What shape of domain minimizes the n-th eigenvalue (frequency) of the Laplacian, for large n? Does the minimizer approach a disk as n tends to infinity? This “isoperimetric” conjecture is supported by a recent discovery of Antunes and Freitas: among rectangular drums of given area, the rectangle minimizing the n-th frequency converges to the square as n tends to infinity.

This result for rectangles is proved via lattice point counting in ellipses, similar to the Gauss circle problem. More generally, we aim to maximize the lattice point count in other convex regions, with respect to area-preserving stretches of the first quadrant. The special case of the p-circle (x^p + y^p = 1) leads to an open problem about right triangles.

[Joint with Shiya Liu, U. of Illinois, and Sinan Ariturk, Pontificia U. Catolica do Rio de Janeiro, Brazil.]