Topics in Cyclic Theory
Topics in Cyclic 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.
- 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
Reviews & endorsements
'The monograph is an excellent introduction to cyclic theory and an absolute must to any academic library, let alone a superb first-hand account and a selfless tribute to the late Daniel G. Quillen.' Igor V. Nikolaev, zbMATH
'These lectures reveal the breadth of Quillen's interests and the depth of the ideas developed. The lectures are clear and careful, rich in detail. The book is an opportunity to be lectured anew by this extraordinary mathematician.' John McCleary, Mathematical Association of America
Product details
July 2020Paperback
9781108790444
328 pages
227 × 152 × 20 mm
0.48kg
60 exercises
Available
Table of Contents
- 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.
