Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 236-241). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Oregon, 2005. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 88-90). Also available for download via the World Wide Web; free to University of Oregon users.

Spectral geometry is the field of mathematics which concerns relationships between geometric structures of manifolds and the spectra of canonical differential operators. Inverse Spectral Geometry in particular concerns the geometric information that can be recovered from the knowledge of such spectra. A deep link between inverse spectral geometry and sampling theory has recently been proposed. Specifically, it has been shown that the very shape of a Riemannian manifold can be discretely sampled and then reconstructed up to a cutoff scale. In the context of Quantum Gravity, this means that, in the presence of a physically motivated ultraviolet cuttoff, spacetime could be regarded as simultaneously continuous and discrete, in the sense that information can. In this thesis, we look into the properties of the Laplace-Beltrami operator on a compact Riemannian manifold with no boundary. We discuss the behaviour of its spectrum regarding a perturbation of the Riemannian structure. Specifically, we concern ourselves with infinitesimal inverse spectral geometry, the inverse spectral problem of locally determining the shape of a Riemannian manifold. We discuss the recenSpectral geometry is the field of mathematics which concerns relationships between geometric structures of manifolds and the spectra of canonical differential operators. Inverse Spectral Geometry in particular concerns the geometric information that can be recovered from the knowledge of such spectra. A deep link between inverse spectral geometry and sampling theory has recently been proposed. Specifically, it has been shown that the very shape of a Riemannian manifold can be discretely sampled and then reconstructed up to a cutoff scale. In the context of Quantum Gravity, this means that, in the presence of a physically motivated ultraviolet cuttoff, spacetime could be regarded as simultaneously continuous and discrete, in the sense that information can. In this thesis, we look into the properties of the Laplace-Beltrami operator on a compact Riemannian manifold with no boundary. We discuss the behaviour of its spectrum regarding a perturbation of the Riemannian structure. Specifically, we concern ourselves with infinitesimal inverse spectral geometry, the inverse spectral problem of locally determining the shape of a Riemannian manifold. We discuss the recently presented idea that, in the presence of a cutoff, a perturbation of a Riemannian manifold could be uniquely determined by the knowledge of the spectra of natural differential operators. We apply this idea to the specific problem of determining perturbations of the two dimensional flat torus through the knowledge of the spectrum of the Laplace-Beltrami operator.tly presented idea that, in the presence of a cutoff, a perturbation of a Riemannian manifold could be uniquely determined by the knowledge of the spectra of natural differential operators. We apply this idea to the specific problem of determining perturbations of the two dimensional flat torus through the knowledge of the spectrum of the Laplace-Beltrami operator.

No description available.

Spectral geometry is a mathematical discipline that studies the relationship between the geometry of Riemannian manifolds and the spectra of natural differential operators defined on them. The spectra of Laplacians are the ones most studied in this context. A sub-field of this discipline, called inverse spectral geometry, studies how much geometric information one can recover from such spectra. The motivation behind our study of inverse spectral geometry is a physical one. It has recently been proposed that inverse spectral geometry could be the missing mathematical link between quantum field theory and general relativity needed to unify those theories into a single theory of quantum gravity. Unfortunately, this proposed link is not well understood. Most of the efforts in inverse spectral geometry were historically concentrated on the generation of counterexamples to the most general formulation of inverse spectral geometry and the few positive results that exist are quite limited. In order to remedy to that, it has been proposed to linearize the problem, and study an infinitesimal version of inverse spectral geometry. In this thesis, I begin by reviewing the theory of pseudodifferential operators and using it to prove the spectral theorem for elliptic operators. I then define the commonly used Laplacians and survey positive and negative results in inverse spectral geometry. Afterwards, I briefly discuss a coordinate free reformulation of Riemannian geometry via the notion of spectral triple. Finally, I introduce a formulation of inverse spectral geometry adapted for numerical implementations and apply it to the inverse spectral geometry of a particular class of star-shaped domains in ℝ².

