AMS-MAA Invited Paper Session: Arithmetic of the Spheres Abstracts

Thursday, August 6, 1:00 PM - 3:50 PM, Marriott Wardman Park, Delaware A

This session deals with topics in number theory, geometry and dynamics related to Farey fractions, circle packings, and dynamical systems where mode locking appears.

William Abram, Hillsdale College
Alex Kontorovich, Rutgers University
Jeffrey Lagarias, University of Michigan

The Apollonian Structure of Imaginary Quadratic Fields

1:00 PM - 1:20 PM
Katherine Stange, University of Colorado Boulder

Let $$K$$ be an imaginary quadratic field with ring of integers $$OK$$. The Schmidt arrangement of $$K$$ is the orbit of the extended real line in the extended complex plane under the Bianchi group $$PSL(2,OK)$$ (realised as Mobius transformations). The arrangement takes the form of a dense collection of intricately nested circles. I'll explain how the number theory of $$K$$ influences the arrangement, and I'll use these arrangements to generalise Apollonian circle packings and define a new collection of thin groups of arithmetic interest.

Circles in the Sand

1:30 PM - 1:50 PM
Lionel Levine, Cornell University

I will describe the role played by an Apollonian circle packing in the scaling limit of the abelian sandpile model on the square grid $$\mathbb{Z}^2$$. The sandpile solves a certain integer optimization problem. Associated to each circle in the packing is a locally optimal solution to that problem. Each locally optimal solution can be described by an infinite periodic pattern of sand, and the patterns associated to any four mutually tangent circles obey an analogue of the Descartes Circle Theorem. Joint work with Wesley Pegden and Charles Smart.

Pythagoras Meets Euclid: A Euclidean Algorithm for Pythagorean Triples

2:00 PM - 2:20 PM
Dan Romik, University of California Davis

It was first discovered by Berggren in 1934 that primitive Pythagorean triples can be arranged in a ternary tree having the "fundamental" triple (3,4,5) at its root, in which each triple appears precisely once; thus to each triple there corresponds a word over a 3-letter alphabet encoding its position on the tree. I will discuss this curious phenomenon, which has at its heart a kind of Euclidean algorithm, explain how this algorithm can be used to define a dynamical system on the positive quadrant of the unit circle analogous to the Gauss continued fraction map, and mention possible extensions of these ideas to other diophantine equations.

Dynamics of Apollonian Circle Packings

2:30 PM - 2:50 PM
Elena Fuchs, University of Illinois Urbana-Champaign

Several years ago, Dan Romik constructed a dynamical system on the unit circle which allowed to prove various interesting theorems about Pythagorean triples. In this talk, we discuss a similar construction in a higher dimensional setting which yields results about Apollonian packings. This is joint work with Sneha Chaubey.

Variations on Apollonian Circle Packing Rules

3:00 PM - 3:20 PM
Steve Butler, Iowa State University

Apollonian circle packings are based on the simple rule that for any three given mutually tangent circles we can insert a unique circle in the space between them and tangent to all three of the original circles. We will see that there are in fact many possible rules that give rise to different properties and behaviors and that will in turn inspire different packings of spheres. By then examining the packings of spheres we will discover new and beautiful packings of circles (so from two dimensions to three dimensions and finally back to two dimensions).

Geometry and Number Theory of Integral Sphere Packings

3:30 PM - 3:50 PM
Kei Nakamura, University of California Davis

Classical Apollonian circle packings are constructed from a quadruple of pairwise tangent circles on a plane by successively inscribing circles into the triangular interstices. We consider variations of this construction, and study other circle packings and sphere packings of higher dimensions. Remarkably, just as in the classical Apollonian circle packings, there are a few circle/sphere packings in which the bends of constituent circles/spheres are integers, giving rise to fascinating questions on the Diophantine properties of the set of bends. We describe examples of integral circle/sphere packings in terms of hyperbolic geometry, Coxeter groups, and quadratic forms, and discuss the "local-global principle" for the set of bends that arise in this context.