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The motion of point vortices constitutes an especially simple class ofsolutions to Euler's equation for two dimensional, inviscid, incompressible,and irrotational fluids. In addition to their intrinsic mathematicalimportance, these solutions are also physically relevant. Rotating superfluidhelium can support rectilinear quantized line vortices, which in certainregimes are accurately modeled by point vortices. Depending on the number ofvortices, it is possible to have either regular integrable motion or chaoticmotion. Thus, the point vortex model is one of the simplest and most tractablefluid models which exhibits some of the attributes of weak turbulence.  The primary aim of this work is to find and classify periodic orbits, aspecial class of solutions to the point vortex problem. To achieve this goal,we introduce a number of algorithms: Lie transforms which ensure that theequations of motion are accurately solved; constrained optimization whichreduces close return orbits to true periodic orbits; object-orientedrepresentations of the braid group which allow for the topological comparisonof periodic orbits. By applying these ideas, we accumulate a large data set ofperiodic orbits and their associated attributes. To render this set tractable,we introduce a topological classification scheme based on a naturaldecomposition of mapping classes. Finally, we consider some of the intriguingpatterns which emerge in the distribution of periodic orbits in phase space.Perhaps the most enduring theme which arises from this investigation is theinterplay between topology and geometry. The topological properties of aperiodic orbit will often force it to have certain geometric properties.

Point Vortices: Finding Periodic Orbits and their Topological Classification