# Asymptotically optimal shapes: drums and oscillators with least n-th frequency

by Richard Laugesen (University of Illinois)

What shape of domain minimizes the n-th eigenvalue (frequency) of the Laplacian, for large n? Antunes and Freitas discovered that among rectangular drums, the one minimizing the n-th frequency converges to a square as n tends to infinity – which suggests the disk might be the asymptotic minimizer among general domains. This "high eigenvalues" conjecture remains open. I will describe recent progress in 2-dimensions: my work with Ariturk and Liu on (shifted) lattice points, work of Marshall and Steinerberger related to the harmonic oscillator, and Larson’s advances on Riesz means. As time permits, I will survey directions for future work that might interest members of the audience.