Mathematics Colloquia and Seminars

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The transport problem introduced by Monge in 1781 has been studied in many works recently. In these works, the cost of a transport mapping or a transport plan is usually an integral of some function of the distance. However, in many real applications, the actual cost of the transport procedures is not necessarily determined by just knowing some optimal mapping from the starting position to the target position. For example in shipping two items from nearby cities to the same far away city, it may be less expensive to first bring them to a common location and put them on a single truck for most of the transport. In this case, a "Y shaped" path is preferable to a "V shaped" path. In both cases, the transport mapping is trivially the same, but the actual transport path naturally gives the total cost. In general, a ramifying structure is more cost efficient than a "linear" structure. This phenomenon of ramified transportation is very common in nature. Trees, railways, airlines, lightning, electric power supply, the circulatory system, the river channel networks, and cardiovascular systems are some common examples. This subject deserves a more general theoretical treatment and thus I built a model for it in a series of papers. In this talk, I am going to discuss my approach to this interesting problem. We will see how to set up the problem in terms of geometric measure theory. Also, we will discuss the nice properties of optimal transport paths. If time permits, I will also provide the description of the dynamic formation of a tree leaf, as an application of this theory.