Spectral Theory In Riemannian Geometry

Author : E. Brian Davies,Yu Safarov,London Mathematical Society,International Centre for Mathematical Sciences
Publisher : Cambridge University Press
Page : 344 pages
File Size : 48,9 Mb
Release : 1999-09-30
Category : Mathematics
ISBN : 9780521777490

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Spectral Theory and Geometry by E. Brian Davies,Yu Safarov,London Mathematical Society,International Centre for Mathematical Sciences Pdf

This volume brings together lectures from an instructional meeting on spectral theory and geometry held under the auspices of the International Centre for Mathematical Sciences in Edinburgh. The contributions here come from world experts and many are much expanded versions of the lectures they gave. Together they survey the core material and go beyond to reach deeper results. For graduate students and experts alike, this book will be a highly useful resource.

Author : Olivier Lablée
Publisher : Erich Schmidt Verlag GmbH & Co. KG
Page : 204 pages
File Size : 48,8 Mb
Release : 2015
Category : Linear operators
ISBN : 3037191511

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Spectral Theory in Riemannian Geometry by Olivier Lablée Pdf

Spectral theory is a diverse area of mathematics that derives its motivations, goals, and impetus from several sources. In particular, the spectral theory of the Laplacian on a compact Riemannian manifold is a central object in differential geometry. From a physical point a view, the Laplacian on a compact Riemannian manifold is a fundamental linear operator which describes numerous propagation phenomena: heat propagation, wave propagation, quantum dynamics, etc. Moreover, the spectrum of the Laplacian contains vast information about the geometry of the manifold. This book gives a self-contained introduction to spectral geometry on compact Riemannian manifolds. Starting with an overview of spectral theory on Hilbert spaces, the book proceeds to a description of the basic notions in Riemannian geometry. Then its makes its way to topics of main interests in spectral geometry. The topics presented include direct and inverse problems. Direct problems are concerned with computing or finding properties on the eigenvalues while the main issue in inverse problems is knowing the spectrum of the Laplacian, can we determine the geometry of the manifold? Addressed to students or young researchers, the present book is a first introduction to spectral theory applied to geometry. For readers interested in pursuing the subject further, this book will provide a basis for understanding principles, concepts, and developments of spectral geometry.

Author : Pierre H. Berard
Publisher : Springer
Page : 284 pages
File Size : 42,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540409588

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Spectral Geometry by Pierre H. Berard Pdf

Author : Urakawa Hajime
Publisher : World Scientific
Page : 312 pages
File Size : 49,9 Mb
Release : 2017-06-02
Category : Mathematics
ISBN : 9789813109100

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Spectral Geometry Of The Laplacian: Spectral Analysis And Differential Geometry Of The Laplacian by Urakawa Hajime Pdf

The totality of the eigenvalues of the Laplacian of a compact Riemannian manifold is called the spectrum. We describe how the spectrum determines a Riemannian manifold. The continuity of the eigenvalue of the Laplacian, Cheeger and Yau's estimate of the first eigenvalue, the Lichnerowicz–Obata's theorem on the first eigenvalue, the Cheng's estimates of the kth eigenvalues, and Payne–Pólya–Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdière, that is, the spectrum determines the totality of all the lengths of closed geodesics is described. We give the V Guillemin and D Kazhdan's theorem which determines the Riemannian manifold of negative curvature.

Author : Werner Müller
Publisher : Springer
Page : 169 pages
File Size : 47,9 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540477624

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Manifolds with Cusps of Rank One by Werner Müller Pdf

The manifolds investigated in this monograph are generalizations of (XX)-rank one locally symmetric spaces. In the first part of the book the author develops spectral theory for the differential Laplacian operator associated to the so-called generalized Dirac operators on manifolds with cusps of rank one. This includes the case of spinor Laplacians on (XX)-rank one locally symmetric spaces. The time-dependent approach to scattering theory is taken to derive the main results about the spectral resolution of these operators. The second part of the book deals with the derivation of an index formula for generalized Dirac operators on manifolds with cusps of rank one. This index formula is used to prove a conjecture of Hirzebruch concerning the relation of signature defects of cusps of Hilbert modular varieties and special values of L-series. This book is intended for readers working in the field of automorphic forms and analysis on non-compact Riemannian manifolds, and assumes a knowledge of PDE, scattering theory and harmonic analysis on semisimple Lie groups.

Author : David Borthwick
Publisher : Springer Nature
Page : 339 pages
File Size : 52,9 Mb
Release : 2020-03-12
Category : Mathematics
ISBN : 9783030380021

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Spectral Theory by David Borthwick Pdf

This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.

Author : Robert Brooks,Michael Entov,Yehuda Pinchover,Michah Sageev
Publisher : American Mathematical Soc.
Page : 275 pages
File Size : 40,6 Mb
Release : 2005
Category : Mathematics
ISBN : 9780821837108

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Geometry, Spectral Theory, Groups, and Dynamics by Robert Brooks,Michael Entov,Yehuda Pinchover,Michah Sageev Pdf

This volume contains articles based on talks given at the Robert Brooks Memorial Conference on Geometry and Spectral Theory and the Workshop on Groups, Geometry and Dynamics held at Technion - the Israel Institute of Technology (Haifa). Robert Brooks' (1952-2002) broad range of mathematical interests is represented in the volume, which is devoted to various aspects of global analysis, spectral theory, the theory of Riemann surfaces, Riemannian and discrete geometry, and number theory. A survey of Brooks' work has been written by his close colleague, Peter Buser. Also included in the volume are articles on analytic topics, such as Szegos theorem, and on geometric topics, such as isoperimetric inequalities and symmetries of manifolds. The book is suitable for graduate students and researchers interested in various aspects of geometry and global analysis.

