**Fall 2011**

**Meetings:** TR 2:10-3:30pm, MSB 3106

**Instructor:** Qinglan Xia

Office: MSB 3216

Email: qlxia@math.ucdavis.edu

Phone: (530)752-1084

**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 problem

References:

1. Some journal articles by Xia and others.

2. Optimal transportation networks; Bernot, Caselles, Morel.

3. Lectures on Geometric measure theory; Leon Simon