Abstract. In complex dynamical systems with general time dependence, transportbarriers are the only directly observable invariant structures. Examples of such systems include oceanic and atmospheric flows, many-body problems, chemical reactions, nuclear fusion dynamics, and crowd dynamics. So far, a unified approach to defining, detecting and controlling transport barriers in these problems has not emerged.

In this talk, we introduce such a unified approach for the case of two-dimensional unsteadyflows. We show that transport barriers can be obtained as minimal geodesics under an appropriate Riemannian metric induced by the flow on the space of initial positions. The barriers are then obtained as solutions of an ordinary differential equation, and hence can be extracted in a smooth, parametrized form from numerical or experimental data. We show how these results reveal previously undetected transport barriers with mathematical rigor in complex model flows and satellite observations of the ocean.