2.7 Discussion Our result on simplicial deformation (Theorem 2.5.1) places the definition of the mul- tiscale simplicial flat norm into clear context. If a current lives in the underlying space of a simplicial complex, we can deform it to be a simplicial current on the simplicial complex, and do so with controlled error. In fact, by subdividing the simplicial complex carefully, we can move this error as close to zero as we like. Since the multiscale simpli- cial flat norm could be computed efficiently when the simplicial complex does not have relative torsion, one could naturally use our approach to compute the flat norm of a large majority of currents in arbitrarily large dimensions. An important open question in this context is whether the multiscale simplicial flat norm of a current on a simplicial com- plex with relative torsion could be approximated efficiently by coarsening the complex so that the relative torsion is removed. For instance, it has been observed recently that edge contractions could remove existing relative torsion while preserving the homology groups of the simplicial complex in certain cases [15]. The multiscale simplicial flat norm problem, similar to the recent results on the optimal bounding chain problem [16], apply notions from algebraic topology and discrete optimization to problems from geometric measure theory such as flat norm of currents and area-minimizing hypersurfaces. What other classes of problems from the broader area of geometric analysis could we tackle using similar approaches? One such question appears to be the following: under what conditions is the flat norm decomposition of an integral current guaranteed to be another integral current? Working in the setting of simplicial complexes, results on the existence of integral optimal solutions for instances of ILPs with integer right-hand side vectors may prove useful in answering this question. While L $^{1}$TV and flat norm computations have been used widely on data in two dimensions, such as images, the multiscale simplicial flat norm opens up the possibility 49