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Topics in Cyclic Theory


Part of London Mathematical Society Student Texts

  • Date Published: July 2020
  • availability: In stock
  • format: Paperback
  • isbn: 9781108790444

£ 33.99

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About the Authors
  • Noncommutative geometry combines themes from algebra, analysis and geometry and has significant applications to physics. This book focuses on cyclic theory, and is based upon the lecture courses by Daniel G. Quillen at the University of Oxford from 1988–92, which developed his own approach to the subject. The basic definitions, examples and exercises provided here allow non-specialists and students with a background in elementary functional analysis, commutative algebra and differential geometry to get to grips with the subject. Quillen's development of cyclic theory emphasizes analogies between commutative and noncommutative theories, in which he reinterpreted classical results of Hamiltonian mechanics, operator algebras and differential graded algebras into a new formalism. In this book, cyclic theory is developed from motivating examples and background towards general results. Themes covered are relevant to current research, including homomorphisms modulo powers of ideals, traces on noncommutative differential forms, quasi-free algebras and Chern characters on connections.

    • Provides background and motivation for topics in cyclic theory; readers can follow the motivating examples and analogies that led to the development of the theory
    • Presents examples and proofs in detail, making the subject accessible to students without specialist knowledge
    • Contains a simplified treatment of the topic, with results appearing in a useable form
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    Product details

    • Date Published: July 2020
    • format: Paperback
    • isbn: 9781108790444
    • length: 328 pages
    • dimensions: 227 x 152 x 20 mm
    • weight: 0.48kg
    • contains: 60 exercises
    • availability: In stock
  • Table of Contents

    1. Background results
    2. Cyclic cocycles and basic operators
    3. Algebras of operators
    4. GNS algebra
    5. Geometrical examples
    6. The algebra of noncommutative differential forms
    7. Hodge decomposition and the Karoubi operator
    8. Connections
    9. Cocycles for a commutative algebra over a manifold
    10. Cyclic cochains
    11. Cyclic cohomology
    12. Periodic cyclic homology
    List of symbols
    Index of notation
    Subject index.

  • Authors

    Daniel G. Quillen, University of Oxford
    Daniel G. Quillen proved Adam's conjecture in topological K-theory, and Serre's conjecture that all projective modules over a polynomial ring are free. He was awarded the Cole Prize in Algebra and the Fields Medal in 1978. He was Waynflete Professor of Pure Mathematics at the University of Oxford, where he lectured on K-theory and cyclic homology.

    Gordon Blower, Lancaster University
    Gordon Blower is Professor of Mathematical Analysis at Lancaster University, with interests in random matrices and applications of operator theory. He attended Quillen's lectures on cyclic theory when he was a junior researcher in Oxford.

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