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Number Theory Seminar

Connections between Integral Sphere Packings and Quaternion Algebras

Arseniy Sheydvasser (Yale)

Location:  Hill 525
Date & time: Tuesday, 17 April 2018 at 2:00PM - 3:00PM

We consider generalizations of the Apollonian circle packing known as crystallographic packings---specifically, we give the problem a number theoritic twist by requiring the bends of spheres to be integers, and we call packings satisfying this restriction integral. While it has been shown that there exist infinitely many integral crystallographic circle packings, the same question for higher dimensional packings is at present unknown. In this talk, we'll show how to use special sub-rings of rational quaternion algebras originally studied by Scharlau in the 1970s to construct examples of integral crystallographic sphere packings.