Topics In Spectral Geometry

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Topics in Spectral Geometry

Author : Michael Levitin,Dan Mangoubi,Iosif Polterovich
Publisher : American Mathematical Society
Page : 346 pages
File Size : 40,6 Mb
Release : 2023-11-30
Category : Mathematics
ISBN : 9781470475253

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Topics in Spectral Geometry by Michael Levitin,Dan Mangoubi,Iosif Polterovich Pdf

It is remarkable that various distinct physical phenomena, such as wave propagation, heat diffusion, electron movement in quantum mechanics, oscillations of fluid in a container, can be described using the same differential operator, the Laplacian. Spectral data (i.e., eigenvalues and eigenfunctions) of the Laplacian depend in a subtle way on the geometry of the underlying object, e.g., a Euclidean domain or a Riemannian manifold, on which the operator is defined. This dependence, or, rather, the interplay between the geometry and the spectrum, is the main subject of spectral geometry. Its roots can be traced to Ernst Chladni's experiments with vibrating plates, Lord Rayleigh's theory of sound, and Mark Kac's celebrated question “Can one hear the shape of a drum?” In the second half of the twentieth century spectral geometry emerged as a separate branch of geometric analysis. Nowadays it is a rapidly developing area of mathematics, with close connections to other fields, such as differential geometry, mathematical physics, partial differential equations, number theory, dynamical systems, and numerical analysis. This book can be used for a graduate or an advanced undergraduate course on spectral geometry, starting from the basics but at the same time covering some of the exciting recent developments which can be explained without too many prerequisites.

Spectral Geometry

Author : Pierre H. Bérard,Pierre H. Berard
Publisher : Lecture Notes in Mathematics
Page : 292 pages
File Size : 52,9 Mb
Release : 1986-08
Category : Mathematics
ISBN : STANFORD:36105032330461

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Spectral Geometry by Pierre H. Bérard,Pierre H. Berard Pdf

Geometric and Computational Spectral Theory

Author : Alexandre Girouard
Publisher : Unknown
Page : 298 pages
File Size : 54,6 Mb
Release : 2017
Category : Geometry, Differential
ISBN : 1470442582

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Geometric and Computational Spectral Theory by Alexandre Girouard Pdf

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.

Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture

Author : Peter B. Gilkey,John V Leahy,JeongHyeong Park
Publisher : CRC Press
Page : 294 pages
File Size : 43,5 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.

Spectral Operator Theory and Related Topics

Author : Vladimir Aleksandrovich Marchenko
Publisher : American Mathematical Soc.
Page : 300 pages
File Size : 47,6 Mb
Release : 1994
Category : Differential operators
ISBN : 082184122X

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Spectral Operator Theory and Related Topics by Vladimir Aleksandrovich Marchenko Pdf

"The collection contains the papers of mathematicians who are participants of the seminar on Mathematical Physics in Kharkov, Ukraine. The papers are mainly devoted to nontraditional problems of spectral theory, of disordered systems, to the spectral aspects of homogenization, and of properties of ergodic dynamical systems."--ABSTRACT.

Spectral Theory

Author : David Borthwick
Publisher : Springer Nature
Page : 339 pages
File Size : 42,6 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.

Geometric and Computational Spectral Theory

Author : Alexandre Girouard,Dmitry Jakobson,Michael Levitin,Nilima Nigam,Iosif Polterovich,Frédéric Rochon
Publisher : American Mathematical Soc.
Page : 284 pages
File Size : 45,7 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.

Spectral Geometry of Graphs

Author : Pavel Kurasov
Publisher : Springer Nature
Page : 644 pages
File Size : 55,8 Mb
Release : 2023-12-09
Category : Science
ISBN : 9783662678725

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Spectral Geometry of Graphs by Pavel Kurasov Pdf

This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph. The book has two central themes: the trace formula and inverse problems. The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book. To solve the inverse problem for Schrödinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions. The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.

