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A new proof of Viazovska's modular form inequalities for sphere packing in dimension 8


Speaker: Dan Romik, UC Davis
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Location: 2112 MSB
Start time: Tue, Nov 28 2023, 2:10PM

Maryna Viazovska in 2016 applied complex analysis to a fundamental problem in geometry, obtaining a solution to the sphere packing problem in dimension 8 through an explicit construction involving classical functions from the theory of modular forms. Viazovska showed that the function she constructed had the properties of a so-called "magic function" conjectured to exist by Cohn and Elkies in 2001, and therefore certified the correct (sharp) bound for the 8-dimensional sphere packing density. Her groundbreaking proof also led in short order to the solution to the sphere packing problem in dimension 24 by her and several collaborators.

After reviewing these remarkable developments, I will present a new proof of a pair of modular form inequalities that formed one component of Viazovska's sphere packing proof. Viazovska's original proof for the inequalities relied on computer calculations. The new proof involves only elementary arguments that can be easily checked by a human. The talk does not assume any prerequisite knowledge of modular forms.