Variational problems in optimal mass transportation and intersection homology theory

University Of California-Davis, Davis CA

Investigators

Abstract

VARIATIONAL PROBLEMS IN MASS TRANSPORTATION AND INTERSECTION HOMOLOGY PI: QINGLAN XIA ABSTRACT: The goal of the proposed projects is to study geometric variational problems derived from optimal mass transportation as well as from the intersection homology theory of singular varieties. The first proposed project is building a model to simulate ``tree shaped'' objects in nature, and apply it to optimal mass transportation problems. The problems proposed here include building a transport theory of rectifiable varifolds, finding connections of this model with other problems, and extending this theory to more general spaces. The second project concerns minimal surfaces that live in singular spaces such as singular complex projective varieties under their intersection homology groups. In his thesis, the PI gave a rectifiable currents' version of intersection homology theory on stratified subanalytic pseudomanifolds. He showed that there exists a modified mass minimizer (which is a rectifiable current) in every intersection homology class. The associated mass minimizers turn out to be almost minimal currents. The mass minimizers the PI considered may intersect (in a controlled fashion) the singular locus of the singular space. In this proposal, the PI will continue his investigation on regularity properties of the associated mass minimizers, and possible link with Hodge theory. One of the main methods used in both projects is geometric measure theory. The phenomenon of ``tree shaped'' path is very common in nature. Trees, railways, airlines, lightning, the circulatory system, and neural networks are common examples. The research of optimal ``tree shaped'' path not only expands the research area of mass transportation, but also of both theoretical and applied interest. Biologists are currently looking for a model to give a fundamental explanation for biological scaling laws which are mathematical expressions of how organisms' biology varies with their size. The model proposed in this research might be the desired one. As Monge's mass transport problem is strongly linked with many areas of mathematics, especially with partial differential equations, it is also possible that this new approach of mass transportation will find its applications in economics, biology, image processing and some other subjects. Soap films are physical model for minimal surfaces. These surfaces play an important role as a tool in the study of topology, geometry and physics. The research of the second project concerns global properties of soap films that live in more general singular spaces and possible links to Hodge theory and optimal mass transportation. It might provides some hint to the famous Hodge conjecture.