112 years of listening to Riemannian manifolds

TL;DR

This paper reviews 112 years of research on the relationship between eigenvalues of the Laplace operator and the geometry of Riemannian manifolds, highlighting key developments and open questions.

Contribution

It provides a comprehensive overview of the historical and mathematical progress in understanding how spectral data relates to geometric properties of manifolds.

Findings

Weyl's law linking eigenvalues and volume

Milnor's example of isospectral non-isometric tori

Kac's question about hearing the shape of a drum

Abstract

In 1910, Hendrik Antoon Lorentz delved into the enigmatic Laplace eigenvalue equation, also known as the Helmholtz equation, pondering to what extent the geometry in which one solves the equation can be recovered from knowledge of the eigenvalues. Lorentz, inspired by physical and musical analogies, conjectured a fundamental relationship between eigenvalues, domain volume, and dimensionality. While his conjecture initially seemed insurmountable, Hermann Weyl's groundbreaking proof in 1912 illuminated the deep connection between eigenvalues and geometric properties. Over the ensuing 112 years, mathematicians and physicists have continued to decipher the intricate interplay between eigenvalues and geometry. From Weyl's law to Milnor's example of isospectral non-isometric flat tori, and Kac's inspiring question about hearing the shape of a drum, the field has witnessed remarkable progress, uncovering spectral invariants and advancing our understanding of geometric properties discernible through eigenvalues. We present an overview of this field amenable to both physicists and mathematicians.