It is a very exciting time to be a graduate student in Mathematics. No one even remotely in touch with recent mathematical realities would deny this, what with the last couple of decades seeing the demise of both Fermat’s Last Theorem and the Poincaré Conjecture, to name but the two most glamorous events along these lines.
It is noteworthy, at a somewhat deeper level, that the strategies finally resulting in victory in both of these titanic achievements were remarkably broad-based. It is not really an exaggeration to say that Andrew Wiles brought in much of twentieth century number theory in order to slay FLT (or Shimura-Taniyama-Weil): he used huge chunks of the theory of modular forms, of the theory of elliptic curves, and of the theory of Galois representations, arranging them in concert and then going well beyond, to prove that every rational elliptic curve is modular. FLT then follows as per Serre and Ribet. And it was all started by Gerhard Frey, really.
Regarding the Poincaré Conjecture (in dimension 3: the remaining ever so sticky wicket), Sasha Perelman saw his way clear to massaging Ricci flows (good heavens: from algebraic and geometric topology to what? hard analysis and the kinds of PDE even the physicists know about? what next? — well, see below!) in order to slay his particular dragon, with his quest being in large part shared with Richard Hamilton.
In any case, these great problems required for their resolution such amalgams of techniques that one sees that borders between, and, more importantly, allegiances to, individual mathematical disciplines had to be flouted with extreme prejudice. One has only to think back to Wiles’ marvelous point in “The Proof” to the effect that at the start of the seven year quest for said proof he systematically set out to familiarize himself carefully with what had been done of late.
This having been said, a pedagogical observation is perhaps in order, also as a segue to what I really should be talking about, the book under review. If you’re a mathematical youth starting off your research in a field with a lot of structure and history, or even a seasoned scholar who is still sufficiently young at heart for this sort of thing, you’re going to have to do a good deal of background work, get the lay of the land, pick weapons, decide on possible strategies, meditate on tactics, and a thousand and one other things you’ll find out about soon enough; be prepared for a cosmopolitan approach — at least in number theory or geometric topology, to stick with the above examples. Moreover, the weapons mentioned briefly above are additionally distinguished by a certain emerging ubiquity: algebraic number theorists cannot desist from wearing an algebraic geometer’s hat (et voilá: arthmetic geometry — of course André Weil knew it all along), geometers and general relativists attend each other’s seminars, and parties thrown by quantum mechanics, to boot. Wonderfully ecumenical stuff! And it is in this spirit that one should approach Geometric and Topological Methods for Quantum Field Theory, which I’ll just refer to as Geo + Topo for QFT, if I may.
The all-important back-cover provides us with the following: “Aimed at graduate students in physics and mathematics, this book provides an introduction to recent developments in several active topics at the interface between algebra, geometry, topology and quantum field theory. The first part … begins with an account of important results in geometric topology … [including] the differential equation aspects of quantum cohomology, before moving on to non-commutative geometry … [Then comes] a further exploration of [QFT] and gauge theory … [and] quantum gravity. The second part [of the book] covers a wide spectrum [their pun, not mine] on the borderline of mathematics and physics, [including] orbifolds … and involving a manifold [gimme a break!] of mathematical tools borrowed from geometry, algebra, and analysis.” Very cool, and, to be sure, very ecumenical.
The point is also made that “[e]ach chapter presents introductory material before moving on to more advanced results. The chapters are self-contained and can be read independently of the rest.” And this is of course very telling: our young buck or doe of a graduate student, or even someone like me, over a quarter of a century out of graduate school but recklessly interested in this stuff, should approach Geo + Topo for QFT as a compendium of possibilities, so to speak, giving a view of what can be had. I’d like to recommend, right off, the following articles: (1) “The impact on QFT on low-dimensional topology,” (2) “Differential equations aspects of quantum cohomology,” (8) “When is a differentiable manifold the boundary of an orbifold?”, and (13) “Heisenberg modules over real multiplication non-commutative tori and related algebraic structures.”
But my choice of these four from the menu of thirteen is quite arbitrary in a more global sense, given that my own recent explorations (as a sometime expatriate number theorist) cause the according resonance. I can very well imagine two if not three of my colleagues in my department cracking Geo + Topo for QFT to p.86 right off the bat so as to get at (3) “Index theory and groupoids” ASAP. And I can’t say that I blame them, either.
What of the quality of the exposition, then? Well, the news is quite good. The expositions are well-written; they look quite sound, mathematically; and, despite their genuine depth (!), they are appropriately tantalizing. The reader will want more, and happily there are good bibliographies given.
Finally, I should point out that of the thirteen chapters in Geo + Topo for QFT the fist six are much longer: they come from invited lecturers (at the 2007 summer school held in Villa de Leyva, Colombia, bearing the same title as the book); the remaining chapters are much shorter.
I intend to read many of these articles a positive number of times. My guess is that many will do the same.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.