OPTIMAL TRANSPORT AND APPLICATIONS
Math 793H, Spring 2018
Instructor: Adrian Tudorascu
Office: Armstrong Hall 410F
· Lecture Schedule
The class will meet twice a week, on Tuesday and Thursday in HOD-D (Hodges Hall) 201 from 10:00 a.m. to 11:15 a.m. We will not meet on official holidays and on dates I need to travel for scientific purposes. On those dates, someone else may substitute; you will be notified in advance.
· Course Description and Goals
Most of the material will be presented in a manner consistent with the presentations in the text. Students are expected to read and study the examples and related material in the text and to work on the assigned problems sets. Similar problems will be used as examples during lectures as preparation for the exams.
Prerequisite: Math 451-452, Math 551.
The problem of optimal allocation of resources was first formulated by Gaspard Monge towards the end of the 18th century in the context of the optimal strategy to transport a given finite physical quantity (say, pile of dirt) into a receptacle of equal capacity (say, excavation with the same volume as the dirt pile). The solution to this problem depends on the cost associated with this transportation, and the problem was open for over two centuries before Yann Brenier solved it in the case of quadratic cost. Before Brenier’s solution, Leonid Kantorovich worked on this problem and solved some weaker (relaxed) formulation which lead to some practical applications to optimal allocation of resources (for which he was awarded the Nobel prize for Economics in 1975).
In this course I will present the Monge-Kantorovich theory and some applications to Fluid Dynamics (such as pressureless gas dynamics) and Economics (such as optimal allocation of resources).
Upon completion of this course, the student should be able to apply these notions and techniques to problem solving. For this, the student is expected to have thoroughly understood the theory linking the concepts.
There is not one particular text to be used for this course. Most of the material will come from “Topics in Optimal Transportation” by C. Villani, “Gradient Flows in Metric Spaces and in the Space of Probability Measures” by L. Ambrosio,
N. Gigli, G. Savaré, “Optimal Transport for Applied Mathematicians” by F. Santambrogio.
· Grading Policy and Evaluation
There will be periodically assigned projects (such as reading a paper and submitting a brief summary; likely two per semester), each accounting for 30% of the course grade, plus one final project/presentation (presenting a paper or book chapter in front of the class during the last two weeks of classes) which accounts for 40% of the course grade. The cutoffs will be 90% for an A, 85% for a B+, 80% for a B, 75% for a C+, 70% for a C and 60% for a D.
· Office Hours
When: Tuesday and Thursday between 2:30--3:30pm
Or by appointment.
Where: My Office.
· Course Policy
Class attendance is encouraged. The Office Hours are to be used only after you will have thoroughly read the material and tried to understand it.
Most likely, no calculators will be allowed on any test.
It is important that you not discard returned tests and graded homework assignments! Not only do they constitute good material for review but they are also the only acceptable proof in case of misrecorded grades.
You may discuss your assignments with each other; however, solutions should be written down individually. You should not read anyone else's completed work or show yours to anyone else.
Exams are to be worked on and written down strictly individually.
How to succeed in this class:
- Attendance will not be taken in this class, however, it is expected that you will attend class regularly. If you do miss a class it is your responsibility to find out what was covered and whether any important announcements were made.
- The single most important thing that you should do is work out at least the assigned homework. You should do the assigned problems, along with an assortment of unassigned problems, as a study aid.
- Collaboration on homework is a good thing. You are encouraged to discuss the homework and to work together on the problems.
- Like all mathematics, the material in the course cannot be learned passively. However reasonable, simple, or rational you may find what you read or hear, you do not understand it if you cannot apply it yourself. Thus it is imperative that you test yourself by doing problems. If you have difficulty with a problem, ask your instructor or your fellow students about it; do not assume that the difficulty will cure itself without treatment.
Have you questions or concern about the course, please see me during the office hours!