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Introduction to Isospectrality

  • Textbook
  • © 2022

Overview

Authors:
  1. Alberto Arabia

    1. CNRS, Institut de Mathématiques de Jussieu, Paris, France

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  • A self-contained account of the key contributions of Sunada, Buser, Bérard, Gordon, Webb and Wolpert
  • Provides a detailed construction of contractible, non-isometric isospectral surfaces
  • Includes 190 figures and illustrations, mostly in color

Part of the book series: Universitext (UTX)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xi
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  2. Chapter 1 Introduction

    • Alberto Arabia
    Pages 1-42

  3. Chapter 2 The Wave Equation on Flat Manifolds

    • Alberto Arabia
    Pages 43-66

  4. Chapter 3 The Sunada–Bérard–Buser Method

    • Alberto Arabia
    Pages 67-106

  5. Chapter 4 The Gordon–Webb–Wolpert Isospectral Domains

    • Alberto Arabia
    Pages 107-135

  6. Appendix A Linear Representations of Finite Groups and Almost-Conjugate Subgroups

    • Alberto Arabia
    Pages 137-164

  7. Appendix B The Laplacian as Isometry-Invariant Differential Operator

    • Alberto Arabia
    Pages 165-192

  8. Appendix C The Path-Distance on a Hausdorf Connected Flat Manifold

    • Alberto Arabia
    Pages 193-202

  9. Appendix D Group Quotients of Flat Manifolds

    • Alberto Arabia
    Pages 203-218

  10. Back Matter

    Pages 219-238
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Keywords

About this book

"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.

Reviews

“This book is accessible to third year university students, which describes a complete and elegant solution to a long-standing mathematical problem.” (Bo Liu, zbMATH 1505.58001, 2023)

Authors and Affiliations

  • CNRS, Institut de Mathématiques de Jussieu, Paris, France

    Alberto Arabia

About the author

Alberto Arabia is a specialist in cohomological theories, especially Equivariant Cohomology and p-adic Cohomology. His publications in Equivariant Cohomology include the book Equivariant Poincaré Duality on G-Manifolds (Lecture Notes in Mathematics 2288, Springer 2021), while in p-adic Cohomology he succeeded with Zoghman Mebkhout in the globalization of the Monsky–Washnitzer cohomology (2010). He has also conducted important research in the field of Configuration Spaces (Mémoires de la SMF 170, 2021).

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