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ICTP Lecture Notes Series, Volume 15

(ISBN 92-95003-21-7) - *September 2003*

## Contemporary Developments in Algebraic K-Theory

**(Dedicated to H. Bass on the occasion of his 70th birthday)**

Editors:

*M. Karoubi*(Univ. Paris 7, France),

*A.O. Kuku*(ICTP, Italy) and

*C. Pedrini*(Univ. Genova, Italy)

**Introduction**

The School and Conference on Algebraic K-theory which took place at ICTP July 8-26, 2002 was a follow-up to the earlier one in 1997, and like its predecessor, the 2002 meeting endeavoured to emphasise the multidisciplinary aspects of the subject. However, one special feature of the 2002 School and Conference is that the whole activity was dedicated to H. Bass, one of the founders of Algebraic K-theory, on the occasion of his seventieth birthday.

The School during the first two weeks, July 8 to 19 was devoted to expository lectures meant to explore and highlight connections between K-theory and several other areas of mathematics - Algebraic Topology, Number theory, Algebraic Geometry, Representation theory, and Non-commutative Geometry. This volume, constituting the Proceedings of the School, is dedicated to H. Bass. The Proceedings of the Conference during the last week July 22 - 26, which will appear in Special issues of K-theory, is also dedicated to H. Bass. The Proceedings of the 1997 School and Conference was published by 'World Scientific' in 1999.

The opening contribution by M. Karoubi to this volume consists of a comprehensive survey of developments in K-theory in the last forty-five years, and covers a very broad spectrum of the subject, including Topological K-theory, Atiyah-Singer index theorem, K-theory of Banach algebras, Higher Algebraic K-theory, Cyclic Homology etc.

J. Berrick's contribution on 'Algebraic K-theory and Algebraic Topology' treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers.

The contributions by M. Kolster titled 'K-theory and Arithmetics' includes such topics as values of zeta functions and relations to K-theory, K-theory of integers in number fields and associated conjectures, Etale cohomology, Iwasawa theory etc.

A.O. Kuku's contributions on 'K-theory and Representation theory' includes K-theory of orders, group-rings and modules over EI categories, Equivariant Higher Algebraic K-theory for finite, profinite and compact Lie group actions together with their relative generalisations and applications.

Topics covered under F. Morel's 'Introduction to *A ^{1}* homotopy theory' include Simplicial sheaves, Quillen's homotopical algebra, Unstable

*A*homotopy theory, Connectivity and

^{1}*A*-localisation, Stable

^{1}*A*homotopy theory of

^{1}*S*-spectra and

^{1}*P*-spectra, etc.

^{1}The contribution by N. Higson titled 'Local index formula in Non-commutative Geometry' includes such topics as Elliptic partial differential operators, cyclic homology theory, Chern characters, homotopy invariants and the index formula.

ACKNOWLEDGEMENT. It is our great pleasure to thank all those who contributed to the success of the School and Conference. Firstly, we are very grateful to ICTP for financial support and the provision of excellent facilities and efficient staff. Furthermore, we thank the staff in the Mathematics Section, especially Ms Alessandra Bergamo, the conference Secretary, for her dedication, efficient and selfless services. Finally, our special thanks go to Ms Dilys Grilli of the Publications Section of ICTP for her cooperation, efficiency and relentless efforts towards the publications of this volume.

*M. Karoubi, A.O. Kuku, C. Pedrini*

September, 2003

**Table of Contents** (PostScript and PDF sources available)

Most (La)TeX & PS sources of Volume 15: lns015.tar.gz (10689158 bytes)

(NB: This tar-compressed file contains LaTeX files, postscript (.ps) and/or encapsulated postscript (.eps) files of each figure plus the postscript and PDF versions of each lecture note)