Operators, Geometry and Quanta

Author : Dmitri Fursaev,Dmitri Vassilevich
Publisher : Springer Science & Business Media
Page : 294 pages
File Size : 45,7 Mb
Release : 2011-06-25
Category : Science
ISBN : 9789400702059

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Operators, Geometry and Quanta by Dmitri Fursaev,Dmitri Vassilevich Pdf

This book gives a detailed and self-contained introduction into the theory of spectral functions, with an emphasis on their applications to quantum field theory. All methods are illustrated with applications to specific physical problems from the forefront of current research, such as finite-temperature field theory, D-branes, quantum solitons and noncommutativity. In the first part of the book, necessary background information on differential geometry and quantization, including less standard material, is collected. The second part of the book contains a detailed description of main spectral functions and methods of their calculation. In the third part, the theory is applied to several examples (D-branes, quantum solitons, anomalies, noncommutativity). This book addresses advanced graduate students and researchers in mathematical physics with basic knowledge of quantum field theory and differential geometry. The aim is to prepare readers to use spectral functions in their own research, in particular in relation to heat kernels and zeta functions.

Discrete Geometric Analysis

Author : Motoko Kotani
Publisher : American Mathematical Soc.
Page : 274 pages
File Size : 48,6 Mb
Release : 2004
Category : Geometry, Differential
ISBN : 9780821833513

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Discrete Geometric Analysis by Motoko Kotani Pdf

This book is a collection of papers from the proceedings of the first symposium of the Japan Association for Mathematical Sciences. Topics covered center around problems of geometric analysis in relation to heat kernels, random walks, and Poisson boundaries on discrete groups, graphs, and other combinatorial objects. The material is suitable for graduate students and research mathematicians interested in heat kernels and random works on groups and graphs.

Spectral Geometry

Author : Pierre H. Berard
Publisher : Unknown
Page : 292 pages
File Size : 43,5 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662162350

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Spectral Geometry by Pierre H. Berard Pdf

Spectral Theory and Geometry

Author : E. Brian Davies,Yu Safarov,London Mathematical Society,International Centre for Mathematical Sciences
Publisher : Cambridge University Press
Page : 344 pages
File Size : 55,7 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.

Spectral Theory and Applications

Author : Alexandre Girouard
Publisher : Unknown
Page : 212 pages
File Size : 48,8 Mb
Release : 2018
Category : Spectral theory (Mathematics)
ISBN : 1470450194

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Spectral Theory and Applications by Alexandre Girouard Pdf

Geometry, Spectral Theory, Groups, and Dynamics

Author : Robert Brooks,Michael Entov,Yehuda Pinchover,Michah Sageev
Publisher : American Mathematical Soc.
Page : 275 pages
File Size : 51,7 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.

Shape Optimization and Spectral Theory

Author : Antoine Henrot
Publisher : De Gruyter Open
Page : 474 pages
File Size : 40,7 Mb
Release : 2017-05-08
Category : Electronic
ISBN : 3110550857

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Shape Optimization and Spectral Theory by Antoine Henrot Pdf

"Shape optimization and spectral theory" is a survey book aiming to give an overview of recent results in spectral geometry and its links with shape optimization. It covers most of the issues which are important for people working in PDE and differential geometry interested in sharp inequalities and qualitative behaviour for eigenvalues of the Laplacian with different kind of boundary conditions (Dirichlet, Robin and Steklov). This includes: existence of optimal shapes, their regularity, the case of special domains like triangles, isospectrality, quantitative form of the isoperimetric inequalities, optimal partitions, universal inequalities and numerical results. Much progress has been made in these extremum problems during the last ten years and this edited volume presents a valuable update to a wide community interested in these topics. List of contributors Antunes Pedro R.S., Ashbaugh Mark, Bonnaillie-Noel Virginie, Brasco Lorenzo, Bucur Dorin, Buttazzo Giuseppe, De Philippis Guido, Freitas Pedro, Girouard Alexandre, Helffer Bernard, Kennedy James, Lamboley Jimmy, Laugesen Richard S., Oudet Edouard, Pierre Michel, Polterovich Iosif, Siudeja Bartlomiej A., Velichkov Bozhidar