By Cortinas G., et al. (eds.)
Seeing that its inception 50 years in the past, K-theory has been a device for realizing a wide-ranging family members of mathematical constructions and their invariants: topological areas, jewelry, algebraic kinds and operator algebras are the dominant examples. The invariants variety from attribute sessions in cohomology, determinants of matrices, Chow teams of types, in addition to strains and indices of elliptic operators. therefore K-theory is impressive for its connections with different branches of arithmetic. Noncommutative geometry develops instruments which enable one to think about noncommutative algebras within the similar footing as commutative ones: as algebras of features on (noncommutative) areas. The algebras in query come from difficulties in a variety of parts of arithmetic and mathematical physics; normal examples contain algebras of pseudodifferential operators, team algebras, and different algebras coming up from quantum box conception. to check noncommutative geometric difficulties one considers invariants of the suitable noncommutative algebras. those invariants comprise algebraic and topological K-theory, and likewise cyclic homology, came across independently by means of Alain Connes and Boris Tsygan, which are seemed either as a noncommutative model of de Rham cohomology and as an additive model of K-theory. There are basic and secondary Chern characters which go from K-theory to cyclic homology. those characters are correct either to noncommutative and commutative difficulties and feature functions starting from index theorems to the detection of singularities of commutative algebraic forms. The contributions to this quantity characterize this diversity of connections among K-theory, noncommmutative geometry, and different branches of arithmetic.
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