Rice University

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Thesis Defense

Graduate and Postdoctoral Studies
Mathematics

Speaker: Carol Downes
Doctoral Candidate

A Mass Minimizing Flow for Real-Valued Flat Chains with Applications to Transport Networks

Friday, April 14, 2017
12:00 PM  to 2:00 PM

427  Herman Brown Hall


An 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 $\mathbf{M}^\alpha$ norm for transportation cost. One then fi nds 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 $\mathbf{M}^\alpha$ mass reducing flow for real-valued flat chains by finding a real current of locally fi nite 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 $\mathbf{M}^\alpha$ mass of the initial chain and is Lipschitz continuous under the flat-$\alpha$ 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.

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