MAT 1839 Optimal Transportation, Dynamics, and Ricci Curvature  Sept 11, 2008 
              http://www.math.toronto.edu/mccann/1839.html

 Prof. Robert McCann  www.math.toronto.edu/mccann   BA 6124  (416) 978-4658

 Lectures:             Thursday    4:10 -  5:30 PM   BA 6183
 (except Oct 30-31)    Friday      4:10 -  5:30 PM   BA 6183

 Office Hours:         Friday      3:10 -  4:00 PM   BA 6124

This course is an introduction to the active research areas surrounding 
optimal transportation and its deep connections to problems in dynamical 
systems, geometry, physics, and nonlinear partial differential equations.
The basic problem is to find the most efficient structure linking two 
continuous distributions of mass --- think of pairing a cloud of electrons 
with a cloud of positrons so as to minimize average distance to annihilation. 
Applications include existence, uniqueness, and regularity of surfaces 
with prescribed Gauss curvature (the underlying PDE is Monge-Ampère), 
geometric inequalities with sharp constants, periodic orbits for dynamical 
systems, long time asymptotics in kinetic theory and nonlinear diffusion, 
and the geometry of fluid motion (Euler's equation and approximations 
appropriate to atmospheric, oceanic, damped and porous medium flows). 
The course builds on a background in analysis, including measure theory, 
but will develop elements as needed from the calculus of variations, 
game theory, convexity, elliptic regularity, dynamical systems 
and fluid mechanics, not to mention physics, economics, and geometry.

Depending on interest and preparation of the students, we will go more or 
less deeply into certain aspects of the theory, such as geometry and 
geometric evolution equations including Ricci flow, the question of when 
the solution to a fully nonlinear degenerate elliptic equation is smooth, 
or into applications such as atmospheric and economic modelling. 
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Text: Cedric Villani "Topics in Optimal Transportation" 
      Providence: AMS 2003 GSM/58 ISBN 0-8218-3312-X

See also further sources periodically linked to the course webpage above.

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Grading Scheme:    Attendance and participation                     20 %
                   Written Project                                  50 %
                   Oral presentation                                30 %


INDEPENDENT PROJECT                                             DEADLINES
                                   Settle on a topic:   Friday,   Oct. 10
                                   Written report:    Thursday,   Nov. 20 

    This project is intended to provide you with an opportunity to 
independently pursue some topic of interest to you within the framework of 
the course.  Those taking this course for credit will be expected to prepare 
a written report on your topic (something on the order of 6-12 pages), and 
give a presentation on the same topic in the second half of the semester.

    One possibility is to choose a paper of interest to you, study it,
and write a summary explaining its interest, importance and key 
contributions.  Chapter 10 of Villani contains suggests many possible
topics in this vein,  and I am happy to discuss others with you.
My webpage and the course webpage are other resources.  In any case you 
should settle on a topic and format in consultation with me by Oct. 10.