Title
Twisted isospectrality, homological wideness, and isometry : a sample of algebraic methods in isospectrality / Gunther Cornelissen, Norbert Peyerimhoff.
ISBN
9783031277047 (electronic bk.)
303127704X (electronic bk.)
9783031277030
3031277031
Published
Cham : Springer, 2023.
Language
English
Description
1 online resource (xvi, 111 pages) : illustrations.
Other Standard Identifiers
10.1007/978-3-031-27704-7 doi
Call Number
QA601
Dewey Decimal Classification
515/.723
Summary
The question of reconstructing a geometric shape from spectra of operators (such as the Laplace operator) is decades old and an active area of research in mathematics and mathematical physics. This book focusses on the case of compact Riemannian manifolds, and, in particular, the question whether one can find finitely many natural operators that determine whether two such manifolds are isometric (coverings). The methods outlined in the book fit into the tradition of the famous work of Sunada on the construction of isospectral, non-isometric manifolds, and thus do not focus on analytic techniques, but rather on algebraic methods: in particular, the analogy with constructions in number theory, methods from representation theory, and from algebraic topology. The main goal of the book is to present the construction of finitely many "twisted" Laplace operators whose spectrum determines covering equivalence of two Riemannian manifolds. The book has a leisure pace and presents details and examples that are hard to find in the literature, concerning: fiber products of manifolds and orbifolds, the distinction between the spectrum and the spectral zeta function for general operators, strong isospectrality, twisted Laplacians, the action of isometry groups on homology groups, monomial structures on group representations, geometric and group-theoretical realisation of coverings with wreath products as covering groups, and "class field theory" for manifolds. The book contains a wealth of worked examples and open problems. After perusing the book, the reader will have a comfortable working knowledge of the algebraic approach to isospectrality. This is an open access book.
Bibliography, etc. Note
Includes bibliographical references and index.
Access Note
Open access.
Source of Description
Online resource; title from PDF title page (SpringerLink, viewed May 18, 2023).
Series
SpringerBriefs in mathematics, 2191-8201
Available in Other Form
Print version: 9783031277030
Chapter. 1. Introduction
Part I: Leitfaden
Chapter. 2. Manifold and orbifold constructions
Chapter. 3. Spectra, group representations and twisted Laplacians
Chapter. 4. Detecting representation isomorphism through twisted spectra
Chapter. 5. Representations with a unique monomial structure
Chapter. 6. Construction of suitable covers and proof of the main theorem
Chapter. 7. Geometric construction of the covering manifold
Chapter. 8. Homological wideness
Chapter. 9. Examples of homologically wide actions
Chapter. 10. Homological wideness, "class field theory" for covers, and a number theoretical analogue
Chapter. 11. Examples concerning the main result
Chapter. 12. Length spectrum
References
Index.
