Integrable systems can be found in a surprising variety of different types of geometric situations, for example:
Right now I'm interested in the role of integrable systems in the geometry of conformally immersed 2-tori in the conformal 4-sphere. In particular, the suggestion is that any such torus with flat normal bundle possesses integrable deformations: typically infinitely many independent ones. For example, if true this applies to any torus in a 3-sphere and hence any torus in Euclidean 3-space - we're talking doughnuts (with holes) and coffee cups. But no-one has succeeded in proving this, and part of the obstacle lies in a lack of understanding about the geometric meaning of the underlying integrable systems. Quaternionic projective geometry seems to play a key role here.
I'm also interested in Lagrangian surfaces in Kaehler 4-manifolds, mostly when the latter are Hermitian symmetric spaces (because that is when integrable systems plays a role). My student Richard Hunter studied HSL tori in the complex projective plane, and there is an interesting reason for studying the same condition in the Grassmannian of real oriented 2-spaces in 4-space (which has the structure of a product of two Riemann spheres).
PHONE: (++ 44 01904) 433094. FAX: (++ 44 1904) 433071
E-MAIL: im7 (append @york.ac.uk from outside the University of York).
POST: Department of Mathematics, University of York, Heslington, York YO10 5DD, UK.