Flow separation--the detachment of fluid from a boundary-- is a major cause of performance loss in engineering 
devices such as diffusers, airfoils and jet engines. In a landmark 1904 paper on boundary layers, Ludwig Prandtl 
derived a criterion for flow separation from no-slip boundaries in steady two-dimensional incompressible flows. 
Despite widespread effort, however, no unsteady or three-dimensional extension of Prandtl's criterion has emerged 
in the fluid dynamics literature.

In this talk, I discuss recent success in extending Prandtl's criterion to unsteady three-dimensional compressible 
flows. This new separation theory relies on nonstandard dynamical systems concepts, such as  nonhyperbolic invariant 
manifold theory and aperiodic averaging. Remarkably, these techniques render exact flow separation criteria that 
cannot be obtained from first principles. Beyond discussing the mathematics behind the new theory, I show numerical 
and experimental results confirming the new separation criteria. I also discuss applications to flow control and 
pollution tracking.