rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Twisted Isospectrality, Homological Wideness, and Isometry
    sub title: A Sample of Algebraic Methods in Isospectrality
  name:
    name part: Cornelissen, Gunther.
    role:
      role term: creator
    role:
      role term: author.
    role:
      role term: aut
    role:
      role term: http://id.loc.gov/vocabulary/relators/aut
  name:
    name part: Peyerimhoff, Norbert.
    role:
      role term: author.
    role:
      role term: aut
    role:
      role term: http://id.loc.gov/vocabulary/relators/aut
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  genre: Electronic books.
  origin info:
    place:
      place term: sz
    date issued: 2023
    edition: 1st ed. 2023.
    issuance: monographic
  language:
    language term: eng
  physical description:
    reformatting quality: access
    form: remote
    extent: 1 online resource (XVI, 111 pages 1 illustrations)
  abstract: 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.
  table 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.
  note: by Gunther Cornelissen, Norbert Peyerimhoff.
  note: Open Access
  subject:
    topic: Global analysis (Mathematics)
  subject:
    topic: Manifolds (Mathematics)
  subject:
    topic: Number theory
  subject:
    topic: Group theory
  subject:
    topic: Algebraic topology
  subject:
    topic: Geometry, Differential
  subject:
    topic: Global Analysis and Analysis on Manifolds
  subject:
    topic: Manifolds and Cell Complexes
  subject:
    topic: Number Theory
  subject:
    topic: Group Theory and Generalizations
  subject:
    topic: Algebraic Topology
  subject:
    topic: Differential Geometry
  classification: QA614-614.97
  classification: 514.74
  related item:
    title info:
      title: Springer Nature eBook
  related item:
    title info:
      title: SpringerBriefs in Mathematics
  identifier: 9783031277047
  identifier: https://doi.org/10.1007/978-3-031-27704-7
  access condition: Open Access
  record info:
    record creation date: 230510
    record change date: 20230907142648.8
    record identifier: in00004737291