Noncommutative Geometry: A Functorial Approach

This unusual book sports a compact Preface in which the author asks a question many of us might echo:

What is the purpose of N[on] C[ommutative] G[eometry]? … Can [it] solve open problems … inaccessible otherwise? … Can [it] benefit a topologist, an algebraic geometer, or a number theorist? What is it anyway?

One expects the second question to be answered in an emphatic affirmative: Alain Connes didn’t get his Fields Medal for nothing, after all. But can we flesh out this answer? Well, yes, of course, and that’s what this book does. In its Part II, titled “noncommutative invariants,” the first chapter is devoted to topology. Topics covered include foliations, Anosov maps, obstruction theory, and (most excitingly, I think) the Jones and HOMFLY polynomials. The next chapter is concerned with algebraic geometry and elliptic curves are everywhere, and there’s a lot more, including material on the mapping class group (of isotopy classes of the orientation preserving diffeomorphisms of a surface of strictly positive genus). Finally, the third (and sixth) chapter of Part I covers number theory, and here we encounter a cornucopia of topics: complex (and real!) multiplication, transcendental number theory, class field theory, noncommutative [!] reciprocity, an instance of Langlands’ conjecture, and Artin L-functions. Quite a sweep.

All this good stuff is followed in Part III by a “Brief Survey of NCG,” where, not surprisingly, Connes bestrides the stage like a colossus, with a chapter titled “Connes geometries” with some topics occurring that have already been introduced earlier. But there’s a lot more there, of course. Part III goes on to address some mainstays of contemporary mathematics: index theory (from Atiyah-Singer, through K-theory, through KK-theory, to Novikov’s conjecture and beyond), more on Jones polynomials, and then quantum groups. The final two chapters are concerned with NC algebraic geometry and trends in NCG.

For completeness we should note that Nikolaev’s Part I is titled “Basics,” and we go from noncommutative tori (and a load of other examples loved by the NCG players) to categories and functors and \(C^*\)-algebras. So the reader, possibly a rookie grad student, possibly a more experienced scholar, has a vade mecum at his disposal as he plows ahead. It’s Nikolaev’s intent to teach: the book has lots of exercises, for example.

Modulo my being an outsider, I think it’s fair to say that the standard source on NCG is the huge book, Noncommutative Geometry, by Connes himself, but we have a number of other texts, exposés, and monographs on this still pretty young subject. In fact Nikolaev echoes this appraisal in his Introduction. It’s a marvel indeed, given the sweep that it has, including themes from physics, number theory, K-theory, topology, and so on. I recall, for example, that when I was a graduate student at UCSD in the 1980s, K-theory maven Max Karoubi was visiting us for a while, and Connes came to visit him. I attended Connes lectures, which were really aimed almost entirely at Karoubi, and recall being struck by how algebraic and algebraic topological it all was: mathematical physics and number theory? Who knew? Obviously Connes did and does: he is still after the Riemann Hypothesis, after all — or I haven’t heard otherwise.

In any case, it is true with a vengeance that NCG is a big and important subject, and the book we are looking at here, with its subtitle of “A functorial approach,” is doubtless a very valuable contribution on any number of counts, not the least of which being that the more functorial we get, the more generality we reap, and the clearer the overarching structure becomes. Nikolaev illustrates his point even in his Preface by noting that in connection with some of the number theoretic considerations in his book,

It turns out that [a non-commutative torus] behaves much like a coordinate ring of a non-singular elliptic curve. In other words, we deal with a functor from the category of elliptic curves to a category of [noncommutative algebras, which] can be used to construct generators of … Abelian extensions of real quadratic fields … and to prove that \(e^{2\pi i \vartheta + \log\log\varepsilon}\) is … algebraic … [when] \(\vartheta\) and \(\varepsilon\) are quadratic irrational[s].

Very cool. Says Nikolaev: “This and other functors are at the heart of our book.” In point of fact, the book under review is the first book on NCG to take a functorial approach as its Leitmotiv, and that is in itself a very solid contribution, pedagogically and mathematically.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.