Mathematics 4FT: Topics in Differential Equations

2012-2013: Applications of the Fourier transform in mathematical physics



Instructor: W. Craig
Meeting times: Monday 14:30 - 16:20 in HH 217 
                          Thursday 11:30 -12:20 in HH 410 (changed from the previous schedule of Friday 14:30 - 15:20 in HH 217)
Office hours: Tuesday 11:00 - 12:30 or by appointment, HH 418



Synopsis: This course will focus on the role that the Fourier transform plays in mathematical physics. We will describe the solution of the principal partial differential equations of physics in terms of the Fourier transform. Concurrently we will introduce the main analytic theory of Fourier series and Fourier integrals, including the theory of the geometry of Hilbert space and its relation to Fourier approximation. As well as the  emphasis on examples from physics, the course will draw motivation from other areas of mathematics, such as number theory and geometry.


There is no required text. Course notes are planned for library reserve.

Recommended texts: 
Dym & McKean, Fourier Series and Integrals
Rauch, Partial Differential Equations
Stein & Shakarchi, Fourier Analisis



Problem sets:
Problem set 1 (due Monday February 11 in class) .pdf file

Problem set 2 (due Monday March 11  in class) .pdf file

Problem set 3 (due Monday April 8 in class)  .pdf file

Takehome Final exam (due Friday April 26)  .pdf


Syllabus:

1) Introduction: Fourier's question     Lecture notes Section 1:

2) Mathematical physics     Lecture notes Section 2:  Lecture notes Section 2 - part 2:

    i)  classical harmonic oscillator
    ii)  linear ODEs: resolvant and path expansions
    iii)  Fourier's law and the heat equation
    iv) Derivation of the heat equation
    v)  Dirichlet's approximation theorem
    vi)  heat kernels and the maximum principle
    vii)  the wave equation
    viii) musical instruments
    ix) Schrödinger's equation

3)  Geometry of Hilbert space    Lecture notes Section 3:  Lecture notes Section 3 - part 2: (new version)
 
    i)  Hilbert space in coordinates
    ii)  Lebesgue integral primer* (background material, if necessary)
    iii)  completeness and the Lebesgue integral
    iv)  Fourier series and L2(T1)
    v)  convolution operators
    vi) differential operators
    vii) Fourier series on Td

4)  Applications of Fourier series to geometry    Lecture notes Section 4:

    i) Wirtinger's inequality
    ii) isoperimetric inequality
    iii) Jacobi's theta-function identity
    iv) equidistribution of irrational orbits
    v) recurrence of random walks

5)  Convergence properties of Fourier series    Lecture notes Section 5: Lecture notes Section 5 - part 2:

    i) Fejer's theorem
    ii) L1(T1)
    iii) Gibbs' phenomenon
    iv) lacunary series
    v) Pinsky's theorem

6)  Fourier integrals

    i)  Schwartz class
    ii) Fourier inversion theorem
    iii)  Hilbert space L2(R1)
    iv)  convolution and differential operators

7) Advanced mathematical physics

    i)  Heisenberg uncertainty principle
    ii)  heat kernel with potential
    iii) central limit theorem
    iv) Dyson expansion and the Feynman - Kac formula

8) Application of Fourier integrals to geometry

    i) Minkowski's theorem
    ii) Poisson summation formula
    iii) prime number theorem
    iv) Dirichlet's theorem on arithmetic progressions
   


last edits February 5 2013