Overview
- This book is open access, which means that you have free and unlimited access
- Offers a solid background on the theory of twisting Laplace operators on Riemannian manifolds
- Includes many examples and several open problems
Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)
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Table of contents (12 chapters)
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Front Matter
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Leitfaden
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Front Matter
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Back Matter
Keywords
About this book
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.
Authors and Affiliations
About the authors
Gunther Cornelissen holds the chair of geometry and number theory at Utrecht University. He earned his PhD from Ghent University in 1997 and has held visiting positions at institutions such as the Max Planck Institute for Mathematics in Bonn, Leuven University, California Institute of Technology and the University of Warwick. His research focuses on Arithmetic Geometry, particularly in positive characteristic, and also branches off into areas such as Spectral Geometry, Undecidability, Algebraic Dynamics and Graph Algorithms.
Norbert Peyerimhoff received his PhD in Mathematics in 1993 from the University of Augsburg. He held postdoctoral positions at the City University of New York, was an Assistant at the University of Basel and at the Ruhr University Bochum, before moving to Durham University (United Kingdom) in 2004. He has been a Professor of Geometry at Durham University since 2013, and his research interests include Differential Geometry, Discrete Geometry,Lie groups, Dynamical Systems, Spectral Theory and X-Ray Crystallography.
Bibliographic Information
Book Title: Twisted Isospectrality, Homological Wideness, and Isometry
Book Subtitle: A Sample of Algebraic Methods in Isospectrality
Authors: Gunther Cornelissen, Norbert Peyerimhoff
Series Title: SpringerBriefs in Mathematics
DOI: https://doi.org/10.1007/978-3-031-27704-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Author(s) 2023
Softcover ISBN: 978-3-031-27703-0Published: 11 May 2023
eBook ISBN: 978-3-031-27704-7Published: 10 May 2023
Series ISSN: 2191-8198
Series E-ISSN: 2191-8201
Edition Number: 1
Number of Pages: XVI, 111
Number of Illustrations: 1 b/w illustrations
Topics: Global Analysis and Analysis on Manifolds, Topology, Number Theory, Group Theory and Generalizations, Algebraic Topology, Differential Geometry