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Speaker:   Spencer Smith (Department of Physics, Mt Holyoke College)

Abstract:  Mixing in non-turbulent 2-dimensional fluid flows is due to the repeated stretching and folding of material lines. The exponential rate of increase in the length of these curves provides a fundamental measure of flow complexity. This presents an interesting challenge in the case where our knowlege of the fluid flow is through sparse data: Given a set of trajectories and an initial curve, find the minimum length curve that is compatible with the motion of these points. A very elegant algorithm by Moussafir, and elaborated on by Thiffeault, solves this by encoding the trajectories as braids, loops as algebraic coordinates (Dynnikov), and specifying the action of braids on loops. I will present two new algorithms which solve this same problem more efficiently. Both use ideas from computational geometry to maintain triangulations of the points as they move, while encoding curves as edge weights. These results also pave the way toward rapid detection of coherent structures in flows and provide a foot-hold to higher-dimensional versions of this problem.

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