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

• Course information handout containing syllabus and grading policies.

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