Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Thanks to the development of tools from Lie theory, algebraic geometry, symplectic geometry, and topology, classical problems are investigated more systematically. New problems are also arising in mathematical physics. A major international conference was held at the University of Tokyo in July 2000. It brought together scientists in all of the areas influenced by integrable systems. This book is the second of three collections of expository and research articles. This volume focuses on topology and physics. The role of zero curvature equations outside of the traditional context of differential geometry has been recognized relatively recently, but it has been an extraordinarily productive one, and most of the articles in this volume make some reference to it.Symplectic geometry, Floer homology, twistor theory, quantum cohomology, and the structure of special equations of mathematical physics, such as the Toda field equations - all of these areas have gained from the integrable systems point of view and contributed to it.
Many of the articles in this volume are written by prominent researchers and will serve as introductions to the topics. It is intended for graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics. The first volume from this conference, also available from the 'AMS', is ""Differential Geometry and Integrable Systems, Volume 308"" in the ""Contemporary Mathematics"" series. The forthcoming third volume will be published by the Mathematical Society of Japan and will be available outside of Japan from the 'AMS' in the ""Advanced Studies in Pure Mathematics"" series.
Twisted Tomei manifolds and the Toda lattices by L. Casian and Y. Kodama Quantization of Benney hierarchies by J.-H. Chang Floer homology for families-A progress report by K. Fukaya Rozansky-Witten invariants of log symplectic manifolds by R. Goto An update on harmonic maps of finite uniton number, via the zero curvature equation by M. A. Guest The lattice Toda field theory for simple Lie algebras by R. Inoue On the cohomology ring of the hyperKahler analogue of the polygon spaces by H. Konno On the theorem of Kim concerning $QH^*(G/B)$ by A.-L. Mare Geometry of the twistor equation and its applications by Y. Nagatomo On the cohomology of theta divisors of hyperelliptic Jacobians by A. Nakayashiki Isomonodromy deformations and twistor theory by Y. Ohyama Simple singularities and symplectic fillings by K. Ono Quantum cohomology of infinite dimensional flag manifolds by T. Otofuji Discrete conjugate nets of strings by S. Saito, N. Suzuki, and H. Yamaguchi Nongeneric flows in the full Kostant-Toda lattice by B. A. Shipman Frobenius manifolds and bi-Hamiltonian structures on discriminant hypersurfaces by I. A. B. Strachan Periodicity conditions for harmonic maps associated to spectral data by T. Taniguchi Higher dimensional parallel transports for Deligne cocycles by Y. Terashima Geometric nonlinear Schrodinger equations by H.-Y. Wang.
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