EMS Series of Congress Reports 2008; 454 pp; hardcover Volume: 2 ISBN10: 3037190604 ISBN13: 9783037190609 List Price: US$124 Member Price: US$99.20 Order Code: EMSSCR/2
 Since its inception 50 years ago, Ktheory has been a tool for understanding a wideranging family of mathematical structures and their invariants: topological spaces, rings, algebraic varieties and operator algebras are the dominant examples. The invariants range from characteristic classes in cohomology, determinants of matrices, Chow groups of varieties, as well as traces and indices of elliptic operators. Thus Ktheory is notable for its connections with other branches of mathematics. Noncommutative geometry develops tools which allow one to think of noncommutative algebras in the same footing as commutative ones: as algebras of functions on (noncommutative) spaces. The algebras in question come from problems in various areas of mathematics and mathematical physics; typical examples include algebras of pseudodifferential operators, group algebras, and other algebras arising from quantum field theory. To study noncommutative geometric problems one considers invariants of the relevant noncommutative algebras. These invariants include algebraic and topological Ktheory, and also cyclic homology, discovered independently by Alain Connes and Boris Tsygan, which can be regarded both as a noncommutative version of de Rham cohomology and as an additive version of Ktheory. There are primary and secondary Chern characters which pass from Ktheory to cyclic homology. These characters are relevant both to noncommutative and commutative problems and have applications ranging from index theorems to the detection of singularities of commutative algebraic varieties. The contributions to this volume represent this range of connections between Ktheory, noncommmutative geometry, and other branches of mathematics. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in Ktheory and noncommutative geometry. Table of Contents  R. Meyer  Categorical aspects of bivariant Ktheory
 A. Bartels, S. Echterhoff, and W. Lück  Inheritance of isomorphism conjectures under colimits
 H. Emerson and R. Meyer  Coarse and equivariant coassembly maps
 F. Muro and A. Tonks  On \(K_1\) of a Waldhausen category
 M. Karoubi  Twisted \(K\)theoryold and new
 C. Voigt  Equivariant cyclic homology for quantum groups
 P. C. Rouse  A Schwartz type algebra for the tangent groupoid
 J. Cuntz  \(C^*\)algebras associated with the \(ax+b\)semigroup over \(\mathbb{N}\)
 W. Werner  On a class of Hilbert \(C^*\)manifolds
 U. Bunke, T. Schick, M. Spitzweck, and A. Thom  Duality for topological abelian group stacks and \(T\)duality
 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan  Deformations of gerbes on smooth manifolds
 G. Garkusha and M. Prest  Torsion classes of finite type and spectra
 T. Geisser  Parshin's conjecture revisited
 C. Weibel  Axioms for the norm residue isomorphism
 List of contributors
 List of participants
