Spectral geometry

Summary

Spectral geometry studies how the geometry of a Riemannian manifold relates to the spectra of natural differential operators on it. The main such operator is the Laplace–Beltrami operator. Direct problems bound or compute eigenvalues from geometric data such as curvature, diameter and volume. Inverse problems ask how much of the geometry is determined by the spectrum, as in the question of hearing the shape of a drum. Its tools include the heat kernel, Weyl's law and trace formulas.