Our geometric concepts evolved first through the discovery of
NonEuclidean
geometry. The discovery of quantum mechanics in the form of the
noncommuting
coordinates on the phase space of atomic systems entails an equally
drastic
evolution.
We describe a basic construction which extends the familiar duality
between
ordinary spaces
and commutative algebras to a duality between Quotient spaces and
Noncommutative algebras.
The basic tools of the theory, K-theory, Cyclic cohomology, Morita
equivalence, Operator theoretic index theorems,
Hopf algebra symmetry are reviewed. They cover the global aspects of
noncommutative spaces, such as the
transformation
$\theta \rightarrow 1/\theta$
for the noncommutative
torus
$Tb_{\theta}^2$ which are unseen
in perturbative
expansions in
$\theta$ such as star or Moyal products. We discuss the
foundational problem of
"what is a manifold in NCG" and explain the fundamental role of Poincare
duality in K-homology
which is the basic reason for the spectral point of view. This leads us,
when specializing to 4-geometries
to a universal algebra called the "Instanton algebra". We describe our
joint
work with G. Landi which
gives noncommutative spheres
$S_{\theta}^4$ from representations of the
Instanton algebra.
We give a survey of several recent developments. First our joint work
with
H. Moscovici on
the transverse geometry of foliations which yields a diffeomorphism
invariant (rather than the usual covariant
one) geometry on the bundle of metrics on a manifold and a natural
extension
of cyclic cohomology
to Hopf algebras. Second, our joint work with D. Kreimer on
renormalization
and the Riemann-Hilbert
problem. Finally we describe the spectral realization of zeros of zeta
and
L-functions from the noncommutative space
of Adele classes on a global field and its relation with the
Athur-Selberg
trace formula in the Langlands program.
We end with a tentalizing connection between the renormalization group
and
the missing Galois theory at
Archimedian places.