Mathematics 4AT: Topics in Analysis

2008-2009: The Fourier transform and its applications



Instructor: W. Craig
Meeting times: Mon 12:30, Tues 13:30 & Fri 12:30 in JHE 329
Office hours: tba, HH418




Course Structure:

    There will be six micro-modules of approximately two weeks duration apiece.
    A problem set is associated with each micro-module.
    There will be a final exam.




Syllabus:

(0) Introduction

(1) Mathematical physics

    i) Fourier's problem
    ii) derivation of the heat equation
    iii) Dirichlet's theory - uniform approximation in C2(T)
    iv) the wave equation
    v) musical instruments
    vi) Schrödinger's equation

(2) Geometry of Hilbert space

    i) Hilbert space in coordinates
    ii) completeness and the Lebesgue integral
       ii)+ Lebesgue integration (optional)
    iii) Fourier series and L2(T)
    iv) convolution
    v) differential operators
    vi) Fourier series on Td

(3) Applications of Fourier series

    i) Geometry
     - Wirtinger's inequality
     - the isoperimetric inequality
     - Jacobi's Theta-function identity
    ii) ergodic theory
     - equidistribution under irrational rotation
     - recurrence of random walks

(4) Convergence properties of Fourier series

    i) Fejer's theorem
    ii) L1(T)
    iii) Gibbs' phenomenon
    iv) lacunary series
    v) diverging Fourier series
    vi) Pinsky's theorem

(5) Fourier integrals

    i) Schwartz class
    ii) the Fourier inversion theorem
    iii) Hilbert space L2(R)
    iii) convolution and differential operators

(6) Applications of Fourier integrals

    i) mathematical physics
      - Heisenberg's uncertainty principle
      - the heat kernel
     - the central limit theorem
    ii) number theory
      - Minkowski's theorem
      - the Poisson summation theorem
      - the prime number theorem
      - Dirichlet's theorem (primes within arithmetic progressions)