Twisted Isospectrality, Homological Wideness, and Isometry
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 on...
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| Materialtyp: | Online |
| Språk: | engelska |
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Springer Nature
2023
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| Länkar: | ONIX_20230516_9783031277047_6 |
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| _version_ | 1869519643330215936 |
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| author | Cornelissen, Gunther Peyerimhoff, Norbert |
| author_browse | Cornelissen, Gunther Peyerimhoff, Norbert |
| author_facet | Cornelissen, Gunther Peyerimhoff, Norbert |
| author_sort | Cornelissen, Gunther |
| collection | Directory of Open Access Books |
| description | 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. |
| format | Online |
| id | doab-20.500.12854ir-100209 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2023 |
| publishDateRange | 2023 |
| publishDateSort | 2023 |
| publisher | Springer Nature |
| publisherStr | Springer Nature |
| record_format | ojs |
| spelling | doab-20.500.12854ir-1002092025-07-17T10:01:00Z Twisted Isospectrality, Homological Wideness, and Isometry Cornelissen, Gunther Peyerimhoff, Norbert Riemannian manifolds twisted Laplacian Sunada theory spectral zeta function finite group actions on manifolds finite group actions on homology monomial representations wreath products thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBH Number theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBG Groups and group theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology::PBPD Algebraic topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBH Number theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBG Groups and group theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology::PBPD Algebraic topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry 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. 2023-05-17T04:02:16Z 2023-05-17T04:02:16Z 2023-05-16T15:05:23Z 2023 book ONIX_20230516_9783031277047_6 https://library.oapen.org/handle/20.500.12657/62957 9783031277047 9783031277030 https://directory.doabooks.org/handle/20.500.12854/100209 eng SpringerBriefs in Mathematics open access image/jpeg image/jpeg n/a n/a https://library.oapen.org/bitstream/20.500.12657/62957/1/978-3-031-27704-7.pdf https://library.oapen.org/bitstream/20.500.12657/62957/1/978-3-031-27704-7.pdf Springer Nature Springer International Publishing 10.1007/978-3-031-27704-7 10.1007/978-3-031-27704-7 9fa3421d-f917-4153-b9ab-fc337c396b5a Nederlandse Organisatie voor Wetenschappelijk Onderzoek da087c60-8432-4f58-b2dd-747fc1a60025 9783031277047 9783031277030 Dutch Research Council (NWO) Springer International Publishing 111 Cham [...] open access |
| spellingShingle | Riemannian manifolds twisted Laplacian Sunada theory spectral zeta function finite group actions on manifolds finite group actions on homology monomial representations wreath products thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBH Number theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBG Groups and group theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology::PBPD Algebraic topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBH Number theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBG Groups and group theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology::PBPD Algebraic topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry Cornelissen, Gunther Peyerimhoff, Norbert Twisted Isospectrality, Homological Wideness, and Isometry |
| title | Twisted Isospectrality, Homological Wideness, and Isometry |
| title_full | Twisted Isospectrality, Homological Wideness, and Isometry |
| title_fullStr | Twisted Isospectrality, Homological Wideness, and Isometry |
| title_full_unstemmed | Twisted Isospectrality, Homological Wideness, and Isometry |
| title_short | Twisted Isospectrality, Homological Wideness, and Isometry |
| title_sort | twisted isospectrality homological wideness and isometry |
| topic | Riemannian manifolds twisted Laplacian Sunada theory spectral zeta function finite group actions on manifolds finite group actions on homology monomial representations wreath products thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBH Number theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBG Groups and group theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology::PBPD Algebraic topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBH Number theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBG Groups and group theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology::PBPD Algebraic topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry |
| topic_facet | Riemannian manifolds twisted Laplacian Sunada theory spectral zeta function finite group actions on manifolds finite group actions on homology monomial representations wreath products thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBH Number theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBG Groups and group theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology::PBPD Algebraic topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBH Number theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBG Groups and group theory thema EDItEUR::P Mathematics and Science::PB Mathematics::PBP Topology::PBPD Algebraic topology thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry |
| url | ONIX_20230516_9783031277047_6 |
| work_keys_str_mv | AT cornelissengunther twistedisospectralityhomologicalwidenessandisometry AT peyerimhoffnorbert twistedisospectralityhomologicalwidenessandisometry |