A short survey of noncommutative geometry

We give a survey of selected topics in noncommutative geometry, with some e mphasis on those directly related to physics, including our recent work wit h Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discu ss at length two issues. The first is the relevance of the paradigm of geom etric space, based on spectral considerations, which is central in the theo ry. 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-sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It is of the f orm [(e - 1/2)[D,e](4)] = gamma(5) and determines both the sphere and all i ts metrics with fixed volume form. The expectation [x] is the projection on the commutant of the algebra of 4 by 4 matrices. We also show, using the n oncommutative analog of the Polyakov action, how to obtain the noncommutati ve metric (in spectral form) on the noncommutative tori from the formal nai ve metric. We conclude with some questions related to string theory. (C) 20 00 American Institute of Physics. [S0022-2488(00)01706-0].