Abstract:

Integrable systems seem to pop up in many different branches of mathematics, and in many disguises. This talk will introduce the audience to arguably the two most classical examples of such systems: the Toda lattice and the KdV equation. These are nonlinear evolution equations. Yet there are many quantities that remain invariant under their flows, so this makes them rather special. This is an indication of the symmetry they contain. Actually, there are certain linear problems associated to both nonlinear problems. For certain cases, one can ``linearize'' the nonlinearity using a class of algebraic curves and related objects. In other cases, one gets ``soliton'' and ``multi-soliton'' solutions of the equations. I hope to scratch the surface of these beatiful topics, explain what these terms mean, and mention some of the more recent instances of popping up. My aim will be to make the talk accessible to all graduate students.