The isospectral problem for flat tori from three perspectives

TL;DR

This paper explores the isospectral problem for flat tori from analytic, geometric, and number theoretic perspectives, highlighting historical context, key results, and open problems in understanding when non-isometric flat tori share the same Laplace spectrum.

Contribution

It provides a comprehensive overview of the isospectral problem for flat tori from three different mathematical perspectives and discusses the resolution and open questions in the field.

Findings

Milnor's example of 16-dimensional isospectral non-isometric tori

The minimal dimension for such examples is unknown but discussed

Schiemann's resolution of the problem in the 1990s

Abstract

Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed. In 1964, Milnor used a construction of Witt to find an example of isospectral non-isometric Riemannian manifolds, a striking and concise result that occupied one page in the Proceedings of the National Academy of Science of the USA. Milnor's example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues are identical, in spite of the fact that these tori are not isometric. A natural question is: what is the \em lowest \em dimension in which such isospectral non-isometric pairs exist? This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We explore this question from all three perspectives and describe its resolution by Schiemann in the 1990s. Moreover, we share a number of open problems.