Applications of Modular Forms to Geometry and Interpolation Problems
Feigenbaum, Ahram Samuel
:
2019-10-23
Abstract
The sphere packing problem asks for the densest collection of non-overlapping con-
gruent spheres in Rn. In 2016, Viazovska proved that the E8 lattice is optimal for n = 8.
Subsequently, she with Cohn, Kumar, Miller, and Radchenko that showed the Leech lat-
tice was optimal for n = 24. Their proofs relied on the theory of weakly holomorphic
and quasi-modular forms to construct Fourier eigenfunctions with prescribed zeros at
distances in the E8 and Leech lattices. Similar ideas were applied by Radchenko and
Viazovska to obtain interpolation formulas for real Schwartz functions and by Cohn and
Gon¸calves to study uncertainty principles in harmonic analysis. In this thesis, we de-
velop a unified approach to the construction of such functions. We show that the weakly
holomorphic and weakly quasi-modular forms behind them are uniquely defined by the
conditions that they be eigenfunctions of the Fourier transform belonging to the Schwartz
class. We construct the Fourier eigenfunctions for all n divisible by 4. We also show an
extension of the interpolation formula given by Radchenko and Viazovska in R to radial
functions in R2 and R