Spectral geometry is a mathematical discipline that studies the relationship between the geometry of Riemannian manifolds and the spectra of natural differential operators defined on them. The spectra of Laplacians are the ones most studied in this context. A sub-field of this discipline, called inverse spectral geometry, studies how much geometric information one can recover from such spectra. The motivation behind our study of inverse spectral geometry is a physical one. It has recently been proposed that inverse spectral geometry could be the missing mathematical link between quantum field theory and general relativity needed to unify those theories into a single theory of quantum gravity. Unfortunately, this proposed link is not well understood. Most of the efforts in inverse spectral geometry were historically concentrated on the generation of counterexamples to the most general formulation of inverse spectral geometry and the few positive results that exist are quite limited. In order to remedy to that, it has been proposed to linearize the problem, and study an infinitesimal version of inverse spectral geometry. In this thesis, I begin by reviewing the theory of pseudodifferential operators and using it to prove the spectral theorem for elliptic operators. I then define the commonly used Laplacians and survey positive and negative results in inverse spectral geometry. Afterwards, I briefly discuss a coordinate free reformulation of Riemannian geometry via the notion of spectral triple. Finally, I introduce a formulation of inverse spectral geometry adapted for numerical implementations and apply it to the inverse spectral geometry of a particular class of star-shaped domains in ℝ².

Spectral geometry is the field of mathematics which concerns relationships between geometric structures of manifolds and the spectra of canonical differential operators. Inverse Spectral Geometry in particular concerns the geometric information that can be recovered from the knowledge of such spectra. A deep link between inverse spectral geometry and sampling theory has recently been proposed. Specifically, it has been shown that the very shape of a Riemannian manifold can be discretely sampled and then reconstructed up to a cutoff scale. In the context of Quantum Gravity, this means that, in the presence of a physically motivated ultraviolet cuttoff, spacetime could be regarded as simultaneously continuous and discrete, in the sense that information can. In this thesis, we look into the properties of the Laplace-Beltrami operator on a compact Riemannian manifold with no boundary. We discuss the behaviour of its spectrum regarding a perturbation of the Riemannian structure. Specifically, we concern ourselves with infinitesimal inverse spectral geometry, the inverse spectral problem of locally determining the shape of a Riemannian manifold. We discuss the recenSpectral geometry is the field of mathematics which concerns relationships between geometric structures of manifolds and the spectra of canonical differential operators. Inverse Spectral Geometry in particular concerns the geometric information that can be recovered from the knowledge of such spectra. A deep link between inverse spectral geometry and sampling theory has recently been proposed. Specifically, it has been shown that the very shape of a Riemannian manifold can be discretely sampled and then reconstructed up to a cutoff scale. In the context of Quantum Gravity, this means that, in the presence of a physically motivated ultraviolet cuttoff, spacetime could be regarded as simultaneously continuous and discrete, in the sense that information can. In this thesis, we look into the properties of the Laplace-Beltrami operator on a compact Riemannian manifold with no boundary. We discuss the behaviour of its spectrum regarding a perturbation of the Riemannian structure. Specifically, we concern ourselves with infinitesimal inverse spectral geometry, the inverse spectral problem of locally determining the shape of a Riemannian manifold. We discuss the recently presented idea that, in the presence of a cutoff, a perturbation of a Riemannian manifold could be uniquely determined by the knowledge of the spectra of natural differential operators. We apply this idea to the specific problem of determining perturbations of the two dimensional flat torus through the knowledge of the spectrum of the Laplace-Beltrami operator.tly presented idea that, in the presence of a cutoff, a perturbation of a Riemannian manifold could be uniquely determined by the knowledge of the spectra of natural differential operators. We apply this idea to the specific problem of determining perturbations of the two dimensional flat torus through the knowledge of the spectrum of the Laplace-Beltrami operator.

