In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space X. Expanders with this girth property are a necessary ingredient in the construction ...

The number N(E) of complex zeros of the Riemann zeta function with positive imaginary part less than E is the sum of a "smooth" function N[over ](E) and a "fluctuation." Berry and Keating have shown that the asymptotic expansion of N[over ](E) counts states of positive energy less than E in a "regularized" semiclassical model with classical Hamiltonian H=xp. For a different regularization, Conn...

The purpose of these notes is to give a feeling of “K-theory”, a new interdisciplinary subject within Mathematics. This theory was invented by Alexander Grothendieck1 [BS] in the 50’s in order to solve some difficult problems in Algebraic Geometry (the letter “K” comes from the German word “Klassen”, the mother tongue of Grothendieck). This idea of K-theory has invaded other parts of Mathematic...

Alain Connes’ construction of the standard model is based on a generalized Dirac-Yukawa operator and the K-cycle (H,D), with H a fermionic Hilbert space. If this construction is reformulated at the level of the differential algebra then a direct comparison with the alternative approach by the Marseille-Mainz group becomes possible. We do this for the case of the toy model based on the structure...

We present the construction of a Chern character in cyclic cohomology, involving an arbitrary number of associative algebras in contravariant or covariant position. This is a generalization of the bivariant Chern character for bornological algebras introduced in a previous paper [35], based on Quillen superconnections and heat-kernel regularization. Then we adapt the formalism to the cyclic coh...

The purpose of these notes is to give a feeling of “K-theory”, a new interdisciplinary subject within Mathematics. This theory was invented by Alexander Grothendieck 1[BS] in the 50’s in order to solve some difficult problems in Algebraic Geometry (the letter “K” comes from the German word “Klassen”, the mother tongue of Grothendieck. This idea of K-theory has invaded other parts of Mathematics...

The fundamental results of A. Connes which determine a complete set of isomorphism classes for most injectlve factors are discussed in detail. After some introductory remarks which lay the foundation for the subsequent discussion, an historical survey of some of the principal lines of the investigation in the classification of factors is presented, culminating in the Connes-Takesakl structure t...