Given a distribution, say, of data or mass, over a space, it is natural to consider a lower dimensional structure that is most “similar” or “close” to it. For example, consider a planning problem for an irrigation system (1-dimensional structure) over an agricultural region (2-dimensional distribution) where one wants to optimize the coverage and effectiveness of the water supply. This type of problem is related to “principal curves” in statistics and “manifold learning” in AI research. We will discuss some recent results in this direction that employ optimal transport approaches. This talk will be based on joint projects with Anton Afanassiev, Jonathan Hayase, Forest Kobayashi, Lucas O’Brien, Geoffrey Schiebinger, and Andrew Warren.