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Topics in Cyclic Theory - Daniel G. Quillen, Gordon Blower

Topics in Cyclic Theory

Buch | Hardcover
328 Seiten
2020
Cambridge University Press (Verlag)
978-1-108-47961-5 (ISBN)
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Noncommutative geometry combines themes from algebra, analysis and geometry and has many applications to physics. This book focuses on cyclic theory, containing background not found in published papers. It is intended for Ph.D. students in analysis and geometry, and researchers using K-theory, cyclic theory, differential geometry and index theory.
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.

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 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.

Introduction; 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; References; List of symbols; Index of notation; Subject index.

Erscheinungsdatum
Reihe/Serie London Mathematical Society Student Texts
Zusatzinfo Worked examples or Exercises
Verlagsort Cambridge
Sprache englisch
Maße 156 x 234 mm
Gewicht 590 g
Themenwelt Mathematik / Informatik Mathematik Geometrie / Topologie
Naturwissenschaften Physik / Astronomie
ISBN-10 1-108-47961-8 / 1108479618
ISBN-13 978-1-108-47961-5 / 9781108479615
Zustand Neuware
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