Symmetry Breaking for the Incompressible Navier-Stokes Equations

The validity of uniquely determined solutions to 3-dimensional incompressible Navier-Stokes equations for smooth initial data remains one of the great challenges since their inception. Jean Leray was among the first investigage the mathematical mysteries embodied by these equations in terms of self-similarities viewed backwards in time. However this proved to be uneventful, and recently shown to be a failed approach. Recent new developments in the mathematical representation of solutions as expected values of stochastic cascades have lead to new insights into the nature of self-similarities in forward in time solutions to these equations. The purpose of this talk is to describe some of the progress and challenges underlying these fundamentally important equations for fluid flow when viewed from this perspective.