Variational problems in intersection homology theory and optimal transport

Abstract

This thesis studies geometric variational problems derived from intersection homology theory of singular varieties as well as from optimal transportation. Part I: Intersection homology theory via rectifiable currents . Here is given a rectifiable currents' version of intersection homology theory on stratified subanalytic pseudomanifolds. This new version enables one to study some variational problems on stratified subanalytic pseudomanifolds. We first achieve an isomorphism between this rectifiable currents' version and the version using subanalytic chains. Then we define a suitably modified mass on the complex of rectifiable currents to ensure that each sequence of subanalytic chains with bounded modified mass has a convergent subsequence and the limit rectifiable current still satisfies the crucial perversity condition of the approximating chains. The associated mass minimizers turn out to be almost minimal currents and this fact leads to some regularity results. Part II: Optimal paths related to transport problems. In transport problems of Monge's types, the total cost of a transport map is usually an integral of some function of the distance, such as | x - y|p. In many real applications, the actual cost may naturally be determined by a transport path. For shipping two items to one location, a "Y shaped" path may be preferable to a "V shaped" path. Here, we show that any probability measure can be transported to another probability measure through a general optimal transport path, which is given by a normal 1-current in our setting. Moreover, we define a new distance on the space of probability measures which in fact metrizies the usual weak * topology of measures. When we take into account the time consumption, we get a Lipschitz flow of probability measures, which helps us to visualize the actual flow of measures as well as the new distance between measures. Relations as well as related problems about transport paths and transport plans are also discussed in the end.