Fourier Interpolation and Extremal Problems

Introduction

In 2006, Cohn and Kumar conjectured that the A2 lattice is universally optimal, meaning that it has the lowest potential energy among all configurations of the same density for all completely monotone potentials. This conjecture has several very important corollaries.

Implications of the Conjecture

Among other consequences, it is known that it implies:

1. A positive solution to the 2D crystallization problem, a major unsolved problem coming from materials science.
2. A conjecture on the emergence of the triangular lattice of Abrikosov vortices in the Landau-Ginzburg theory of superconductivity.

Recent Developments

Recently, the 8 and 24-dimensional cases of the Cohn-Kumar conjecture have been positively resolved using novel interpolation formulas for radial Schwartz functions. This formula recovers a radial function from the data of it and its Fourier transform on a discrete set of radii, and its construction uses classical modular and quasi-modular forms.

Project Goals

In this project, we will prove a significant generalization of these interpolation formulas with a view towards applications in extremal problems in Fourier analysis.

Methodology

To prove these formulas, we will develop new analytic and numerical techniques for solving certain types of functional equations in one complex variable.

Conclusion

Finally, based on these proposed interpolation formulas, we will give a refinement of the Cohn-Kumar conjecture in dimension 2 and use it to attack the full conjecture in this case.