Year: 2020
Type: dissertation
Abstract: Interest in the field of spectral geometry, the study of how analytic and geometric properties of\nmanifolds are related, was sparked when Marc Kac in 1966 asked the question “can one hear the\nshape of a drum?”. One of the problems that garnered attention because of this was whether the\nLaplace spectrum of a flat torus determines its shape, even though it was not new. The final\nanswer to this question is due to Alexander Schiemann and it turns out to be yes if and only if\nthe dimension of the flat torus is 3 or lower. His results are not widely known in today’s thriving\nspectral geometry community and there are two main reasons for this. Firstly, his published\nthesis and article are entirely number theoretical and never mention the related spectral geometry.\nSecondly, the thesis is written in german and the proof is quite technical.\nThe reason why the spectral geometry of flat tori is particularly interesting is its connection\nto the geometry of lattices and the number theory of positive definite forms over the integers.\nIn this thesis we aim to present this subject and its different perspectives. We especially focus\non the details of Schiemann’s proof that ternary positive definite forms are determined by their\nrepresentation numbers over the integers. Building on his techniques, we finally discuss some open\nproblems and ideas for how to solve them.
Author: Felix Rydell
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3081660880