Author : Peter B. Gilkey,John V Leahy,JeongHyeong Park
Publisher : CRC Press
Page : 294 pages
File Size : 42,5 Mb
Release : 1999-07-27
Category : Mathematics
ISBN : 0849382777

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Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture by Peter B. Gilkey,John V Leahy,JeongHyeong Park Pdf

This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum of Dp on Y and Dp on Z, given that Dp is the p form valued Laplacian and pi: Z ® Y is a Riemannian submersion. After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure that the pull back of every eigenform on Y is an eigenform on Z so the eigenvalues do not change, then show that if a single eigensection is preserved, the eigenvalues do not change for the scalar or Bochner Laplacians. For the form valued Laplacian, they show that if an eigenform is preserved, then the corresponding eigenvalue can only increase. They generalize these results to the complex setting as well. However, the spinor setting is quite different. For a manifold with non-trivial boundary and imposed Neumann boundary conditions, the result is surprising-the eigenvalues can change. Although this is a relatively rare phenomenon, the authors give examples-a circle bundle or, more generally, a principal bundle with structure group G where the first cohomology group H1(G;R) is non trivial. They show similar results in the complex setting, show that eigenvalues can decrease in the spinor setting, and offer a list of unsolved problems in this area. Moving to some related topics involving questions of positive curvature, for the first time in mathematical literature the authors establish a link between the spectral geometry of Riemannian submersions and the Gromov-Lawson conjecture. Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture addresses a hot research area and promises to set a standard for the field. Researchers and applied mathematicians interested in mathematical physics and relativity will find this work both fascinating and important.

Author : Alex Barnett
Publisher : American Mathematical Soc.
Page : 354 pages
File Size : 49,8 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821853191

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Spectral Geometry by Alex Barnett Pdf

This volume contains the proceedings of the International Conference on Spectral Geometry, held July 19-23, 2010, at Dartmouth College, Dartmouth, New Hampshire. Eigenvalue problems involving the Laplace operator on manifolds have proven to be a consistently fertile area of geometric analysis with deep connections to number theory, physics, and applied mathematics. Key questions include the measures to which eigenfunctions of the Laplacian on a Riemannian manifold condense in the limit of large eigenvalue, and the extent to which the eigenvalues and eigenfunctions of a manifold encode its geometry. In this volume, research and expository articles, including those of the plenary speakers Peter Sarnak and Victor Guillemin, address the flurry of recent progress in such areas as quantum unique ergodicity, isospectrality, semiclassical measures, the geometry of nodal lines of eigenfunctions, methods of numerical computation, and spectra of quantum graphs. This volume also contains mini-courses on spectral theory for hyperbolic surfaces, semiclassical analysis, and orbifold spectral geometry that prepared the participants, especially graduate students and young researchers, for conference lectures.

Author : Steven Rosenberg
Publisher : Cambridge University Press
Page : 190 pages
File Size : 42,8 Mb
Release : 1997-01-09
Category : Mathematics
ISBN : 0521468310

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The Laplacian on a Riemannian Manifold by Steven Rosenberg Pdf

This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Author : Alexandre Girouard,Dmitry Jakobson,Michael Levitin,Nilima Nigam,Iosif Polterovich,Frédéric Rochon
Publisher : American Mathematical Soc.
Page : 284 pages
File Size : 40,8 Mb
Release : 2017-10-30
Category : Geometry, Differential
ISBN : 9781470426651

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Geometric and Computational Spectral Theory by Alexandre Girouard,Dmitry Jakobson,Michael Levitin,Nilima Nigam,Iosif Polterovich,Frédéric Rochon Pdf

A co-publication of the AMS and Centre de Recherches Mathématiques The book is a collection of lecture notes and survey papers based on the mini-courses given by leading experts at the 2015 Séminaire de Mathématiques Supérieures on Geometric and Computational Spectral Theory, held from June 15–26, 2015, at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. The volume covers a broad variety of topics in spectral theory, highlighting its connections to differential geometry, mathematical physics and numerical analysis, bringing together the theoretical and computational approaches to spectral theory, and emphasizing the interplay between the two.

Author : Peter Buser
Publisher : Springer Science & Business Media
Page : 456 pages
File Size : 42,6 Mb
Release : 2010-10-29
Category : Mathematics
ISBN : 9780817649920

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Geometry and Spectra of Compact Riemann Surfaces by Peter Buser Pdf

This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature –1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.

Author : Rongwei Yang
Publisher : Springer Nature
Page : 277 pages
File Size : 40,7 Mb
Release : 2024-06-16
Category : Electronic
ISBN : 9783031516054

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A Spectral Theory Of Noncommuting Operators by Rongwei Yang Pdf

Author : Marcel Berger
Publisher : Springer Science & Business Media
Page : 824 pages
File Size : 42,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642182457

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A Panoramic View of Riemannian Geometry by Marcel Berger Pdf

This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS

Author : Hiroshi Isozaki
Publisher : American Mathematical Soc.
Page : 243 pages
File Size : 41,9 Mb
Release : 2004
Category : Mathematics
ISBN : 9780821834213

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Inverse Problems and Spectral Theory by Hiroshi Isozaki Pdf

This volume grew out of a workshop on spectral theory of differential operators and inverse problems held at the Research Institute for Mathematical Sciences (Kyoto University). The gathering of nearly 100 participants at the conference suggests the increasing interest in this field of research. The focus of the book is on spectral theory for differential operators and related inverse problems. It includes selected topics from the following areas: electromagnetism, elasticity, the Schrodinger equation, differential geometry, and numerical analysis. The material is suitable for graduate students and researchers interested in inverse problems and their applications.