Math 542 - Topics in Harmonic Analysis - Spring 2018
Instructor: Malabika Pramanik
Office: 214 Mathematics Building
E-mail: malabika at math dot ubc dot ca
Lectures: Tuesday and Thursday 12:30 PM to 2:00 PM in Room 203 of Mathematics Building.
Office hours: Tuesday 10-11, Thursday 2-3 or by appointment.
Course information
Week-by-week course outline
This section contains a summary of the material covered in class, arranged by week. Supplementary reading material is mentioned as a reference and as a study guide. The treatment of these topics in lecture may vary somewhat from that of the text.
- Week 1:
- The polynomial method: an overview
- Parameter counting and vanishing lemmas
- Week 2:
- Finite field Kakeya problem
- Finite field Nikodym problem
- The joints problem
- Week 3:
- A brief tour of problems in geometric measure theory
- Notions of dimension
- Hausdorff, Minkowski and Fourier dimensions
- Week 4:
- Marstrand projection theorem
- Properties of Hausdorff dimension
- Energy integrals and Frostman's lemma
- Week 5:
- Proof of Frostman's lemma for compact Borel sets
- Distance sets and Falconer's conjecture
- Week 6:
- Borel subrings of the reals
- Restriction and extension results
- Restriction for the sphere: the Knapp example
- Week 7:
- Stein-Tomas restriction theorem for measures
- Restriction conjecture
- Week 9:
- Existence of Kakeya-Besicovitch sets
- Kakeya conjecture
- Fourier dimension of Kakeya sets
- Week 10:
- Kakeya maximal function
- Kakeya maximal conjecture and known bounds
- Dual formulation of the Kakeya maximal conjecture
- Week 11:
- Restriction implies Kakeya
- Week 12: Student-led presentations
- Finer properties of the exceptional set in Marstrand projection theorem
- Fourier dimension of Brownian images of sets