Three Perspectives of Schiemann’s Theorem

Abstract

Interest in the field of spectral geometry, the study of how analytic and geometric properties of manifolds are related, was sparked when Marc Kac in 1966 asked the question “can one hear the shape of a drum?”. One of the problems that garnered attention because of this was whether the Laplace spectrum of a flat torus determines its shape, even though it was not new. The final answer to this question is due to Alexander Schiemann and it turns out to be yes if and only if the dimension of the flat torus is 3 or lower. His results are not widely known in today’s thriving spectral geometry community and there are two main reasons for this. Firstly, his published thesis and article are entirely number theoretical and never mention the related spectral geometry. Secondly, the thesis is written in german and the proof is quite technical. The reason why the spectral geometry of flat tori is particularly interesting is its connection to the geometry of lattices and the number theory of positive definite forms over the integers. In this thesis we aim to present this subject and its different perspectives. We especially focus on the details of Schiemann’s proof that ternary positive definite forms are determined by their representation numbers over the integers. Building on his techniques, we finally discuss some open problems and ideas for how to solve them.