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Twisted isospectrality, homological wideness, and isometrya sample of algebraic methods in isospectrality /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Twisted isospectrality, homological wideness, and isometryby Gunther Cornelissen, Norbert Peyerimhoff.
Reminder of title:	a sample of algebraic methods in isospectrality /
Author:	Cornelissen, Gunther.
other author:	Peyerimhoff, Norbert.
Published:	Cham :Springer International Publishing :2023.
Description:	xvi, 111 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Isometrics (Mathematics)
Online resource:	https://doi.org/10.1007/978-3-031-27704-7
ISBN:	9783031277047$q(electronic bk.)

Twisted isospectrality, homological wideness, and isometrya sample of algebraic methods in isospectrality /
Cornelissen, Gunther.

Twisted isospectrality, homological wideness, and isometrya sample of algebraic methods in isospectrality /[electronic resource] :by Gunther Cornelissen, Norbert Peyerimhoff. - Cham :Springer International Publishing :2023. - xvi, 111 p. :ill., digital ;24 cm. - SpringerBriefs in mathematics,2191-8201. - SpringerBriefs in mathematics..

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.

Open access.

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.

ISBN: 9783031277047$q(electronic bk.)

Standard No.: 10.1007/978-3-031-27704-7doiSubjects--Topical Terms:

574461
Isometrics (Mathematics)

LC Class. No.: QA601 / .C67 2023

Dewey Class. No.: 515.723

Twisted isospectrality, homological wideness, and isometrya sample of algebraic methods in isospectrality /
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505 0 $a 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.
506 $a Open access.
520 $a 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.
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650 2 4 $a Algebraic Topology. $3 273784
650 2 4 $a Differential Geometry. $3 273785
700 1 $a Peyerimhoff, Norbert. $3 973442
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