Hardt, Robert2017-08-022017-08-022017-052017-04-21May 2017Downes, Carol Ann. "A Mass Minimizing Flow for Real-Valued Flat Chains with Applications to Transport Networks." (2017) Diss., Rice University. <a href="https://hdl.handle.net/1911/96173">https://hdl.handle.net/1911/96173</a>.https://hdl.handle.net/1911/96173An oriented transportation network can be modeled by a 1-dimensional chain whose boundary is the difference between the demand and supply distributions, represented by weighted sums of point masses. To accommodate efficiencies of scale into the model, one uses a suitable Mα norm for transportation cost. One then finds that the minimal cost network has a branching structure since the norm favors higher multiplicity edges, representing shared transport. In this thesis, we construct a continuous flow that evolves some initial such network to reduce transport cost without altering its supply and demand distributions. Instead of limiting our scope to transport networks, we construct this M^α mass reducing flow for real-valued flat chains by finding a real current of locally finite mass with the property that its restrictions are flat chains; the slices of such a restriction dictate the flow. Keeping the boundary fixed, this flow reduces the M^α mass of the initial chain and is Lipschitz continuous under the flat-α norm. To complete the thesis, we apply this flow to transportation networks, showing that the flow indeed evolves branching transport networks to be more cost efficient.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Geometric Measure TheoryGeometric FlowsOptimal Transport TheoryThesis2017-08-02