Twisted Isospectrality, Homological Wideness, and Isometry
A Sample of Algebraic Methods in Isospectrality
Series: SpringerBriefs in Mathematics;
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Product details:
- Edition number 1st ed. 2023
- Publisher Springer
- Date of Publication 11 May 2023
- Number of Volumes 1 pieces, Book
- ISBN 9783031277030
- Binding Paperback
- No. of pages111 pages
- Size 235x155 mm
- Weight 209 g
- Language English
- Illustrations 1 Illustrations, black & white 511
Categories
Long 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 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.
?Most of the chapters contain some Open Problems or Projects. ... Most of the presented results are given with their complete proofs; this fact increases the value of the book and makes it an excellent scientific material for the researchers. ... one must remark once more the very well organization and extremely clarity of this SpringerBriefs monograph.? (Adela-Gabriela Mihai, zbMATH 1535.58001, 2024)
MoreTable of Contents:
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.
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Twisted Isospectrality, Homological Wideness, and Isometry: A Sample of Algebraic Methods in Isospectrality
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