The goal of this proposal is to study some geometric variational problems derived from optimal transportation. The PI proposes projects related to this goal with an emphasis on the following: (a) studying the optimal allocation problem arisen in mathematical economics with ramified transport technology; (b) modeling the dynamic formation of mud cracking using Monge-Kantorovich optimal transportation; and (c) modeling vascular tree structures of the placentas using ramified optimal transportation.
The optimal transportation problem aims at finding an efficient allocation plan for transporting some commodity from sources to targets. In particular, ramified optimal transportation is used to model the transport economy of scale in group transportation observed widely in both nature (e.g. trees, river channel networks) and efficiently designed transport systems of branching structures (e.g. railway configurations and postage delivery networks). The proposed optimal allocation problem is the prototype for a class of problems (e.g. optimal storage problem) arising in mathematical economics. It aims at finding an optimal allocation plan as well as an associated optimal allocation path to minimize overall cost of transporting commodity from factories to households. Also, the proposed dynamic model for mud cracking not only increases our understanding about mud cracking but also may be useful in predicting the position of future cracking of a similar material (e.g. a prediction of an earth quake), and hence provide increased public safety. Furthermore, the quantities found in the proposed model of placentas using ramified transportation can potentially be used as part of diagnostic tools in predicting healthy problem for newborns with abnormal placentas.