Course project

  • Y. Barsheshat - The Yamabe problem
  • S. Lu - Lin's local isometric embedding theorem
  • G. Martine-La Boissonière - Approximations in quantum mechanics
  • M. Stevenson - The heat kernel for forms


  • Assignment 1 [tex] due Wednesday January 29
  • Assignment 2 [tex] due Wednesday February 12
  • Assignment 3 [tex] due Monday March 10
  • Reading assignment: §6 and §10 of Distributions notes, by Monday March 10
  • Assignment 4 [tex] due Monday March 24
  • Assignment 5 [tex] due Wednesday April 9

    Lecture notes

  • Distributions [updated on 14.03.07]

    Class schedule

  • MW 13:05–14:25, Burnside Hall 1234

    Date Topics
    M 1/6 The Euclidean heat kernel. Parabolic maximum principles. Tychonov's uniqueness theorem
    W 1/8 Heat equations with variable coefficients. Heat kernels. The Laplace-Beltrami operator
    M 1/13 The Minakshisundaram-Pleijel recursion formula and heat parametrices
    W 1/15 Duhamel's formula. Construction of the heat kernel
    M 1/20 Heat trace asymptotics. Weyl's law. The zeta function and the determinant of the Laplacian
    W 1/22 The Hodge theorem
    M 1/27 The Cauchy-Kovalevskaya theorem. The Janet-Cartan isometric embedding theorem
    W 1/29 Proof of the Cauchy-Kovalevskaya theorem
    M 2/3 Characteristic surfaces
    W 2/5 Holmgren's theorem
    M 2/10 Locally convex spaces
    W 2/12 Fréchet spaces. Embeddings
    M 2/17 Test functions and distributions
    W 2/19 Subspaces of distributions. Basic operations
    M 2/24 Compactly supported distributions. Local structure
    W 2/26 More on compactly supported distributions
    M 3/10 Complex valued distributions. Fundamental solutions
    W 3/12 Convolutions
    M 3/17 Schwartz's theorem on hypoellipticity
    W 3/19 The Fourier transform. Paley-Wiener theorem
    M 3/24 Tempered distributions. Liouville's theorem
    W 3/26 Laurent expansion. Malgrange-Ehrenpreis theorem
    M 3/31 Hypoelliptic polynomials. The Seidenberg-Tarski theorem
    W 4/2 Hörmander's theorem
    M 4/7 No class
    W 4/9 Petrowsky's theorem. Fundamental solution of the Cauchy problem

    Student seminar

  • Thursdays 11:30–13:00, Burnside Hall 1214

    Date Topics Speaker
    Th 1/23 Hahn-Banach theorem
    Th 1/30 Basic Riemannian geometry
    Th 2/6 Baire's theorem and its consequences
    Th 2/13 Fredholm operators
    Th 2/20 Weyl's law
    Th 2/27 Weyl's law
    Th 3/13 No seminar
    Th 3/20 Fredholm property of elliptic operators
    Th 3/27 Optimal transport problem
    Th 4/3 The heat kernel for forms
    Th 4/10 Spectral theorem

    Course outline

    Instructor: Dr. Gantumur Tsogtgerel

    Prerequisite: MATH 580 (PDE1), MATH 355 (Honours Analysis 4) or equivalent

    Note: If you plan to take this course without taking MATH 580, please consult with the instructor.

    Topics: After covering some topics related to the heat equation and the Cauchy-Kovalevskaya theorem, the main focus of the course is going to be on the theory of distributions, general constant coefficient operators, semilinear evolution equations, and strongly elliptic systems. The approach will be from the viewpoint of a general theory, rather than as apparently ad hoc treatments of specific examples. As such, distributions, the Fourier transform, and functional analytic methods will be heavily used. More precisely, the planned topics are

  • The heat equation: maximum principles, the heat kernel, uniqueness theorems
  • Nirenberg-Nishida, Cauchy-Kovalevskaya, and Holmgren theorems
  • Distributions, convolution, Fourier transform
  • Constant coefficient operators, fundamental solution, hypoellipticity
  • Malgrange-Ehrenpreis theorem
  • Hörmander's characterization of hypoelliptic polynomials
  • Introduction to wave front sets, microlocal regularity
  • Problems in half-space, shades of hyperbolicity, parabolicity, and ellipticity
  • Fourier analytic treatment of Sobolev spaces
  • Semilinear evolution equations, Duhamel's principle
  • The Navier-Stokes equations
  • Overview of elliptic theory, Lopatinsky-Shapiro condition
  • Strongly elliptic systems, Gårding inequality
  • Elliptic regularity
  • Introduction to pseudodifferential operators (if time permits)
  • Semilinear elliptic problems with critical exponents (if time permits)

    Calendar description: Systems of conservation laws and Riemann invariants. Cauchy-Kowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.

    Books: There is no required textbook. The followings are recommended.

  • Gerald Budge Folland, Introduction to partial differential equations. Princeton 1995.
  • Gregory Il'ich Eskin, Lectures on linear partial differential equations. AMS 2011.
  • Joseph Theodor Wloka, Partial differential equations. Cambridge 1987.

    Homework: Assigned and graded roughly every other week.

    Weakly seminars: We will organize weekly seminars on standard results from analysis and geometry, and other stuff related to the course.

    Course project: The course project consists of the student reading a paper or monograph on an advanced topic, typing up notes, and giving a lecture.

    Grading: The final grade will be the weighted average of homework 20%, the take-home midterm exam 30%, and the course project 50%.

    Online resources

  • Lecture notes by Bruce Driver (UCSD)
  • Teaching page of John Hunter (UC Davis)