Integrable Systems, Geometry, and Topology

This book contains a written version of (some of) the lecture series given in the program on integrable systems and differential geometry organized in Taiwan in 1999. In the book, the reader can find review papers in five different fields. The first paper (F. Burstall) describes the classical theory of isothermic surfaces in Rn from the point of view of integrable systems. Classical geometric transformations are constructed using a loop group action. The second contribution (M. Guest) introduces the reader to quantum cohomology. In particular, quantum cohomology is computed for the flag manifold and for Hirzebruch surfaces. This part also contains a description of the Givental quantum differential equation and a construction of the Dubrovin connection. The fourth article (E. Heintze, X. Liu and C. Olmos) describes a generalization of the Chevalley restriction theorem to the case of an isometric submanifold. The last two articles are devoted to relations between Yang-Mills fields and harmonic maps (M. Mukai-Hidano and Y. Ohnita) and to Schrödinger flow on Grassmannians (K. Uhlenbeck).

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