The isospectral problem for flat tori from three perspectives

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 nonisometric 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 lowest dimension in which such isospectral nonisometric 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.