
Thematic Program on Partial Differential Equations
Graduate Courses
Starting Fall Semester
Starts Sept. 10  Course on Partial Differential
Equations (Fall and Winter Term)
Starts Sept. 8  Course on Optimal Transportation
& Nonlinear Dynamics
Starts Sept. 8  Course on Wave Propagation
Starting Winter Semester
Starts Jan. 16  Course on Asymptotic Methods
for PDE
Starts Jan. 15  Course on Applied
Nonlinear Equations
Starts Jan. 13  Course on Inverse Problems

September 10, 2003  April 14, 2004
Wednesday 10:30  1:00
Course on Partial Differential Equations
Instructor: W. Craig
This is a one year course that is intended to be a graduate
level introduction to the theory of partial diferential equations
(PDEs). The course material will start with an overview of the basic
properties of the wave equation, Laplace's equation and the heat
equation; introducing Fourier transform techniques, distributions,
Green's functions, and some of the basic notions of the theory of
PDE. We will proceed to cover the general theory of PDE, including
first order theory, the Cauchy Kowalevskaya theorem and generalizations,
the Malgrange  Ehrenpreis theorem, and subsequent counterexamples
to existence. On a more general level, we will take up the theory
of elliptic equations and their regularity, symmetric hyperbolic
systems and energy estimates, and parabolic systems. We will then
move to the study of more advanced techniques, such as the development
of Brownian motion and Wiener measure, pseudodifferential and Fourier
integral operators, and methods of nonlinear functional analysis.
Throughout this course, an attempt will be made to connect the theory
to relevant examples of current research interest in mathematics
and its physical applications.
______________________________

September 8  December 11, 2003
Tuesdays 1:00  3:00, Thursdays 4:00  6:00
Course on Optimal Transportation & Nonlinear Dynamics
Instructor: R. McCann
* Thursdays, starting Sept. 11, 2003  Guest Lecturer:
John Urbas (Australian National University & The Fields Institute)
Fully Nonlinear Elliptic PDE: A Graduate Level Introduction
The optimal transportation problem of Monge and Kantorovich is now
understood to be a mathematical crossroads, where problems from economics,
fluid mechanics, and physics meet geometry and nonlinear PDE. This
course gives a survey of these unexpected developments, using variational
methods and duality to address free boundary problems, nonlinear elliptic
equations (MongeAmpere), regularity, geometric inequalities with
sharp constants, metric and Riemannian geometry of probability measures,
nonlinear diffusion, fluid mixing, and atmospheric flows.
References:
C. Villani. Topics in Optimal Transportation. Providence: AMS 2003.
GSM/58 ISBN 082183312X $59 ($47 AMS members)
______________________________

September 8, 2003  December 11, 2003
Mondays 10:00  11:30, Tuesdays 11:00  12:30
Course on Wave Propagation
Instructor: C. Sulem
1. Derivation of canonical equations of mathematical physics.
2. Small amplitude dispersive waves: Basic concepts, the nonlinear
Schr\"odinger (NLS) equation as an envelope equation.
3. Structural properties of the NLS equation: Lagrangian and Hamiltonian
structure, Noether theorem, invariances and conservation laws
4. The initial value problem: Existence theory, finitetime blowup,
Stability/instability of solitary waves; longtime dynamics
5. Analysis of the blowup: selfsimilarity; modulation analysis;
rate of blowup.
Winter/Spring Semester
 September 2003  April 14, 2004
Wednesday 10:30  1:00
Course on Partial Differential Equations (cont'd)
Instructor: W. Craig
This is a one year course that is intended to be a graduate level
introduction to the theory of partial diferential equations (PDEs).
The course material will start with an overview of the basic properties
of the wave equation, Laplace's equation and the heat equation; introducing
Fourier transform techniques, distributions, Green's functions, and
some of the basic notions of the theory of PDE. We will proceed to
cover the general theory of PDE, including first order theory, the
Cauchy Kowalevskaya theorem and generalizations, the Malgrange  Ehrenpreis
theorem, and subsequent counterexamples to existence. On a more general
level, we will take up the theory of elliptic equations and their
regularity, symmetric hyperbolic systems and energy estimates, and
parabolic systems. We will then move to the study of more advanced
techniques, such as the development of Brownian motion and Wiener
measure, pseudodifferential and Fourier integral operators, and methods
of nonlinear functional analysis. Throughout this course, an attempt
will be made to connect the theory to relevant examples of current
research interest in mathematics and its physical applications.

January 16  April 16, 2004
Course on Asymptotic Methods for PDE
Friday 10:00  1:00
Instructor: V. Buslaev
1.WKB asymptotics for ODE's. Turning points. Uniform asymptotic
representations.
2. Oscillating solutions of stationary wavetype equations: formal
asymptotic solutions, eikonal equation, wave fronts, rays, asymptotical
properties of formal solutions
3. Oscillating solutions of nonstationary wavetype equations.
Oscillating solutions of Schroedingertype equation (semiclassical
approximation). Uniform global asymptotic representations
4. Generalized solutions of PDE's. Singular solutions of wavetype
equations. Propagation of singularities.
 January 15  April 8, 2004
Thursday 12:00  3:00
Course on Applied Nonlinear Equations
Instructor: R. McCann
(MAT 1508S / APM 446S) An introduction to nonlinear partial differential
equations as they arise in physics, geometry, and optimization. A
key theme will be the development of techniques for studying nonsmooth
solutions to these equations, in which the nature of the nonsmoothness
or its absence is often the phenomenon of interest. The course will
begin with a survey at the level of Evans' textbook, followed by an
excursion into the mathematics of fluids.
References:
L.C. Evans "Partial Differential Equations" GSM 19 Providence
AMS 1998, GSM 19
ISBN 0821807722 $75 (\$60 AMS members)
A.J. Majda and A.L. Bertozzi "Vorticity and Incompressible Flow"
Cambridge Univ. Press 2002. ISBN 0521639484 \$40
 NOTES:
Exercise 1, Exercise
2, Exercise 3, Assignment
1 Example
 January 13  April 20, 2004
Tuesday 10:00  1:00
Course on Inverse Problems
Instructor: A. Nachman
Taking the Institute's Courses for Credit
As graduate students at any of the Institute's University Partners,
you may discuss the possibility of obtaining a credit for one or more
courses in this lecture series with your home university graduate officer
and the course instructor. Assigned reading and related projects may
be arranged for the benefit of students requiring these courses for
credit.
Financial Assistance
As part of the Affiliation agreement with some Canadian Universities,
graduate students are eligible to apply for financial assistance to
attend graduate courses. To apply for funding, apply
here
Two types of support are available:
 Students outside the greater Toronto area may apply for travel support.
Please submit a proposed budget outlining expected costs if public
transit is involved, otherwise a mileage rate is used to reimburse
travel costs. We recommend that groups coming from one university
travel together, or arrange for car pooling (or car rental if applicable).
 Students outside the commuting distance of Toronto may submit an
application for a term fellowship. Support is offered up to $1000
per month.
For more details on the thematic year, see Program
Page or contact thematic@fields.utoronto.ca

