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Twisted Isospectrality, Homological Wideness, and Isometry : A Sample of Algebraic Methods in Isospectrality

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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.

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Detail Information

Series Title

SpringerBriefs in Mathematics (BRIEFSMATH)

Call Number

510 COR t

Publisher

Cham : Springer Cham., 2023

Collation

XVI, 111

Language

English

ISBN/ISSN

978-3-031-27704-7

Classification

510

Content Type

text

Media Type

computer

Carrier Type

online resource

Edition

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Subject(s)

Global Analysis and Analysis on Manifolds

Specific Detail Info

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Statement of Responsibility

Gunther Cornelissen, Norbert Peyerimhoff

Other Information

Cataloger

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Source

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Validator

Maya

Digital Object Identifier (DOI)

https://doi.org/10.1007/978-3-031-27704-7

Journal Volume

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Journal Issue

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Subtitle

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Parallel Title

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