Donut choirs and Schiemann's symphony: An imaginative investigation of the isospectral problem for flat tori
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 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 16dimensional 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? Do you know the answer to this question? The isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We take this opportunity to explore this question in all three formulations and describe its resolution by Schiemann in the 1990s. We explain the different facets of this area; the number theory, the analysis, and the geometry that lie at the core of it and invite readers from all backgrounds to learn through exercises. Moreover, there are still a wide array of open problems that we share here. In the spirit of Mark Kac and John Horton Conway, we introduce a playful description of the mathematical objects, not only to convey the concepts but also to inspire the reader's imagination, as Kac and Conway have inspired us.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.09457
 Bibcode:
 2021arXiv211009457N
 Keywords:

 Mathematics  Spectral Theory;
 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 Mathematics  Number Theory;
 Primary 58C40;
 11H55;
 11H06;
 Secondary 11H50;
 11H71;
 94B05;
 11F11