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On the foundations of noncommutative geometry. (English) Zbl 1106.58004
Etingof, Pavel (ed.) et al., The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31–September 4, 2003. Boston, MA: Birkhäuser (ISBN 0-8176-4076-2/hbk). Progress in Mathematics 244, 173-204 (2006).
The purpose of this remarkable survey is to outline the philosophy of noncommutative geometry and to describe “a few of the open frontiers and problems” in this field.
The author describes the extensions of classical concepts to the noncommutative framework for measure theory, topology, differential geometry and Riemannian geometry. The paper has the following sections: The framework of noncommutative geometry; Measure theory; Topology; Differential geometry; Quantized calculus; Metric geometry and spectral action; Metric geometry, the local index formula; Renormalization, residues, and locality; Modular forms and the space of \(\mathbb Q\)-lattices. It also has a rich bibliography which is useful for researchers who are interested in one of the most fascinating branches of mathematics.
For the entire collection see [Zbl 1083.00015].

58B34 Noncommutative geometry (à la Connes)
58J42 Noncommutative global analysis, noncommutative residues
58-02 Research exposition (monographs, survey articles) pertaining to global analysis