[PAST EVENT] Mathematics Colloquium: Ian Whitehead (Macalester College)
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Mathematics Colloquium: Ian Whitehead (Macalester College)
Title: Apollonian Packings and Number Theory
Abstract: Start with four circles, all tangent to one another; then fill in the gaps between them with additional tangent circles. If you keep filling the gaps with smaller and smaller circles, you will generate an Apollonian circle packing. This picture has a rich history, from ancient Greece to Ren? Descartes, to Japanese temple geometry. Amazingly, if you start with four circles whose curvatures are integers, then all the circles in the packing have this property. In my talk I'll describe some of the number theory that's been inspired by Apollonian packings in the last 20 years: theorems and conjectures about the growth of curvatures in a packing, and which integers can appear as curvatures. I will finish with my own work on the domain of multivariable power series defined by Apollonian packings.