When I was a graduate student at UCSD in the early 1980s we had the good fortune of sporting a bevy of Fields Medalists on our faculty. In fact, one year Yau, Freedman, and Connes were all on the premises and, accordingly, geometry and topology were in the air everywhere. Karoubi was visiting, too, and he and Connes launched something of a shared missionary effort to spread their style of K-theory and algebraic topology to their charges. I remember vividly the experience of a lecture given by Connes, with Karoubi in the front row; I fear that, to me, the most memorable feature was the fact that I couldn’t quite follow the otherwise impeccable English of the speaker because of its Gallic speed and cadence! Given the level of sophistication of the material being discussed it might not have mattered anyway: this was apparently real insider stuff.
Nowadays Connes can be found online, in streaming video format, lecturing on non-commutative geometry at what seems to me a much mellower pace, and still in wonderfully expressive and perfect English. These online lectures, given in shifts with Mathilde Marcolli, are an experience well worth having, especially in light of both their intrinsic clarity and what they are talking about: non-commutative geometry is utterly fascinating stuff, with a very broad sweep, and with a revolutionary promise — just read any interview with Connes on this count, e.g. his EMS interview with Skandalis and Goldstein available on Connes’ web-page, http://www.alainconnes.org/en/.
What is this revolutionary promise, then? It is of course very well known that Connes has nothing less than the Riemann hypothesis in his sights, but this is really only a small part of the picture (!). Says Khalkhali, in the Introduction to the book under review: “To understand the basic ideas on noncommutative geometry one should… first come to grips with the idea of a noncommutative space. What is a noncommutative space? The answer to this question is based on one of the most profound ideas in mathematics, namely a duality or correspondence between algebra and geometry… according to which every concept or statement in Algebra corresponds to, and can be equally formulated by, a similar concept and statement in Geometry.”
To be sure, this philosophical position has historical antecedents possibly going back to well before even the ancient Greeks and including the according Weltanschauung of such figures as “Descartes (analytic geometry), Hilbert (affine varieties and commutative algebras), Gelfand [and] Naimark (locally compact spaces and C*-algebras), and Grothendieck (affine schemes and commutative rings),” as per Khalkhali’s account.
Next, getting down to some specific brass tacks, here is an extremely evocative and revealing passage from the author: “[NCG, in Connes’ slang,] has as its … limiting case … classical geometry, but geometry expressed in algebraic terms. In some respects this should be compared with the celebrated correspondence principle in quantum mechanics where classical mechanics appears as a limit of quantum mechanics for large quantum numbers or small values of Planck’s constant.” Wow!
Subsequently, a few pages later, Khalkhali presents a list of nineteen pairs of “commutative” notions coupled with their noncommutative counterparts; e.g., measure spaces parallel von Neumann algabras, and, as already mentioned, LC spaces parallel C*-algebras; and vector fields pair up with derivations, as Lie group theory tells us. But things get pretty wild pretty quick: “complex variable” matches to “operator on a Hilbert space” (shades of Fourier analysis + QM?), “integral” matches to “trace” (well, I guess we’re talking representation theory here, maybe à la Gel’fand, Graev, Piatetskii-Shapiro), “de Rham cohomology” goes with “cyclic homology,” and (most tellingly) “group, Lie algebra” is matched to “Hopf algebra, quantum group.” Obviously NCG covers a vast landscape indeed.
With this background in place, Khalkhali goes on to point out that he intends for his audience to have a pretty broad background in modern mathematics even before they open the book: “While the idea was to write a primer for the novice to the subject, some acquaintance with functional analysis, differential geometry, and algebraic geometry is assumed.” And even with such prerequisites in place, the reader had better be prepared to do some outside (“commutative”) reading: Morita equivalence appears in chapter two, chapter four is devoted to the Connes-Chern character, and the book’s appendices include coverage of the Gelfand-Naimark theorems (from commutative Banach algebras), Fredholm theory, and abstract index theory.
I find the point of view taken by Connes and his followers extremely attractive, in fact, irresistible. Accordingly I believe that this book by Khalkhali is both well-conceived and well-timed. Additionally Basic Noncommutative Geometry is well-written, if necessarily compact (but certainly not closed!), and looks to be quite accessible — again modulo the reader’s willingness to go at the book the way a serious mathematics monograph should be approached, namely, with pen(cil) and paper at the ready. But, given what’s being offered by NCG, it’ll all be well worth the effort.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.