DATA-INSPIRED ADVANCES IN GEOMETRIC MEASURE THEORY: GENERALIZED SURFACE AND SHAPE METRICS Abstract by Sharif N. Ibrahim, Ph.D. Washington State University August 2014 Chair: Kevin R. Vixie Modern geometric measure theory, developed largely to solve the Plateau problem, has generated a great deal of technical machinery which is unfortunately regarded as inaccessible by outsiders. Consequently, its ideas have not been incorporated into other fields as effectively as possible. Some of these tools (e.g., distance and decompositions in generalized surface space using the flat norm) hold interest from a theoretical perspective but computational infeasibility prevented practical use. Others, like nonasymptotic densities as shape signatures, have been developed independently as useful data analysis tools (e.g., the integral area invariant). Here, geometric measure theory has promise to help close the gaps in our understanding of these ideas. The flat norm measures distance between currents (or generalized surfaces) by de- composing them in a way that is robust to noise. One new result here is that the flat norm can be suitably discretized and approximated on a simplicial complex by means of a simplicial deformation theorem. While not surprising given the classical (cubical) deformation theorem or, indeed, Sullivan’s convex cellular deformation theorem (which includes simplicial deformation as a special case), the bounds on the deformation can be made smaller and more practical by focusing on the simplicial case. Computationally, the discretized flat norm can be expressed as a linear programming problem and thus solved in polynomial time. Furthermore, the solution is guaranteed ii