Marston Morse Selected Papers

Marston Morse was a central figure in American mathematics throughout most of the twentieth century, being primarily associated with Harvard and the Institute for Advanced Study. His impact was international and pervasive in that his contributions in the area of analysis situs, specifically his famous works on the calculus of variations “in the large,” rapidly took center stage in topology and geometry, soon to gel, so to speak, into what has come to be called Morse theory. Over the years and certainly even up to the present moment, Morse theory has evolved and expanded, perhaps even mutated, in the sense that we now count, for instance, Novikov’s work on singular behavior of certain differential forms as Morse theory, at least in a broad sense.

Interestingly, as Bott points out in “Marston Morse and his mathematical works,” which opens the book under review, Morse was at odds with the algebraic-topological character that “his beloved analysis situs” soon assumed, but it is undeniable that the presentation of the main themes of Morse theory are best couched in algebraic-topological language. Certainly the Morse inequalities, which compare so-called Morse indices of functions on a smooth manifold with the manifold’s Betti numbers, would be unheimlich in any other context. But it may just be that we are the beneficiaries — or maybe victims — of something like a historical imperialism in that the general smashing success of algebraic topology has made it unavoidable: so be it — nothing succeeds like success.

What then are some of the main themes of Morse theory? Well, if, by definition, a (real valued) Morse function f on a smooth manifold M is characterized by the fact that it has no non-degenerate critical points (df = 0 but the Hessian does not vanish), say that the Morse index λ of an f-critical point p on M is the number of negative eigenvalues of that Hessian; equivalently it is the dimension of the Hessian’s negative eigenvalue subspace. One shows (see the most important and, arguably, the most elegant book on Morse theory ever written, namely Milnor’s classic Morse Theory) that in a neighborhood of p the Morse function f looks, up to a constant, like a quadratic form with coefficients of 1 or –1, with λ occurrences of the latter. This is usually called Morse’s Lemma.

And then there are the famous Morse Inequalities. If, for instance, we consider the classic example of a donut (2-torus of genus 1) standing on a table and take “height” as our Morse function, then the four critical points of the height function are: the spot where the donut hits the table, the bottom and top of the hole, and the top of the donut itself. Locally, in the obvious coordinate system with up as up (giving the z-axis) and the table top as the Cartesian plane, at the bottom the donut is like a saucer or an upward paraboloid: z = x² + y², and as per Morse’s Lemma, the absence of negative coefficients gives λ = 0. The hole’s bottom vertex is a saddle looking like z = x² – y², say, making the hole’s top vertex a saddle looking like z = –x² + y²: in both cases we get one negative coefficient, so in each case λ = 1. Finally the top critical point of the donut is like a cap, i.e. z = –x² – y², making for λ = 2. This translates to 1 (= B₀) critical point of index 0, 2 (= B₁) critical points of index 1, and 1 (= B₂) critical point of index 2, where the B’s are in fact the Betti numbers for the torus: the ranks of the indicated homology groups. So, in this case it turns out that the Morse indices and Betti numbers agree: this does not always happen, since generally what we get is the aforementioned Morse Inequalities. These beautiful inequalities are in fact expressible in a “strong form,” namely the assertion that the difference between the Morse polynomial (whose coefficients are the respective numbers of the M’s f-critical points of a given Morse index) and M’s Poincaré series (also a polynomial: its coefficients are the Betti numbers) is a polynomial all of whose coefficients are positive integers.

Indeed, another mainstay of Morse theory is the fact that M itself is homotopically equivalent to a CW-complex with a cell of dimension λ attached for each f-critical point of Morse index λ: this is the thrust of what is usually called the Fundamental Theorem of Morse Theory and makes for a point of entry for a historical development of truly momentous proportions. This topological behavior, of how the manifold’s topology changes locally (ultimately expressed modulo homotopy equivalence, although local homeomorphisms are in fact in the game) as the Morse function sweeps across it passing through its critical points, was the focus of work by Thom and especially Smale, on the order of fifty years ago. That yields a rather different way of looking at the whole subject: dynamical systems began to dominate and indeed transformed everything. This flow of events is described in an unparalleled way by Bott in his famous article, “Morse Theory indomitable” where he in fact proceeds from a (fantastic) discussion of Smale’s way of doing Morse theory to the next revolution in the subject, namely, Witten’s ground breaking 1982 paper “Supersymmetry and Morse theory.”

Well, what would Morse say about that? The “calculus of variations in the large,” his original description for what he was doing, was first usurped by algebraic topology, then by dynamical systems (which can still be classed as algebraic topology, or perhaps geometric topology, if we are going to go that route), and now by particle physics. What’s next? Perhaps this is all disingenuous, however: after all, what’s in a name? There is no doubt whatsoever that what Morse wrought has had, and continues to have, effects of seismic proportions throughout mathematics and it is currently part of nothing less an extremely fecund interplay between physics and geometry (both in their broadest senses) showing no signs of letting up.

Therefore Morse’s Selected Papers are of huge, and indeed more than historical, interest. The papers in the book under review are by and large concerned with the calculus of variations interpreted in the sense indicated above: “in the large.” They present a particular geometric and analytic perspective on the subject, which is by no means anachronistic and covers a lot of ground indeed. Additionally there are a number of papers on such themes as minimal surfaces, the Schoenflies problem, “unending chess,” and even the relationship between mathematics and the arts.

Finally, there is the introductory article that Bott first published in the Bulletin of AMS in 1980: it is always irresistible to read Bott on just about anything, and this is no exception. He gives a wonderful sketch of Morse the mathematician and Morse the man, as can really only be done by someone who knew the subject personally — of course, Bott also knew Morse’s mathematics intimately, so his appraisal of Morse’s work is particularly meaningful. Bott spends a lot of time talking about the nature and impact of Morse’s contributions, with 20/20 hindsight, and we learn about particular aspects of Morse’s work in considerable (and welcome) detail: Bott includes, for example, themes from minimal surfaces, complex analysis, harmonic functions, and differential topology (cf. the Schoenflies problem mentioned above) in his discussion. All in all it is very evocative and very enlightening and testifies to the enduring significance of Morse’s work. 

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