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Author Cornelissen, Gunther, 1971- author.

Title Twisted isospectrality, homological wideness, and isometry : a sample of algebraic methods in isospectrality / Gunther Cornelissen, Norbert Peyerimhoff.

Publication Info. Cham : Springer, 2023.


Location Call No. Status
 University of Saint Joseph: Pope Pius XII Library - Internet  WORLD WIDE WEB E-BOOK Springer    Downloadable
Please click here to access this Springer resource
Description 1 online resource (xvi, 111 pages) : illustrations.
Series SpringerBriefs in mathematics, 2191-8201
SpringerBriefs in mathematics, 2191-8201
Contents 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.
Access Open access. GW5XE
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 Includes bibliographical references and index.
Note Online resource; title from PDF title page (SpringerLink, viewed May 18, 2023).
Local Note Springer Nature Springer Nature - SpringerLink eBooks - Fully Open Access
Subject Isometrics (Mathematics)
Riemannian manifolds.
Isometrics (Mathematics)
Riemannian manifolds
Added Author Peyerimhoff, Norbert, 1964- author.
Other Form: Original 3031277031 9783031277030 (OCoLC)1369513602
ISBN 9783031277047 electronic book
303127704X electronic book
Standard No. 10.1007/978-3-031-27704-7 doi
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