K-Theory and Noncommutative Geometry: (v. 2 EMS Series of Congress Reports)

Since its inception 50 years ago, K-theory has been a tool for understanding a wide-ranging 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 K-theory is notable for its connections with other branches of mathematics. Non commutative geometry develops tools which allow one to think of non commutative algebras in the same footing as commutative ones: as algebras ofnon commutative(non commutative) spaces. The algebras in question come from problems in various areas of mathematics and mathematical physics; typical examples includes pseudo differentialudodifferential operators, group algebras, and other algebras arising from quantum field thnon commutative non commutative geometric problems one considers invariants onon commutative non commutative algebras. These invariants include algebraic and topological K-theory, and also cyclic homology, discovered independently by Alain Connes and Boris Tsygan, which can be reganon commutative non commutative version of de Rham cohomology and as an additive version of K-theory. There are primary and secondary Chern characters which pass from K-theory to cyclic homology. These characters are renon commutative non commutative 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 betnon commutative non commutative geometry, and other branches of mathematics. This is a publication of the European Mathematical Society (EMS). It is distributed within the Americas by the American Mathematical Society.

Table of Contents:
R. Meyer -- Categorical aspects of bivariant K-theory; A. Bartels, S. Echterhoff, and W. Luck -- Inheritance of isomorphism conjectures under colimits; H. Emerson and R. Meyer -- Coarse and equivariant co-assembly maps; F. Muro and A. Tonks -- On $K_1$ of a Waldhausen category; M. Karoubi -- Twisted $K$-theory--old 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.