  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. This equation, foundational in physics, puzzled scholars for
decades due to its complexities when seeking a solution in an arbitrarily
shaped bounded domain. 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 listen to Riemannian manifolds,
deciphering the intricate interplay between eigenvalues and geometry. From
Weyl's law to Milnor's example of isospectral non-isometric flat tori, and Mark
Kac's influential work, the field has witnessed remarkable progress, uncovering
spectral invariants and advancing our understanding of geometric properties
discernible through eigenvalues. This article offers a comprehensive survey of
these developments, catering to both novices and experts alike. By elucidating
key proofs and exploring diverse research niches, we trace the evolution of
eigenvalue studies, culminating in a glimpse of the future directions this rich
field may take.
