Mathematics Colloquia and Seminars

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In 1643, Ren\'{e} Descartes discovered a formula relating curvatures of circles in Apollonian circle packings, constructed by Apollonius of Perga in 200 BC. This formula has recently led to a connection between the construction of Apollonius and orbits of a certain so-called \emph{thin} subgroup $\Gamma$ of $\textrm{GL}_4(\mathbb Z)$. This connection is key in recent results on the arithmetic of Apollonian packings, which I will describe in this talk. A crucial ingredient in the proofs is the spectral gap coming from families of expander graphs associated to $\Gamma$ -- this gap is far less understood in the case of thin groups than that of non-thin groups. Motivated by this problem, I will then discuss the ubiquity of thin groups and present results on thinness of monodromy groups of hypergeometric equations in the case where these groups act on hyperbolic space.

Department Tea immediately following in the Alder Room.