Integrable equations, Bott-Virasoro group and reduction by stage

Gerard Misiolek, CDS, Caltech - Visiting Associate

I will show that the Korteweg-de Vries and Camassa-Holm equations are geodesic equations of certain right-invariant metrics on the one dimensional central extension of the group of diffeomorphisms of the circle (the Bott-Virasoro group). I will show how the equations are hamiltonian with respect to two different Poisson structures and, time permitting, will interpret the Lie algebra 2-cocycle of the Bott-Virasoro group (the Gelfand-Fuchs cocycle) as curvature of a mechanical connection arising in the context of reduction-by-stages.