This project concerns geometric variational problems derived from optimal transportation as well as from the intersection homology theory of singular varieties. The first part of the project studies optimal ramified (i.e. branching) transportation, which was modeled after many branching structures in nature such as trees, river channel networks etc. A central concept there is the notion of optimal transport path, which mathematically plays the role of a "geodesic" between two probability measures. Graphically, it has a "tree-shaped" branching structure similar to fractals. Some of the problems to be addressed within the project are the following: (1) Optimal ramified transportation in metric measure spaces; (2) Optimal ramified transportation of higher dimensional geometric objects (3) Flow of surfaces driven by curvature and transportation (including both ramified type transportation and Monge-Kantorovich type transportation), which are modeled after the growth of tree leaves, flowers and mud-cracking; (4) Dimensional distance between measures and sets. The second part of the project studies properties of "minimal surfaces" lived in singular varieties, using geometric measure theory. The PI proved that the intersection homology theory of MacPherson and Gorsky on singular varieties can be reformulated in terms of rectifiable currents, and also studied existence as well as partial regularities of minimizers under suitable modified masses within each intersection homology group. The PI aims at further investigations of these minimizers using analytic tools developed in geometric measure theory.
One of the main purposes of mathematics is to explore the beauty and mechanisms of many naturally generated shapes such as soap films, trees, leaves, and mud cracking. Many times, optimization process plays a very important role in the formation of these shapes. For instance, soap films come from minimizing surface area, while trees adopt branching transport systems to minimize transportation cost. This project aims at studying geometric shapes whose formation is driven by some optimization process. The first part of this project studies "tree-type" branching systems, which are commonly found in living and non-living systems such as trees, railways, river channel networks, lightning, the circulatory system, and neural networks. Nature has selected these branching transport systems partially because they are comparably cost efficient. As a result, to study mechanisms behind the formation of these branching systems, one may simply start with the study of optimal transport systems. In the last few years, the PI and many others have developed geometric variational methods for studying optimal ramified transportation, and also found its application in modeling the formation of tree leaves. In this project, the principal investigator is interested in developing the existing theory in a more general setting, and then applies it to transport specific geometric objects. Also, by considering geometric flow of surfaces driven by curvature and transportation, one may use them to model the formation of leaves, flowers, and some other fractal type objects. On the other hand, soap films are physical model for minimal surfaces, which play an important role as a tool in the study of topology, geometry and physics. The second part of the project aims at studying general properties of "soap films" lived in singular varieties, which are very nice but usually contains some singularities, within their intersection homology groups. We will mainly use geometric measure theory as our analytic tools.
