Twisted Isospectrality, Homological Wideness, and Isometry
A Sample of Algebraic Methods in Isospectrality
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Beschreibung
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.
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
Autor*in
Gunther Cornelissen
Themen in »Twisted Isospectrality, Homological Wideness, and Isometry«
Riemannian manifolds twisted Laplacian Sunada theory spectral zeta function finite group actions on manifolds finite group actions on homology monomial representations wreath products Open Access
Stimmen zu »Twisted Isospectrality, Homological Wideness, and Isometry«
“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)
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