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 L

v) convolution operators

vi) differential operators

vii) Fourier series on T

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) L

iii) Gibbs' phenomenon

iv) lacunary series

v) Pinsky's theorem

6) Fourier integrals

i) Schwartz class

ii) Fourier inversion theorem

iii) Hilbert space L

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