by He Zhaokui. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 81-82). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminaries --- p.8 / Chapter 2.1 --- The Laplacian of a compact manifold --- p.8 / Chapter 2.2 --- The Laplacian of a graph --- p.9 / Chapter 2.3 --- Some basic facts about the eigenvalues of a graph --- p.13 / Chapter 3 --- Bound of the first non-zero eigenvalue in terms of Cheeger constant --- p.18 / Chapter 3.1 --- The Cheeger constant --- p.18 / Chapter 3.2 --- The Cheeger inequality of a compact manifold --- p.19 / Chapter 3.3 --- The Cheeger inequality of a graph --- p.23 / Chapter 4 --- Diameters and eigenvalues --- p.27 / Chapter 4.1 --- Some facts --- p.27 / Chapter 4.2 --- Estimate the eigenvalues of graphs --- p.29 / Chapter 4.3 --- The heat kernel of compact manifolds --- p.34 / Chapter 4.4 --- Estimate the eigenvalues of manifolds --- p.35 / Chapter 5 --- Harnack inequality and eigenvalues on homogeneous graphs --- p.40 / Chapter 5.1 --- Preliminaries --- p.40 / Chapter 5.2 --- The Neumann eigenvalue of a subgraph --- p.41 / Chapter 5.3 --- The Harnack inequality --- p.44 / Chapter 5.4 --- A lower bound of the first non-zero eigenvalue --- p.52 / Chapter 6 --- Harnack inequality and eigenvalues on compact man- ifolds --- p.54 / Chapter 6.1 --- Gradient estimate --- p.54 / Chapter 6.2 --- Lower bounds for the first non-zero eigenvalue --- p.59 / Chapter 7 --- Heat kernel and eigenvalues of graphs --- p.63 / Chapter 7.1 --- The heat kernel of a graph --- p.54 / Chapter 7.2 --- Lower bounds for eigenvalues --- p.70 / Chapter 8 --- Estimate the eigenvalues of a compact manifold --- p.73 / Chapter 8.1 --- An isoperimetric constant --- p.75 / Chapter 8.2 --- A lower estimate for the (m + l)-st eigenvalue --- p.77

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge - de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.

Le contexte est la géométrie discrète dans Zn. Il s'agit de décrire les courbes et surfaces discrètes composées de voxels: les définitions usuelles de droites et plans discrets épais se comportent mal quand on passe à des ensembles courbes. Comment garantir un bon comportement topologique, les connexités requises, dans une situation qui généralise les droites et plans discrets?Le calcul de données sur ces courbes, normales, tangentes, courbure, ou des fonctions plus générales, fait appel à des moyennes utilisant des masques. Une question est la pertinence théorique et pratique de ces masques. Une voie explorée, est le calcul de masques fondés sur la marche aléatoire. Une marche aléatoire partant d'un centre donné sur une courbe ou une surface discrète, permet d'affecter à chaque autre voxel un poids, le temps moyen de visite. Ce noyau permet de calculer des moyennes et par là, des dérivées. L'étude du comportement de ce processus de diffusion, a permis de retrouver des outils classiques de géométrie sur des surfaces maillées, et de fournir des estimateurs de tangente et de courbure performants. La diversité du champs d'applications de ce processus de diffusion a été mise en avant, retrouvant ainsi des méthodes classiques mais avec une base théorique identique.} motsclefs{Processus Markovien, Géométrie discrète, Estimateur tangentes, normales, courbure, Noyau de diffusion, Analyse d'images / The context of discrete geometry is in Zn. We propose to discribe discrete curves and surfaces composed of voxels: how to compute classical notions of analysis as tangent and normals ? Computation of data on discrete curves use average mask. A large amount of works proposed to study the pertinence of those masks. We propose to compute an average mask based on random walk. A random walk starting from a point of a curve or a surface, allow to give a weight, the time passed on each point. This kernel allow us to compute average and derivative. The studied of this digital process allow us to recover classical notions of geometry on meshes surfaces, and give accuracy estimator of tangent and curvature. We propose a large field of applications of this approach recovering classical tools using in transversal communauty of discrete geometry, with a same theorical base.