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Ramified Optimal Transportation is the study of transporting `mass' from one measure, or `mass distribution', to another along branched transport paths. The transport cost of different paths is determined by the $M_\alpha$ cost functional, with different values of $\alpha \in [0,1]$ favoring different geometries and different optimal branching structures. The landscape function defined on an optimal transport path represents its marginal transportation costs, which can have many interesting connections and applications. Two different frameworks allow for defining landscape functions on two classes of transport paths, but unfortunately many transport paths are not yet supported by the existing theory.

I will introduce branched transport paths, the $M_\alpha$ functional, and some applications of ramified optimal transport (with pictures) before transitioning to focus on landscape functions. We will discuss what they are, the two current frameworks for defining them, and how the theory might be extended to cover more general classes of transport paths.

https://docs.google.com/spreadsheets/d/1wyOmPJvaqsSngBuIcjnN3vLAWeRFTyru47HFaT0wlMc/edit#gid=99207826