Motion by mean curvature for networks of surfaces arises in a

variety of applications, such as the dynamics of foam and the evolution

of microstructure in polycrystalline materials. It is steepest descent

(gradient flow) for an energy: the sum of the areas of the surfaces

constituting the network.

During the evolution, surfaces may collide and junctions (where three or

more surfaces meet) may merge and split off in myriad ways as the

network coarsens in the process of decreasing its energy. The first idea

that comes to mind for simulating this evolution -- parametrizing the

surfaces and explicitly specifying rules for cutting and pasting when

collisions occur -- gets hopelessly complicated. Instead, one looks for

algorithms that generate the correct motion, including all the necessary

topological changes, indirectly but automatically via just a couple of

simple operations.

A remarkably elegant such algorithm, known as threshold dynamics, was

proposed by Merriman, Bence, and Osher in 1992. Extending this

algorithm, while preserving its simplicity, to more general energies

where each surface in the network is measured by a different, possibly

anisotropic, notion of area requires new mathematical understanding of

the original version, which then elucidates a systematic path to new

algorithms. Based on joint works, first with Felix Otto, and then with

Matt Elsey, Matt Jacobs, and Pengbo Zhang.