This seems to be a banner year for books on noncommutative geometry. Leading the pack is “Noncommutative Geometry, Quantum Fields. and Motives” by Alain and Matilde which will be published by the American Mathematical Society in January 2008. Earlier this year we introduced two other books.

Now two other new books that just appeared in the market. Topological and Bivariant K-Theory by Joachim Cuntz, Ralf Meyer, and Jonathan Rosenberg has just been published by Birkhaeuser. In publisher’s introduction we read:

“Topological K-theory is one of the most important invariants for noncommutative algebras equipped with a suitable topology or bornology. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory.

We describe a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, we discuss other approaches to bivariant K-theories for operator algebras. As applications, we study K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem. “

The second book is Local and Analytic Cyclic Homology by Ralf Meyer and is published by the European Mathematical Society Publication House. Here is an excerpt by the publisher describing the book: “Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as most Banach algebras or C*-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras. In this book the author develops and compares these theories, emphasising their homological properties. This includes the excision theorem, invariance under passage to certain dense subalgebras, a Universal Coefficient Theorem that relates them to K-theory, and the Chern–Connes character for K-theory and K-homology”

A note to our readers and authors: if you know of any other books on the subject that is published this year, or will be published soon, please let one of us know. I know of at least one, but we should wait a bit!

Do any non-commutative geometers have an opinion on “Geometric Complexity Theory” ? Non-commutative geometry seems relevant to it since Quantum groups are used there and there are connections to versions of Riemann Hypotheses too. The papers are at http://ramakrishnadas.cs.uchicago.edu/