THEMATIC PROGRAMS

April 18, 2014

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.
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  • 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 (Monge-Ampere), 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 0-8218-3312-X $59 ($47 AMS members)
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  • 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, finite-time blowup, Stability/instability of solitary waves; long-time dynamics
5. Analysis of the blow-up: self-similarity; modulation analysis; rate of blow-up.


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 wave-type equations: formal asymptotic solutions, eikonal equation, wave fronts, rays, asymptotical properties of formal solutions
    3. Oscillating solutions of non-stationary wave-type equations. Oscillating solutions of Schroedinger-type equation (semiclassical approximation). Uniform global asymptotic representations
    4. Generalized solutions of PDE's. Singular solutions of wave-type 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 non-smooth solutions to these equations, in which the nature of the non-smoothness 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 0-8218-0772-2 $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