The flow of large crowds of pedestrians

Despite popular belief the motion of a crowd is governed by well-defined rules of behaviour. These rules imply a set of coupled, non-linear, partial differential equations for the density and velocity potential for each type of pedestrian in the crowd. As may be expected, the solution of these equations may, in different regions of space, be supercritical or subcritical with the possibility of a shock wave separating the regions. Less predictable is the remarkable finding that these coupled, non-linear, time dependent equations are conformally mappable and this finding enables solutions to be obtained easily for both supercritical and subcritical flows.