Noncommutative geometry - Year 2000

Our geometric concepts evolved first through the discovery of Non-Euclidean geometry. The discovery of quantum mechanics in the form of the noncommuti ng coordinates on the phase space of atomic systems entails an equally dras tic evolution. We describe a basic construction which extends the familiar duality between ordinary spaces and commutative algebras to a duality betwe en Quotient spaces and Noncommutative algebras. The basic tools of the theo ry: K-theory, Cyclic cohomology, Morita equivalence, Operator theoretic ind ex theorems, Hopf algebra symmetry are reviewed. They cover the global aspe cts of noncommutative spaces, such as the transformation theta --> 1/theta for the noncommutative torus inverted perpendicular (2)(theta) which are un seen in perturbative expansions in theta such as star or Moyal products. We discuss the foundational problem of "what is a manifold in NCG" and explai n the fundamental role of Poincare duality in K-homology: which is the basi c reason for the spectral point of view. This leads us, when specializing t o 4-geometries to a universal algebra called the "Instanton algebra". We de scribe our joint work with G. Landi which gives noncommutative spheres S-th eta(4) from representations of the Instanton algebra. We show that any comp act Riemannian spin manifold whose isometry group has rank r greater than o r equal to 2 admits isospectral deformations to noncommutative geometries. We give a survey of several recent developments. First our joint work with H. Moscovici on the transverse geometry of foliations which yields a diffeo morphism invariant (rather than the usual covariant one) geometry on the bu ndle 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 realizat ion of zeros of zeta and L-functions from the noncommutative space of Adele classes on a global field and its relation with the Arthur-Selberg trace f ormula in the Langlands program. We end with a tantalizing connection betwe en the renormalization group and the missing Galois theory at Archimedean p laces.