Symplectic geometry is a marvelous part of contemporary mathematics, straddling different subfields and having connections to everything from physics to number theory. In very broad terms, the idea behind symplectic structure is to impart a skew-symmetric bilinear form to an even-dimensional vector space. This bilinear form is then used to create bases for this vector space that obey a version of Heisenberg’s commutation relations for quantum mechanics. Exploiting the interplay between linear duality and Pontryagin duality afforded by the presence of the skew-symmetric form, also called a symplectic form, one is in a position, first, to introduce natural Lie groups and Lie algebras into the game, specifically the Heisenberg (Lie) group and algebra, and, subsequently, to proceed to the attendant unitary representation theory, reaching, e.g., the Schrödinger representation of the Heisenberg group in mimicry of what the physicists did in the 1920s and ’30s starting from the original archetype of the Heisenberg commutation relations in quantum mechanics (QM).
It is this representation theoretic perspective that drives the train as far as, for instance, analytic number theoretic applications go: the Schrödinger representation of the Heisenberg group can be realized as an induced representation of a character naturally attached to the Lie group of a Lagrangian plane. Here a Lagrangian plane is, by definition, a subspace of the symplectic space maximal with respect to the requirement of being its own annihilator (under the symplectic form). Such a subspace is said to be maximally isotropic and must be of half the dimension of the ambient symplectic space. The aforementioned character arises as a central character and one can employ the formalism of the Stone-Von Neumann theorem to produce a projective representation of the underlying Heisenberg’s group’s isotropy group: the symplectic group of the given symplectic structure. This projective representation is called by many names, as a function of who’s doing the naming: physicists tend to call it the Segal-Shale representation, after D. Shale, I. M. Segal’s student, who first displayed the representation in connection with the behavior of bosons (particles obeying Bose-Einstein statistics). Others call it the oscillator representation (for transparent quantum mechanical reasons). Number theorists call it the Weil representation, since André Weil developed it in exquisite detail in his famous work in the 1960s on Siegel’s analytic theory of quadratic forms. Nowadays a lot of (presumably ecumenically minded) people use the term Segal-Shale-Weil representation.
The presence of Lagrangian planes on the stage is the main reason for the geometrical angle on the whole business. It is no surprise, for linear algebraic reasons, that the class of Lagrangian planes sitting inside a given symplectic space has a lot of geometric structure. Indeed, this class constitutes a variety and a manifold: the Lagrangian Grassmannian, and we find ourselves at the intersection of at least four methodologies: algebraic and differential geometry/topology, the representation theory of algebraic groups, the dynamics arising from physics’ point of view (i.e., the presence of PDE in the shadows), and, of course, Lie theory. Truly a cornucopia.
The authors of book under review, Helmut Hofer and Eduard Zehnder, are genuine heavy hitters in this area: Hofer is regarded as one of the founders of symplectic topology, and Zehnder is, among other things, a major force in the area of dynamical systems whose name is also connected to the solution (in 1983, with C. C. Conley) of a form of the Arnol’d conjecture (about which more below). What they have produced in Symplectic Invariants and Hamiltonian Dynamics is a marvel of mathematical scholarship, far reaching in its scope, and timely in its appearance — or reappearance: the book under review is a re-issue of the 1994 original, now as part of the “Modern Birkhäuser Classics” series.
The book’s preface starts off with the observation that “[t]he discoveries of the past decades have opened up new perspectives for the old field of Hamiltonian systems and led to the creation of a new field: symplectic topology.” The authors then proceed to note that ostensible mysteries surrounding symplectic mappings are connected at a deep level to (and even resolved by) “global periodic phenomena in Hamiltonian systems,” with the so-called symplectic capacities coming into play in a critical manner. Regarding these we read, “These invariants are the main theme of this book.”
Although the book is advertised to be accessible to graduate students, the latter should approach it with their loins girded: “The introductory chapter presents in a rather unsystematic way some background material… [:] symplectic manifolds and symplectic mappings and [we] briefly recall the Hamiltonian formalism. … Cartan’s calculus is used. The classification of 2-dimensional symplectic manifolds by the Euler characteristic and the total volume is proved.” And then, “the direct method of the calculus of variations [is used] to establish a periodic orbit on a convex energy surface of a Hamiltonian system in R2n.” And on it goes: beautiful but non-trivial stuff.
The mathematics presented here is unabashedly avant garde, and the discussion is unquestionably research-oriented. A graduate student going this route should obviously be already pretty well-versed in differential geometry, certain clearly indicated parts of topology, as well as, possibly, some neo-classical themes from physics.
Chapters 2–6 accordingly take the reader from a treatment of symplectic capacities to closed characteristics, the geometry of compactly supported symplectic mappings in R2n, and finally a discussion of the Arnol’d conjecture and the work of the late Andreas Floer. The Arnol’d conjecture states that “[e]very Hamiltonian diffeomorphism … of a compact symplectic manifold … possesses at least as many fixed points as a [function from the manifold to R] possesses critical points” (cf. p. 199); in somewhat more familiar parlance, that of Wikipedia, it states that states that a Hamiltonian symplectomorphism on a compact symplectic manifold must have at least as many fixed points as a generic smooth function on the manifold has critical points (in the Morse theoretic sense).
Regarding Floer homology, suffice it to cite the introduction to § 6.5:
We shall sketch Floer’s seminal approach to solving the Arnol’d conjecture in the non-degenerate case where one assumes that all the 1-periodic solutions of the Hamiltonian system on [the given symplectic manifold] are non-degenerate. Since [the manifold] is compact, there are only finitely many such 1-periodic solutions. Floer studies the set of bounded orbits of the unregularized gradient equation defined by the action functional on the space of contractible loops of [the manifold]. The bounded orbits are special solutions of a distinguished system of first order elliptic equations of Cauchy-Riemann type…
And it just gets better from there: a wonderful way to finish a wonderful book.
Note, though, that at this late stage in the game, both Morse theory à la Andreas Floer and the theory of (gradient) flows on a manifold have begun to figure prominently: this underscores our earlier warning and should be borne in mind by any one diving into this material — it’s pretty sophisticated and pulls in a lot of modern material. For the relative novice, outside reading is indicated, but it’s well worth it. The pay-off is quite substantial.
Symplectic Invariants and Hamiltonian Dynamics is obviously a work of central importance in the field and is required reading for all would-be players in this game. Happily, it is very well written and sports a lot of very useful commentary by the authors; the sections introducing the individual chapters are particularly well done: they lead the reader to the upcoming material in just the right way, with historical perspectives in place, the right amount of mathematical foreshadowing, and the right perspectives in place. It goes without saying that everything is on the money as far as rigor goes: the book abounds with propositions and theorems and their proofs, again with lots of historical connections in place, as well as references, mainly to recent works, this subject being so young. It is all fine scholarship in an exciting and fertile area.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.