nLab
hearing the shape of a drum

Contents

Contents

Idea

Physical oscillations (of elastic bodies) are described via a wave equation and hence by the spectrum of a Laplace operator on a Riemannian manifold. The phrase hearing the shape of a drum in this context refers to the issue of recovering a Riemannian manifold from just the spectrum of the Laplace operator on it (“spectral geometry”).

In plain Riemannian geometry there are counter-examples to the possibility of recovering Riemannian manifolds from just the spectrum of their Laplace operator. But from a little bit more of information, enhancing the Laplace operator to a spectral triple it becomes possible in general. This theorem due to Alain Connes is the motivation for the formulation of all of metric geometry via spectral triples, a program which is known as Connes’ noncommutative geometry.

References

Last revised on October 3, 2018 at 22:40:20. See the history of this page for a list of all contributions to it.