Approximation of Higher Degree Spectra Results for Twisted Laplace Operators

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dc.contributor.advisorLeistner, Thomas
dc.contributor.advisorHochs, Peter
dc.contributor.advisorBuchdahl, Nicholas
dc.contributor.authorFresacher, Matthias Eduard
dc.contributor.schoolSchool of Mathematical Sciencesen
dc.date.issued2021
dc.description.abstractThis thesis examines the eigenvalues of the connection Laplacian acting on differential forms with values in a Hermitian vector bundle with connection over a closed Riemannian manifold. Specifically, building upon previous work by Whitney, Dodziuk, Patodi and Zahariev, a combinatorial analogue of the connection Laplacian is defined via triangulations of the manifold whereby differential forms are associated to cochains. Using the min-max principle as a key ingredient, this reduces the infinite dimensional analytic eigenvalue problem to a finite dimensional combinatorial one. In theory, this allows the eigenvalues to be calculated with numerical methods and sufficient computational power. In this thesis, I prove that the eigenvalues of the analytic Laplacian are bounded below by the eigenvalues of the combinatorial Laplacian for differential forms and cochains of arbitrary degree with values in a trivial complex line bundle provided an assumption is met. This is achieved via an explicit calculation of the growth rate of the Whitney map under standard subdivisions.en
dc.description.dissertationThesis (MPhil) -- University of Adelaide, School of Mathematical Sciences, 2021en
dc.identifier.urihttps://hdl.handle.net/2440/133623
dc.language.isoenen
dc.provenanceThis electronic version is made publicly available by the University of Adelaide in accordance with its open access policy for student theses. Copyright in this thesis remains with the author. This thesis may incorporate third party material which has been used by the author pursuant to Fair Dealing exceptions. If you are the owner of any included third party copyright material you wish to be removed from this electronic version, please complete the take down form located at: http://www.adelaide.edu.au/legalsen
dc.subjectLaplace Operatoren
dc.subjectCombinatorial Approximationen
dc.subjectRiemannian Geometryen
dc.titleApproximation of Higher Degree Spectra Results for Twisted Laplace Operatorsen
dc.typeThesisen

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