Meetings: TR 2:10-3:30pm, MSB 3106
Instructor: Qinglan Xia
Office: MSB 3216
Description: This topic course aims at introducing ramified optimal transportation to graduate students in both pure and applied mathematics. Ramified transportation formally formulates the concept of transport economy of scale in group transportation observed widely in both nature (e.g. trees, blood vessels, river channel networks, lighning) and efficiently designed transport systems of branching structures (e.g. railway configurations and postage delivery networks). Motivations of this theory originate from the study of minimal surfaces in geometry as well as designing optimal communication networks. During the course, we will first study mathematical formulations of the problem. Our attention will be focused on studying analytical as well as geometric properties of an optimal transport path. Then, we will study its applications in both pure mathematics (e.g. metric geometry) and applied mathematics (e.g. formation of tree leaves, fractals, mathematical economics). Suitable audiences include those who are interested in geometry, combinatorics, analysis, optimization, complex structures, fractals, math biology, mathematical economics etc, and in particularly those who are amazed by the beauty of the nature.
Outline of topics (Draft):
Part I: Ramified optimal transportation between atomic measures
1. Motivations. Mathematical models of optimal transport networks;
2. Optimal transport paths between atomic measures;
3. Geometric structures of optimal transport paths;
4. Analysis on optimal transport paths;
5. Numerical simulations of optimal transport paths;
Part II: Optimal transport paths between probability measures in general.
1. Basic concepts in geometric measure theory;
2. Interior regularity of optimal transport paths;
3. Boundary regularity of optimal transport paths;
4. Other models;
Part III: Applications in theoretical mathematics
1. Introduction to Quasi-metric spaces;
2. Optimal transport path viewed as geodesics in quasi-metric spaces.
3. Transport dimension of measures
4. Ramified optimal transportation in geodesic metric spaces.
5. p-harmonic maps on weighted graphs, and its connection to ramified transportation.
Part IV: Applications in applied mathematics
1. The formation of a tree leaf;
2. Diffusion-limited aggregation driven by optimal transportation.
3. The exchange value embedded in a transport system
4. On the ramified optimal allocation problemTextbook: No textbook is required, but I plan to write one using the lecture notes generated during this course. I will make available detailed lecture notes covering the course material.
1. Some journal articles by Xia and others.
2. Optimal transportation networks; Bernot, Caselles, Morel.
3. Lectures on Geometric measure theory; Leon Simon