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Tim Carson
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Differential forms are used in geometric measure theory to define currents. One use of currents is as a generalization of submanifolds, with better compactness properties. That is, it's easier to show that a subsequence of currents converge to a current. This can give one approach to the minimal surface problem; see this encyclopediaofmath.org article.

Maybe someone else can give some better detail or more uses of currents.

Tim Carson
  • 856
  • 7
  • 12