Mathematics 748: Topics in Mathematical Physics

2008-2009: The Fourier transform in PDE and harmonic analysis



Instructor: W. Craig
Meeting times: Mon & Fri 9:30 - 11:00 in HH 410
Office hours: Tues 2:00 - 3:30, HH418




Syllabus:

0) Introduction

1) Fourier transform

    i) transport equations
    ii) Schwartz class
    iii) the wave equation and Paley - Wiener theory
    iv) Schroedinger's equation and wave packets
    v) various Sobolev spaces
    vi) embedding theorems and Schauder spaces
    vii) BMO - the limiting case

2) pseudo-differential calculus

    i) examples
    ii) tempered distributions
    iii) pseudo-differential operators
    iv) operator composition
    v) elliptic operators
    vi) the wave front set
    vii) Egorov's theorem

3) Fourier integral operators

    i) hyperbolic equations
    ii) propagation of singularities
    iii) geometrical optics

4) Littlewood - Paley theory

    i) decompositions in dyadic annuli
    ii) Besov spaces and their embeddings
    iii) Fourier multipliers

5) para-differential calculus

    i) Bony decompositions
    ii) para-linearization theorem
    iii) calculus, composition, and commutation