We develop the notion of a Kleinian Sphere Packing, a generalization of “crystallographic”(
Apollonian-like) sphere packings defined in [A. Kontorovich and K. Nakamura, Geometry …

In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one
of six or eight residue classes modulo 24. The local-global conjecture states that every …

We introduce the notion of a “crystallographic sphere packing,” defined to be one whose
limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension …

In the last decade, it has become increasingly possible for researchers to augment their
experience of abstract mathematics with sensory exploration: 3D-printed or CNC-milled …

For a real projective variety X, the cone \Sigma_X of sums of squares of linear forms plays a
fundamental role in real algebraic geometry. The dual cone \Sigma_X^* is a spectrahedron …

We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral
Apollonian gaskets, Invent. Math. 196 (2014), 589–650] and Zhang [On the local-global …

We study the orbit of under the Bianchi group, where is an imaginary quadratic field. The
orbit, called a Schmidt arrangement, is a geometric realization, as an intricate circle packing …

We explore a three-dimensional counterpart of the Farey tessellation and its relations to
Penner's lambda lengths and $ SL_2 $-tilings. In particular, we prove a three-dimensional …

We introduce an analog of Bianchi groups for rational quaternion algebras and use it to
construct sphere packings that are analogs of the Apollonian circle packing known as …

These notes cover and expand upon the material for two summer schools: The first, which
was held at CIRM, Marseille, France, July 10-14, 2023, as part of" Renormalization and …