Peakons arising as particle paths beneath small-amplitude water waves

We present a new kind of particle path in constant vorticity water of finite depth, within the framework of small-amplitude waves

Riemannian Hilbert manifolds

In this article we collect results obtained by the authors jointly with other authors and we discuss old and new ideas. In particular we discuss singularities of the exponential map, completeness and homogeneity for Riemannian Hilbert quotient manifolds. We also extend a Theorem due to Nomizu and Ozeki to infinite dimensional Riemannian Hilbert manifolds