You are here

Integrable Systems and Algebraic Geometry

Ron Donagi and Tony Shaska, editors
Cambridge University Press
Publication Date: 
Number of Pages: 
Lond Mathematical Society Lecture Note Series
[Reviewed by
Alex Kasman
, on
In their 1895 paper, Diederik Korteweg and Gustav de Vries used elliptic curves to explain the existence of a controversial kind of wave that had been observed on canals.  This surprising result was an early clue to a connection between algebraic geometry and nonlinear wave equations. However, the full implications were not realized until the late 20th century when it became clear that although most nonlinear partial differential equations cannot be solved exactly, a certain special class of equations can be integrated using a miraculous combination of algebraic, analytic, and geometric methods.  These equations are infinite-dimensional analogues of classical integrable systems (2n-dimensional dynamical systems which are solvable by virtue of having n independent conservation laws) and they have solutions called “solitons” which behave in a particle-like manner.  An ever growing list of algebro-geometric objects (Jacobian varieties of arbitrary algebraic curves, Plücker coordinates of an infinite-dimensional Grassmannian manifold, etc.) and nonlinear PDEs (the KdV equation named after Korteweg and de Vries, the nonlinear Schrödinger equation, the KP hierarchy, Self-Dual Yang-Mills equations, etc.) are now linked in a deep and useful way by "soliton theory".
The two volume set under review collects new articles on algebraic geometry and mathematical physics.  It was compiled in honor of the 65th birthday of Emma Previato, an algebraic geometer whose 1983 PhD thesis on solitons of the nonlinear Schrödinger equation and hyper-elliptic curves is a seminal work in the field and was just the first of her many important contributions.
Some of the articles include new theorems, and Mauleshova and Mironov’s article provides the proofs of previously announced results on an algebro-geometric construction for commutative rings of difference operators involving only positive shifts.  However, in this reviewer’s opinion new results are really not the reason one should be interested in this collection.  These articles together serve as a thorough and well-written exposition on many prior results at or near the intersection of integrable systems and algebraic geometry.  In some cases, such as Spera’s article on the role of theta-functions in understanding so-called “topological” phenomena in mathematical physics, it is a review of the author’s own recent work.  In others, like Magri’s re-derivation of the integrability of Kovalevky’s Top, an article presents a new look at some foundational results in the field.
Many of the articles would be accessible to a reader with expertise in either mathematical physics or algebraic geometry and could form a “bridge” for that reader to the other area.  I can also recommend this collection to anyone already working at the intersection of integrable systems and algebraic geometry.  All of the articles would be accessible to such experts in the field, and because the topics covered are so diverse there is certain to be a lot of new information in it even for them.
However, I feel that it is important for me to emphasize that these are not tutorials suitable for the “absolute beginner”.   Several of the survey article are able to make a big topic very accessible by focusing only on one or two key examples.  For instance, in the lovely paper by Rayan, Stanley and Szmigielski the significance of the spectral curve (which here arises as a cover of the Riemann surface with a Higgs bundle) and the (slightly generalized) Lax pair are illustrated using only the example of the Calogero-Françoise equation.  But, even those papers require a good deal of prior knowledge in at least one of the two areas.
The division of the collection into two separate volumes, Volume 1 on “integrable systems” and Volume 2 on “algebraic geometry”, would seem to have practical value.  I initially imagined that a reader with more experience in one of those two areas could reasonable choose to look only at the volume in that area of interest.  However, I no longer think that is true.  
For one thing, whatever one’s interest, there are going to be articles in both volumes that one will want to consult.   In part, this is because the distinction is not always clear.  Mauleshova and Mironov’s previously mentioned contribution that produces commutative rings of difference operators using hyper-elliptic curves appears in Volume 1. Bates and Churchill’s “A Primer on Lax Pairs” which concerns matrix solutions to a differential equation (with respect to an abstract algebraic derivative) appears in Volume 2.  Both articles are wonderful and belong in this collection, but I would personally have chosen to put each in the other volume.  
Moreover, the primary interest lies not in those two separate subjects but in what happens when they come together.  There are many articles which arguably should appear in both volumes, like Vanhaecke’s contribution to Volume 1 which determines the divisor needed to complete the fiber of the momentum map of the 6-particle Kac-van Moerbeke system or Dubrovin’s article in Volume 2 which uses tau-functions of the integrable n-wave system to prove a fact about the rationality of an expression involving theta functions.  
Even articles in Volume 2 that only explicitly discuss algebraic geometry (like the article by Zarhin on half-points in hyper-elliptic Jacobians and the article by Clingher and Malmendier on normal forms for Kummer surfaces, which both build on the work of Emma Previato’s thesis advisor David Mumford) seem likely to have as yet undiscovered connections to integrable systems.  Consequently, despite the fact that they are supposedly divided up into two volumes by topic, I would recommend that any individual or library considering buying one should obtain both volumes rather than selecting one based on this categorization scheme. 
I compliment the authors for the fact that the articles are all well-written and very interesting.  However, the consistent high-quality throughout the collection suggests that the editors and the researcher to whom it is dedicated also deserve to share some of the credit.  This two volume set captures a fascinating snapshot of the current state of this (literally) dynamic area of algebraic geometry research.  It is highly recommended as a reference and an inspiration for anyone interested in this subject.
Alex Kasman is Professor of Mathematics at the College of Charleston and the author of Glimpses of Soliton Theory (AMS, 2010).