Sixth (bonus) exercises online.
Fifth exercises online.
Fourth exercises and solutions online.
Third exercises and solutions online.
Second exercises and solutions online.
Suggestions for presentation topics now online.
First exercises and solutions online.
Webpage and course materials updated.
Thursdays and Fridays 10-12 in MaD245.
First lecture: Oct 27.
This is an introductory course to Fourier restriction theory. The central question is whether the Fourier transform of an Lp function can be "restricted" to curves and surfaces in a meaningful way - and for which values of "p" this is true. Answering this question leads to profound (and mostly open) geometric problems, such as the Kakeya conjecture. Our specific topics will include:
Understanding the elementary geometry of the Fourier transform
Restriction and Kakeya conjectures and their relationship
Dimension of planar Kakeya sets
Khintchine's inequality and applications
Tomas-Stein restriction theorem
Wave packet decomposition
Reverse square function inequality
Sharp restriction theorem for the circle
The multiplier problem for the ball
We will mainly follow these lecture notes. For additional reading, see the books and/or lecture notes by
Fourier analysis I, although we will not need distribution theory. So, it suffices if you know Fourier series, Plancherel's theorem, and how the Fourier transform acts on Schwartz and Lp functions. This material is covered on these video lectures nos. 1-15. Some basics of Real analysis will also be used, e.g. what is meant by Borel measures -- othen than Lebesgue measure -- on Euclidean spaces.
The plan is that the course is passed by solving exercises and preparing a written + oral presentation (length: roughly 60min). There is a chance, however, that there are too many participants to have time for this, and then there will be an exam instead of the oral presentations. Here is a tentative list of topics for presentations (the written parts should be 5-10 pages):
Riesz-Thorin interpolation theorem (Grafakos: Classical Fourier Analysis -- reserved for Miro A.)
Change-of-variables formula for Lebesgue measure (Rudin: Real and Complex Analysis, Theorem 7.26 -- reserved for Lukas F.)
Sharper Lebesgue differentiation theorem for Sobolev functions (Mattila: Fourier Analysis and Hausdorff Dimension -- reserved for Jesse K.)
Finite field Kakeya and Nikodym conjectures via the polynomial method (Mattila: Fourier Analysis and Hausdorff Dimension + Tao: Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence geometry, and number theory -- reserved for Antti K.)
The following two topics are interrelated, and coordination is advised:
Fourier transform of the unit sphere (Mattila: Fourier Analysis and Hausdorff Dimension OR Stein: Harmonic Analysis -- reserved for Jani N.)
Fourier transform and Riesz energies, with applications to distance sets (Mattila: Fourier Analysis and Hausdorff Dimension)
The following two topics are interrelated, and coordination is advised:
Salem sets and Brownian motion (Mattila: Fourier Analysis and Hausdorff Dimension)
Fourier dimension of planar Kakeya sets (Mattila: Fourier Analysis and Hausdorff Dimension + Oberlin: Restricted Radon transforms and unions of hyperplanes)
Exercise session: Thursdays at 8.30 in MaD355. First session: November 3.
(Assistant: Damian Dabrowski).