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