The famous saying by Gauss, that “Mathematics is the Queen of the Sciences, and Arithmetic [i.e., number theory] is the Queen of Mathematics,” immediately raises the question of who, or what, the respective kings might be. The former question, of the coupling of Mathematics to her spouse, actually has an established, if nowadays little known, answer: the King of the Sciences is traditionally Metaphysics, in accord with Aristotle’s hierarchy (cf. Jacques Maritain’s The Range of Reason).
Physics, at least according to Aristotle, is placed below both metaphysics and mathematics largely due to its proper objects of study being correspondingly less abstract: even a meson is much more concrete than a morphism, and spacetime is certainly more concrete than a Riemannian 4-manifold with a – – – + signature.
Or is it? To quote the redoubtable Albert Einstein: “Aber ist daß wirklich so?” How would Aristotle’s eminently reasonable taxonomy play in the current climate of physics as a playground for everything from unitary group representations to differential geometry and low-dimensional topology? Indeed, if the proof is in the pudding, we already have the answer: in today’s mind-set (whatever that means) it is awfully difficult to separate the study of spacetime by the physicists from what’s done by so many geometers of different confessions. The distinction, of course, remains: physics describes, mathematics proves. So, if not the rightful King of the Sciences, in this day and age, as in so many others, physics is once again making certain overtures to the Queen and she is not being unresponsive.
Regarding the book under review, in the Foreword by Jacques Bros we read, in a perhaps more conciliatory tone,“[m]any times throughout the course of their history, theoretical physics and mathematics have been brought together by grand structural ideas which have proved to be a fertile source of inspiration for both subjects.”
Bros goes on to say that
[t]he importance of the structure of holomorphic functions of several variables became apparent [to physicists: mathematicians never doubted it …] around 1960, with the mathematical formulation of the quantum theory of fields and particles (…) their complex singularities, known as “Landau” singularities, form a whole universe, and their physical interpretation is part of a particle physicist’s basic conceptual toolkit.
On the pure mathematics side, one can safely say that this branch of theoretical physics genuinely contributed to the birth of the theory of hyperfunctions and microlocal analysis.
Thus, we have a priori connections with the mathematics of Sato and Kashiwara, right off the bat, and this certainly makes for a serious mathemtaical imprimatur.
But the context of Singularities of Integrals is actually somewhat more in the direction of physics. Bros closes his foreword with the following remark: “… the present work offers us a panoply of results from which a mathematical physicist should be able to draw great benefit.”
The material the author deals with is proper to quantum relativistic field theory: “the methods developed by Frédéric Pham have had wide-ranging impact in the non-perturbative approach approach to this subject, by allowing one to study holomorphic solutions to integral equations in complex varieties with varying cycles.” (Think of this as something of an alternative to the yoga of Feynman path integrals.)
Things get even better:
… on a general, non-perturbative level, the problems of renormalization still appear to be at the very source of the problem of the existence of non-trivial field theories in four-dimensional space-time … This “existential problem” was brought to light by [Lev] Landau in 1960 in the resummation of renormalized perturbation series in quantum electrodynamics and is exhibited … by the simplest scalar field theory with quartic interaction term …
We can really only handle quadratic terms, i.e. the Gaussian case (Yes, he really is everywhere …), completely satisfactorily; higher degree phase functions (e.g. Landau’s 4th degree beast) require sophisticated techniques. This is where, in Feynman’s approach, his diagrams come into play. Plus ça change, plus c’est la même chose.
Well, enough physics. Taking note of the warning that Pham is writing for physicists, as he says in his Introduction that he has “endeavored to present a coherent whole which is understandable to the non-mathematical reader” and that he will only assume that “the reader is familiar with the basic rudiments of general Topology,” we find Singularities of Integrals split into two parts, “Introduction to a topological study of Landau singularities,” and “Introduction to the study of singular integrals and hyperfunctions.”
It is of course objectively false that some one without a good deal of mathematics under his belt will be able to make his way through even book’s first chapter, “Differentiable manifolds.” However, if you’re, say, a combinatorialist who finds differential geometry and differential topology unheimlich and nothing from either undergraduate or graduate school along these lines resonates more than vaguely, you’re in business. Going very light on proofs, it’s pretty much all there. Physicists should also be happy, though I wonder how they’re going to react to a citing of the Nulstellensatz on p.27, in the context of a discussion of complex analytic sets. And then de Rham makes his appearance on p. 39. So, let’s not kid ourselves: the physicist readers had better be prepared to do some real mathematics, even if it were more a matter of computing examples than proving theorems.
Pham then goes on to discuss Leray’s souped up residue theory: “This theory generalizes Cauchy’s theory of residues to complex analytic manifolds.” After this it’s on to Thom’s isotopy theorem and Landau’s varieties and singularities. Part I ends with consideration of integrals depending on a parameter and the problem of “[r]amification of an integral whose integrand is itself ramified.”
Part II deals with a treatment of Nilsson’s theorem (preceded by a good deal of geometry and analysis) and then hits hyperfunctions. The last chapter of the book is devoted to an “[i]ntroduction to Sato’s microlocal analysis.” This is an amalgamation of several complex variables, rather serious differential geometry, and sheaf theory. (In fact I first encountered the phrase “microlocal analysis” in Sheaves on Manifolds by Masaki Kashiwara and Pierre Schapira: certainly not recommended for a “non-mathematical” audience!)
Singularities of Integrals is a very valuable book in that it deals with serious and important material, and does so in a way accessible to some one wanting to get into the field (be they mathematicians or physicists — though I feel much more sanguine recommending this book to the former). However, the target student should really have at least a solid advanced undergraduate background in topology and differential geometry (and I’d really feel happier with some kid who has just been through the according graduate sequences), and a bit of sheaf theory and hard analysis, to boot. Modulo these prerequisites it’s all good stuff.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.