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＜E-Book＞
Introduction to Isospectrality / by Alberto Arabia
(Universitext. ISSN:21916675)

Edition	1st ed. 2022.
Publisher	(Cham : Springer International Publishing : Imprint: Springer)
Year	2022
Language	English
Size	XI, 238 p. 154 illus., 142 illus. in color : online resource
Authors	*Arabia, Alberto author SpringerLink (Online service)
Subjects	LCSH:Global analysis (Mathematics) LCSH:Manifolds (Mathematics) LCSH:Graph theory LCSH:Mathematical physics FREE:Global Analysis and Analysis on Manifolds FREE:Graph Theory FREE:Mathematical Methods in Physics
Notes	1 Introduction -- 2 The Wave Equation on Flat Manifolds -- 3 The Sunada–Bérard–Buser Method -- 4 The Gordon–Webb–Wolpert Isospectral Domains -- A Linear Representations of Finite Groups and Almost-Conjugate Subgroups -- B The Laplacian as Isometry-Invariant Differential Operator -- C The Path-Distance on a Hausdorff Connected Flat Manifold -- D Group Quotients of Flat Manifolds -- References -- Glossary -- Index "Can one hear the shape of a drum?" This striking question, made famous by Mark Kac, conceals a precise mathematical problem, whose study led to sophisticated mathematics. This textbook presents the theory underlying the problem, for the first time in a form accessible to students. Specifically, this book provides a detailed presentation of Sunada's method and the construction of non-isometric yet isospectral drum membranes, as first discovered by Gordon–Webb–Wolpert. The book begins with an introductory chapter on Spectral Geometry, emphasizing isospectrality and providing a panoramic view (without proofs) of the Sunada–Bérard–Buser strategy. The rest of the book consists of three chapters. Chapter 2 gives an elementary treatment of flat surfaces and describes Buser's combinatorial method to construct a flat surface with a given group of isometries (a Buser surface). Chapter 3 proves the main isospectrality theorems and describes the transplantation technique on Buser surfaces. Chapter 4 builds Gordon–Webb–Wolpert domains from Buser surfaces and establishes their isospectrality. Richly illustrated and supported by four substantial appendices, this book is suitable for lecture courses to students having completed introductory graduate courses in algebra, analysis, differential geometry and topology. It also offers researchers an elegant, self-contained reference on the topic of isospectrality HTTP:URL=https://doi.org/10.1007/978-3-031-17123-9

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