Applications of Modular Forms to Geometry and Interpolation Problems

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