Donut choirs and Schiemann's symphony

Magazine article, 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.  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?  This question can be equivalently formulated in analytic, geometric, and number theoretic language.  We will explore all three formulations and describe Schiemann's solution to this question, as well as some new results and open problems.  This talk is based on joint work with Erik Nilsson and Felix Rydell (PhD students at KTH, Stockholm).