Analysis Seminar: "The Simplicial Multiscale Flatnorm and Deformation Theorem"
4:10pm; Neill Hall 5W
In geometric measure theory, the flat norm is defined for currents (generalized oriented surfaces) and finds theoretical use in minimal surface problems and practical use in image analysis. Unfortunately, aside from some (useful!) special cases, it is not known how to compute the flat norm directly. This talk discusses a discretization of the flat norm restricted to currents which are oriented chains on a simplicial complex. This simplicial flat norm can be modeled by an integer linear optimization problem which can be solved in polynomial time when the complex has a desirable topological condition (absence of relative torsion). The simplicial deformation theorem is our extension of a classical result for cubical grids and shows that arbitrary currents can be well-approximated in a precise sense on sufficiently fine simplicial complexes. This is a nontrivial result; the input current might be dense in the underlying space of the simplicial complex and be challenging to represent as required on a lower dimensional skeleton of the complex.