Integrable Systems, Geometry, and Topology
About this Title
Chuu-Lian Terng, University of California, Irvine, Irvine, CA, Editor
Publication: AMS/IP Studies in Advanced Mathematics
Publication Year: 2006; Volume 36
ISBNs: 978-0-8218-4048-1 (print); 978-1-4704-3825-8 (online)
MathSciNet review: MR2216237
MSC: Primary 53-06
The articles in this volume are based on lectures from a program on integrable systems and differential geometry held at Taiwan's National Center for Theoretical Sciences. As is well-known, for many soliton equations, the solutions have interpretations as differential geometric objects, and thereby techniques of soliton equations have been successfully applied to the study of geometric problems.
The article by Burstall gives a beautiful exposition on isothermic surfaces and their relations to integrable systems, and the two articles by Guest give an introduction to quantum cohomology, carry out explicit computations of the quantum cohomology of flag manifolds and Hirzebruch surfaces, and give a survey of Givental's quantum differential equations. The article by Heintze, Liu, and Olmos is on the theory of isoparametric submanifolds in an arbitrary Riemannian manifold, which is related to the n-wave equation when the ambient manifold is Euclidean. Mukai-Hidano and Ohnita present a survey on the moduli space of Yang-Mills-Higgs equations on Riemann surfaces. The article by Terng and Uhlenbeck explains the gauge equivalence of the matrix non-linear Schrödinger equation, the Schrödinger flow on Grassmanian, and the Heisenberg Feromagnetic model.
The book provides an introduction to integrable systems and their relation to differential geometry. It is suitable for advanced graduate students and research mathematicians.
Research mathematicians interested in integrable systems.
Table of Contents
- Isothermic surfaces: Conformal geometry, Clifford algebras and integrable systems
- Introduction to homological geometry: part I
- Introduction to homological geometry: part II
- Isoparametric submanifolds and a Chevalley-type restriction theorem
- Gauge-Theoretic approach to harmonic maps and subspaces in moduli spaces
- Schrödinger flows on Grassmannians