Mathematics 542 (Harmonic Analysis II)
Instructor: I. Laba (Math Bldg 200, 604-822-4457,
Lectures: MWF 12-1, MATX 1118.
Office hours: Mon 2-3, Wed 1-2, and by appointment.
This course is a continuation of
MATH 541 and will cover more advanced topics in
harmonic analysis on Euclidean spaces.
Your course grade will be based on 4 problem sets, tentatively due on January 26,
February 9, March 9 and March 30.
Each problem set will be worth 25% of your grade. There will be no final exam.
Alternatively, you may elect to give a 1 hour in-class presentation on a topic related to the
course material. You would also have to prepare a written exposition to be handed out in class.
A presentation would be fairly substantial (a research paper
rather than just a section of a textbook) and would replace 3 of the homeworks.
Here are a few possible topics:
- Dvir's proof of the finite field Kakeya conjecture (Z. Dvir, On the size of Kakeya sets in
finite fields J. Amer. Math. Soc. 22 (2009), 1093-1097) (Rhoda Sollazzo)
- Bourgain's circular maximal theorem (J. Bourgain, Averages in the plane over convex curves and
maximal operators, J. Anal. Math. 47 (1986) 69-85. This would replace all 4 homeworks if anyone
wants to do it)a (Marc Carnovale)
- Salem's construction of Salem sets (R. Salem, On singular monotonic functions whose spectrum has
a given Hausdorff dimension, Ark. Mat. 1, (1951). 353-365) (Tatchai Titichetrakun)
- Fefferman's counterexample to the ball multiplier problem and an introduction
to the Bochner-Riesz conjecture (C. Fefferman, The multiplier problem for the ball, Ann. Math. 94 (1971), 330-336, plus some expository reference on Bochner-Riesz) (Vince Chan)
- Maximal functions over a Cantor set of directions (M. Bateman, N. Katz, Kakeya
sets in Cantor directions, Math. Res. Lett. 15 (2008), 73-81 (Ed Kroc)
- Maximal functions and differentiation theorems
- The Hardy-Littlewood maximal function
- The spherical maximal function in higher dimensions
- Kakeya sets and the Kakeya maximal function
- The restriction problem for the sphere
- The Tomas-Stein theorem
- The restriction conjecture and the Kakeya problem
- Fourier transforms of singular measures
- Hausdorff measure and Hausdorff dimension
- The energy inequality and Fourier dimension
- Salem sets
- Applications to geometric measure theory: convolutions, projections, distances
- Fourier Analysis, J. Duoandikoetxea, American Mathematical Society, 2001
- Harmonic Analysis: Real-Variable Methods, Orthogonality, and
Oscillatory Integrals, E.M. Stein, Princeton Univ. Press, 1993
- Real Analysis, E.M. Stein and R. Shakarchi, Princeton Univ. Press, 2003
- Lectures on Harmonic Analysis, T. Wolff, American Mathematical Society, 2003
Prerequisites: MATH 541, or equivalent background in basic harmonic analysis.