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Twisted Isospectrality, Homological Wideness, and Isometry
Gunther Cornelissen and Norbert Peyerimhoff
2023
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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. This is an open access book.
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Keywords
- Algebraic topology
- Calculus & mathematical analysis
- Differential & Riemannian geometry
- finite group actions on homology
- finite group actions on manifolds
- Geometry
- Groups & group theory
- Mathematics
- Mathematics & science
- monomial representations
- number theory
- Numerical analysis
- Riemannian manifolds
- spectral zeta function
- Sunada theory
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBG Groups and group theory
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBH Number theory
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology::PBPD Algebraic topology
- Topology
- twisted Laplacian
- wreath products
Links
DOI: 10.1007/978-3-031-27704-7Editions