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# Mathematics > Spectral Theory

# Title: Donut choirs and Schiemann's symphony: An imaginative investigation of the isospectral problem for flat tori

(Submitted on 18 Oct 2021)

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 lowest dimension in which such isospectral non-isometric 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.

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