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Novikov-Shubin Invariants of Nilpotent Lie Groups

Relations to Random Walks and Fibre Bundles

dc.contributor.advisorSchick, Thomas Prof. Dr.
dc.contributor.authorHöpfner, Tim Martin
dc.date.accessioned2023-05-25T06:44:51Z
dc.date.available2023-06-01T00:50:12Z
dc.date.issued2023-05-25
dc.identifier.urihttp://resolver.sub.uni-goettingen.de/purl?ediss-11858/14681
dc.identifier.urihttp://dx.doi.org/10.53846/goediss-9907
dc.format.extentXXX Seitende
dc.language.isodeude
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/
dc.subject.ddc510de
dc.titleNovikov-Shubin Invariants of Nilpotent Lie Groupsde
dc.title.alternativeRelations to Random Walks and Fibre Bundlesde
dc.typedoctoralThesisde
dc.contributor.refereeSchick, Thomas Prof. Dr.
dc.date.examination2023-05-02de
dc.description.abstractengNovikov-Shubin invariants are so-called L2-invariants of non-compact manifolds. They are defined using the Laplace operators and measure the density of their spectra near zero. This near-zero part of the spectrum is particularly interesting, as it gives us a lot of information about the Laplace operator itself, for example, whether it is invertible. It turns out that it also can be used to describe the long-term asymptotic behaviour of solutions to the heat equation. While these invariants are very interesting, in general, they are rather hard to compute. In this thesis, we approach the computation of Novikov-Shubin invariants from different angles: Firstly, we show that for suitable CW complexes, they can be understood as quantities arising from stochastic processes, described explicitly in terms of the CW structure. This gives us a new potential way of computing Novikov-Shubin invariants by understanding these stochastic processes better. Secondly, we take a closer look at an approach by M. Rumin that allows us to estimate Novikov-Shubin invariants of nilpotent Lie groups in some cases. We compute these estimates for low-dimensional nilpotent Lie groups up to dimension six with the help of a Python program and give some further remarks on this approach. Thirdly, we develop a more detailed approach to Novikov-Shubin invariants on fibre bundles. This leads to the definition of a new, two-parameter version of Novikov-Shubin invariants. The goal is to better understand how the fibre and the basis contribute separately to the Novikov-Shubin invariants of the total space. We show that these newly defined numbers satisfy multiple suitable invariance properties and compute them in an explicit example of the three-dimensional Heisenberg group.de
dc.contributor.coRefereeMeyer, Ralf Prof. Dr.
dc.subject.engL2-invariantsde
dc.subject.engNovikov-Shubin invariantsde
dc.subject.engrandom walksde
dc.subject.engheat decayde
dc.subject.engnilpotent Lie groupsde
dc.subject.engfibre bundlesde
dc.subject.engfiber bundlesde
dc.identifier.urnurn:nbn:de:gbv:7-ediss-14681-8
dc.affiliation.instituteFakultät für Mathematik und Informatikde
dc.subject.gokfullMathematics (PPN61756535X)de
dc.description.embargoed2023-06-01de
dc.identifier.ppn1846808677
dc.notes.confirmationsentConfirmation sent 2023-05-25T06:45:02de


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