (Yale University)

3:00 PM - 4:00 PM

Hylan 1106A

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.

Event contact: hazel dot mcknight at rochester dot edu

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