Mathematics 541: Harmonic Analysis, Fall 2005
MWF 11-12, MATX 1102
Instructor: I. Laba.
Office: Math Bldg 239.
Phone: 822 2450.
Office hours: Mon 3-4, Tue 11-12, Fri 10-11.
Homework 1, due on Friday, October 7
Homework 2, due on Friday, November 18
This course will provide an introduction to Euclidean harmonic analysis
as a whole, followed by a closer look at selected areas of current
research, with particular focus on the Kakeya and restriction problems.
The latter are central problems in harmonic analysis which have also
been linked to deep questions in number theory and combinatorics,
and which have (at least indirectly) stimulated the recent exciting
developments related to Szemer\'edi's theorem on arithmetic progressions.
The goal is to provide a good foundation for further reading and
research, with technicalities kept to a minimum. I hope that the course
will be accessible and interesting to students with a variety of backgrounds.
Completion of Math 420/507 and 421/510, or equivalent familiarity with
basic real and functional analysis. Many essential
analysis topics are usually not included in Math 507 and 510:
Riesz representation theorem, Sobolev spaces, distributions,
interpolation, operator theory (eg. dual and unitary operators), to name just
a few. Such topics will be introduced as needed - typically, I will
only state the required results without proof.
- Basics: the Fourier transform, convolution, Plancherel theorem, Fourier
inversion, Hausdorff-Young inequality.
- Scaling and the uncertainty principle.
- The stationary phase method.
- Restriction, Tomas-Stein theorem.
- Selected research topics: Fourier transforms of singular measures, projections
and distance sets, the Kakeya problem.
T. Wolff, Lectures
on Harmonic Analysis, AMS 2003.
- The applications of the stationary phase estimates to the Schroedinger
equations were from Reed and Simon, "Methods of Modern Mathematical Physics",
vol. III: Scattering theory.
- For more on geometric measure theory (Hausdorff measure and dimension,
Fourier-analytic methods, etc) see P. Mattila, "Geometry of Sets and Measures
in Euclidean Spaces), or K. Falconer, "The Geometry of
Fractal Sets" (both published by Cambridge Univ.). In particular, my
presentation of the
construction of a Besicovitch set of measure 0 followed Falconer.