This book contains the proceedings of a satellite ICM 2006 meeting on K-theory and noncommutative geometry held in Valladolid, Spain, in 2006. During the meeting, there were four courses (three lectures each), given by J. Cuntz, B. Keller, B. Tsygan and Ch. Weibel. These courses were complemented by invited talks and short communications. Written versions of 14 of them are included in the book. They cover many aspects of the field, including applications to other branches of mathematics. Three of them are devoted to isomorphism conjectures (R. Meyer; A. Bartels, S. Echterhoff and W. Lück; and H. Emerson and R. Meyer). The paper by F. Muro and A. Tonks discusses K-theory of the Waldhausen category. M. Karoubi has written a review paper on twisted K-theory (including new results on Thom isomorphism and new cohomology operations). Equivariant cyclic homology for quantum groups is discussed in the paper by C. Voigt. Index theory on singular spaces is studied by P. Carrillo Rouse, C*-algebra of a certain type is described by J. Cuntz, C*-algebra methods of infinite dimensional geometry are presented by W. Werner, and an extension of the Pontrjagin duality is described in the paper by U. Bunke, T. Schick, M. Spitzweck and A. Thom. The 2-groupoid of deformations of a germ of a C∞ manifold is treated in the paper by P. Bressler, A. Gorokhovsky, R. Nest and B. Tsygan. Torsion classes of finite type are discussed by G. Garkusha and M. Prest. The last two papers are devoted to the Parshin conjecture (a paper written by T. Geisser) and axioms for the norm residue isomorphism (written by Ch. Weibel). The book offers a nice overview of a broad range of topics studied in the field.

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