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Mathematics Analysis Seminar: Flat norm decompositions of integral currents
4:10, Neill Hall 5W
The flat norm is a natural way to measure currents (generalized surfaces with orientations and multiplicities) by decomposing a d-dimensional current optimally into pieces that are a) boundaries of d+1-dimensional currents and b) any d-dimensional stuff that is left. d+Here, the optimal decomposition has the minimum mass sum of the two parts where part (a) uses d+1-dimensional mass. If the input current has only integer multiplicities, then in general it is not known whether the optimal decomposition can be taken to be integral as well. In the particular case of codimension 1 boundaries, the flat norm is the same as the L1TV functional and this is known to be true. Here, I investigate the case of 1-currents (not required to be boundaries) in 2 dimensions (with some possible extensions to higher dimensions). This builds on previous work with the simplicial deformation theorem for pushing currents to simplicial complexes.