We consider classical Helmholtz-Kirchoff point vortices, an idealization of a two-dimensional incompressible fluid. In the mean-field regime where the  magnitudes of the vortex circulations are inversely proportional to the number of vortices, which is very large, we expect the evolution to effectively be described by the vorticity formulation of the 2D incompressible Euler equation. We will present a result on this approximation problem when the limiting vorticity is only in $L^\infty$, a scaling-critical function space for the well-posedness of the equation, and where multiplicative noise of transport-type is added to the dynamics. Time permitting, we will discuss some of the goals and challenges of going beyond mean-field theory.

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