This text acquaints the reader with the essential ideas of analytic K-homology and develops some of its applications. It includes an introduction to the necessary functional analysis, followed by a detailed treatment of C*-algebra extension theory and an exploration of the connections between K-homology and operator theory, coarse geometry, index theory and assembly maps.

You can view the introduction online.

Proceedings from the Clay Mathematics Institute Instructional Symposium, held in conjection with the AMS-IMS-SIAM Joint Summer Research Conference on Noncommutative Geometry, June 18-29, 2000, Mount Holyoke College, South Hadley, MA. The meeting centered around several series of expository lectures that were intended to introduce key topics in noncommutative geometry to mathematicians unfamiliar with the subject.

Topics covered include: various applications of noncommutative geometry to problems in geometry and topology; the Riemann hypothesis and the possible application of the methods of noncommutative geometry in number theory; and the residue index theorem of Connes and Moscovici. In detail, the book's contents are as follows:

Preface by Nigel Higson and John Roe
A Minicourse on Applications of Non-Commutative Geometry to Topology by Jonathan Rosenberg
On Novikov-Type Conjectures by Stanley Chang and Shmuel Weinberger
The Residue Index Theorem of Connes and Moscovici by Nigel Higson
The Riemann Hypothesis: Arithmetic and Geometry by Jeffrey Lagarias
Noncommutative Geometry and Number Theory by Paula Tretkoff

This book will present lectures that were delivered to the 2004 Spring Institute on Noncommutative Geometry and Operator Algebras at Vanderbilt University. The book aims to give a complete account of the Atiyah-Singer index theorem, and at the same time introduce a number of important concepts of noncommutative geometry, including groupoid algebras, K-theory for foliations, asymptotic morphisms and assembly maps.