## 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