The question of how dense d-dimensional balls can be packed into d-dimensional space is a classical problem. Until now, only the solution for dimensions one, two and three were known, but Dr. Maryna Viazovska, the Berlin Mathematical School Dirichlet Postdoctoral Fellow, has now solved the hypersphere packing problem for dimension eight.

Previously, the packing of balls in this dimension with their centers lying on the E8 lattice resulted in covering no more than 25.367 percent of the space. In 2003, Henry Cohn and Noam Elkies reported that any hypersphere packing in dimension eight (whether the balls are lying on a lattice or not) can be at most 1/1000000 denser than the lattice packing based on E8. Using methods from number theory (modular forms), Viazovska has now pushed this boundary to zero.

Maryna has already published her results on arXiv, an open-access service managed by Cornell University Library, which allows scientists to share their research before it is formally published and officially recognized via a peer review process.

Maryna said recently that “this result became possible thanks to the postdoctoral fellowship and great working conditions at the BMS and at the HU”.

Born in the Ukraine, Maryna completed her PhD at the Max Planck Institute for Mathematics in Bonn in 2013 under the supervision of Prof. Don Zagier. In October of the same year, she became a visiting researcher at the Institut des Hautes Études Scientifiques in France before beginning her fellowship at the BMS in August 2014. Maryna is based at HU Berlin, where she is a member of Prof. Jürg Kramer's arithmetic geometry research group. Her research interests include number theory and discrete geometry.

Congratulations on your success, Maryna!


Source:
https://dmv.mathematik.de/index.php/forum/nachrichten/512-kugelpackungsproblem-in-dimension-8-geloest#586

Detailed article in English:
https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions/