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Function Theory on Symplectic Manifolds

Leonid Polterovich and Daniel Rosen
American Mathematical Society
Publication Date: 
Number of Pages: 
CRM Monograph Series 34
[Reviewed by
Michael Berg
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A symplectic manifold is, by definition, a smooth connected 2n-dimensional manifold equipped with a symplectic form, the latter being simply a closed 2-form with the property that its n-fold wedge power does not vanish. So this n-th wedge power of the symplectic form is in fact a top form, whence a volume form, and we not only have orientation but a great deal of wonderful differential geometry and even algebraic topology to play with — indeed it’s their interplay that is in a way the most exciting aspect of the whole business.

One very highfalutin way to get an idea of why this same business in so tremendous is to observe that symplectic manifolds are a major component of nothing less than homological mirror symmetry: a manifold, as above, is dealt both a symplectic structure, courtesy of the aforementioned 2-form, and an almost complex structure (\(J^2=-1\) or, more precisely, \(J^2=-\mathrm{id}\), and so on, generalizing what happens for the manifold C). Mirror symmetry consists in the interplay between these models, arranged in so-called mirror pairs; marvelous invariants (e.g. Gromov-Witten invariants) and structural parallels (or mirrored properties) are the order of the day, and this subject forms an umbrella for e.g. the famous work by Maxim Kontsevich including his marvelous formula. (In this connection, see e.g. p. 99 of Kock, Vainsencher, An Invitation to Quantum Cohomology.) And so, now that the quantum cohomological cat is out of the bag (with apologies to Schrödinger, I guess), it should be no surprise that the book under review is also ultimately concerned with the very sexy stuff from quantum field theory and string theory that has come to permeate so much of contemporary differential geometry. On the other hand, Polterovich and Rosen present this physics in a somewhat sub rosa fashion: it’s all mathematician friendly. Nonetheless, Chapter 9 is titled “Geometry of Covers and Quantum Noise.” But then Chapter 10 takes up what, from one point of view, is the raison d’être of this subject, the trajectory from Morse theory to Morse-Novikov theory and then to Floer theory: the analysis of singular behavior of manifolds and the (suitable) functions and forms (i.e. closed 1-forms, in Novikov’s set up) that live on them.

Well, this is already very far along in the book under review: Floer appears in the penultimate chapter and then the final chapter deals with such things as quantum homology, its non-Archimedean structure, and spectral invariants in Floer theory. What of the earlier parts of the book? Suffice it to say that it’s both very tantalizing and substantial. The book starts off with “three wonders of symplectic geometry,” namely, C0-rigidity, Arnol’d’s conjecture, and Hofer’s metric. Why are these such wonders? Answer: respectively, the indicated symplectic group is C0-closed in the group of diffeomorphisms with compact support on the manifold; the number of fixed points of a Hamiltonian diffeomeorphism (shades of mechanics, of course) is bounded below by the sum of the dimensions of the manifold’s real cohomology; and Hofer’s metric (cf. p.10) is “essentially the unique non-degenerate Finsler metric on the group [of Hamiltonian diffeomeorphisms]” (and a Finsler metric is “a smooth assignment of a norm on each tangent space” of the manifold). The intervening chapters, connecting the first chapter’s discussion of these “wonders” to the exciting stuff mentioned above, is concerned with a lot of heavy machinery, but it’s all presented in a streamlined fashion: quasi-morphisms, spectral invariants, and symplectic approximation theory make an appearance, for example.

In my review of the above book by Kock and Vainsencher I said that it wasn’t for raw recruits. The same is true for the present book. But it is very exciting and deep mathematics: it’s really altogether irresistible.

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