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Knots and dynamics

The trajectories of a vector field in 3-space can be very entangled; the flow can
swirl, spiral, create vortices etc. Periodic orbits define knots whose topology can sometimes be
very complicated. In this talk, I will survey some advances in the qualitative and quantitative
description of this kind of phenomenon. The first part will be devoted to vorticity, helicity, and
asymptotic cycles for flows. The second part will deal with various notions of rotation and spin
for surface diffeomorphisms. Finally, I will describe the important example of the geodesic
flow on the modular surface, where the linking between geodesics turns out to be related to
well-known arithmetical functions.