# K-Theory and Noncommutative Geometry

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*Guillermo Cortiñas; Joachim Cuntz; Max Karoubi; Ryszard Nest; Charles A. Weibel*

A publication of the European Mathematical Society

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.

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 K-theory, 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 K-theory. There are
primary and secondary Chern characters which pass from K-theory 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
K-theory, 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 K-theory and noncommutative geometry.