Sphere packing problem for dimensions 8 and 24 solved
February 27, 2017 by Maths Statistics Science Tutor
Maryna Viazovska, during her time as a postdoctoral researcher at the Berlin Mathematical School and the Humboldt University of Berlin, solved the centuries-old "sphere packing" problem in dimensions 8 and 24 (the latter dimension in collaboration with other researchers).
Interest in dimensions 8 and 24 stem from the fact that in these dimensions, there exist symmetric sphere packings called the E8 (for the 8th dimension) and the Leech lattice (for the 24th dimension) that pack spheres with a higher average density than the best candidates known to mathematicians in other dimensions. The E8 and the Leech lattice also have connections to a wide range of mathematical subjects, including number theory, combinatorics, hyperbolic geometry and even areas of physics such as string theory.
The Ukrainian mathematician proved that the E8 and the Leech lattice are the best sphere packings in their respective dimensions.
The theorem proved by Maryna Viazovska is a higher-dimensional analogue of the original Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler. The Kepler Conjecture, which was settled in 1998 by Thomas Hales of the University of Pittsburgh using computer-assisted "proof by exhaustion", states that the densest way to pack equal-sized spheres in 3-dimensional Euclidean space is the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.
Viazovska's uses modular forms to find a correctly auxiliary function that can calculate the largest allowable sphere density. Her 2013 doctoral dissertation was also on modular forms, but she has additional expertise in discrete optimization, one of the fields that are central to the sphere-packing problem. Thus, a few years ago when Viazovska's friend Andrii Bondarenko, of the Norwegian University of Science and Technology in Trondheim, suggested that they work together on the 8-dimensional sphere-packing problem, Viazovska agreed. Later this effort become a solo endeavour, and after two years of intense work, Viazovska succeeded in coming up with the right auxiliary function for E8 and proving that it is correct.
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