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.

‘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 Source: 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 Source: Mathematical Association of America