Integrable Systems: The Verdier Memorial Conference Actes du Colloque International de Luminy(v. 115 Progress in Mathematics)

This book contains fifteen articles by eminent specialists in the theory of completely integrable systems, bringing together the diverse approaches to classical and quantum integrable systems and covering the principal current research developments. In the first part of the book, which contains seven papers, the emphasis is on the algebro-geometric methods and the tau-functions. Essential use of Riemann surfaces and their theta functions is made in order to construct classes of solutions of integrable systems. The five articles in the second part of the book are mainly based on Hamiltonian methods, illustrating their interplay with the methods of algebraic geometry, the study of Hamiltonian actions, and the role of the bihamiltonian formalism in the theory of soliton equations. The two papers in the third part deal with the theory of two-dimensional lattice models, in particular with the symmetries of the quantum Yang-Baxter equation. In the fourth and final part, the integrability of the hierarchies of Hamiltonian systems and topological field theory are shown to be strongly interrelated. In the overview that introduces the articles, Bennequin surveys the evolution of the subject from Abel to the most recent developments, and analyzes the important contributions of J.-L. Verdier to whose memory the book is dedicated. This book will be a valuable reference for mathematicians and mathematical physicists.

Table of Contents:
Introduction: Hommage a Jean-Louis Verdier- au jardin des systemes integrables, D. Bennequin; PART I Algebro-Geometric Methods and Tau-functions: Compactified Jacobians of Tangential Covers, A. Treibich; Heisenberg Action and Verlinde Formulas, B. van Geemen and E. Previato; Hyperelliptic Curves that Generate Constant Mean Curvature Tori in R3, N. Ercolani, H. Knorrer, and E. Trubowitz; Modular Forms as Tau-Functions for Certain Integrable Reductions of the Yang-Mills Equations, L. A. Takhtajan; The r-functions of the gAKNS equations, G. Wilson; On Segal-Wilson's definition of the r-function and hierarchies AKNS-D and mcKP, L. A. Dickey; The boundary of isospectral manifolds, Backlund transformations and regularization, P. van Moerbeke; PART 2 Hamiltonian Methods: The Geometry of the Full Toda Lattice, N. M. Ercolani, H. Flaschka, and S. Singer; Deformations of a Hamiltonian action of a Compact Lie Group, V. Guillemin; Linear-Quadratic Metrics "approximate" any nondegenerate, integrable Riemannian metric on the 2-Sphere and the 2-Torus, A. T Fomenko; Canonical Forms for BiHamiltonian Systems, P. J. Olver; BiHamiltonian manifolds and Sato's Equations, P. Casati, F. Magri, and M. Pedroni. PART 3 Solvable Lattice Models: Generalized Chiral Potts Models and Minimal Cyclic Representations of Uq(gl(n,C)), E. Date; Infinite Discrete Symmetry Group for the Yang-Baxter Equations and their Higher-Dimensional Generalizations, M. Bellon, J.-M. Maillard, and C. M. Viallet. PART 4 Topological Field Theory: Integrable Systems and Classification of 2-dimensional Topological Field Theories, B. Dubrovin; List of Participants;