## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, August 1, 2019 — 3:30 PM EDT

**Arthur Baragar, University of Nevada **

“Apollonian packings and Enriques surfaces”

A hypersphere packing is an Apollonian packing if the spheres do not intersect except tangentially (externally), they fill the space up to a set of measure zero, and the packing includes a maximal configuration of mutually tangent hyperspheres. (There are other properties one should want, like ”lattice” based.) Apollonian hypersphere packings were thought to not exist in dimensions greater than three, owing to a Mathematical Review of one of David Boyd’s papers. It turns out, not only do they exist, but George Maxwell (Boyd’s colleague at UBC) described them in a follow up paper, but apparently was unaware that they are Apollonian. (They were among a large set of examples of sphere packings, most of which are not Apollonian.) Now, suppose X is an Enriques surface that contains a smooth rational curve but is otherwise generic. Let Λ be its Picard group modulo torsion. Then Λ is a lattice in a Lorentz space with intersection pairing of signature (1, 9), so naturally includes a model of nine dimensional hyperbolic space H9. Any smooth rational curve C on X satisfies C · C = −2, so represents a plane in H9, and hence a hypersphere in R8, thought of as the boundary of the Poincar ́e upper halfspace model of H9. In this way, the set of all smooth rational curves on X gives a configuration of hyperspheres in R8. That configuration is an Apollonian packing! Furthermore, it is related to the E8 lattice.

M3-3103

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.