Universally Optimal Periodic Configurations in the Plane

We develop lower bounds for the energy of configurations in ${\mathbb{R}}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial interpolation problem. We use this framework to show a scaling of the equitriangular lattice $A_2$ is universally optimal among all configurations of the form ${\omega }_4+A_2$ where ${\omega }_4$ is a 4-point configuration in ${\mathbb{R}}^2$. Likewise, we show a scaling and rotation of $A_2$ is universally optimal among all configurations of the form ${\omega }_6+L$ where ${\omega }_6$ is a 6-point configuration in ${\mathbb{R}}^2$ and $L=\mathbb{Z}\times \sqrt{3}\mathbb{Z}$.