A Short Survey of Noncommutative Geometry

We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four dimensional sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It expresses the gamma five matrix as the pairing between the operator theoretic chern characters of e and D. It is of degree five in the idempotent and four in the Dirac operator which only appears through its commutant with the idempotent. It determines both the sphere and all its metrics with fixed volume form. We also show using the noncommutative analogue of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori from the formal naive metric. We conclude on some questions related to string theory.