Philadelphia Area Number Theory Seminar

Wednesday, February 5, 2020 - 2:15pm

Ian Whitehead

Swarthmore College

Fix four mutually tangent circles; if you fill in the gaps between them with smaller tangent circles, and repeat the process infinitely, you will obtain an Apollonian circle packing. Moreover, if the four starting circles have integer curvatures, then every circle in the packing has an integer curvatures. The question of which integers appear as curvatures in a given packing has inspired exciting work in number theory over the last 20 years. In my talk, I will connect this question to the algebra of infinite root systems. I will describe the convergence domain of a generating function attached to each circle packing, which is also the Tits cone for a particular Kac-Moody root system. The geometry of this domain reflects the geometry of circle packings.