Mathematics 601 (Topics in Analysis)
Proposed for 2013-14 but rejected by the department
The Harmonic Analysis group plan is to offer the Math 541/542 series every other year (Malabika Pramanik is
teaching it this year, 2012-13) and a third more "discrete" course in the remaining years. The proposed Math 543 has never
been approved even though I submitted the paperwork several times, so I'm proposing this course as Math 601
instead. [Update, August 2013: Math 543 has now been added to the UBC calendar.]
The subject of the course will be polynomial and algebraic-geometric methods in harmonic analysis and additive
combinatorics. The main goal is to present the recent solution solution (by Larry Guth and Nets Katz) of the
Erdos distance set problem, along with the necessary background, related results and other applications of
algebraic-geometric methods. Specific highlights will include:
- Introduction to the polynomial method: Dvir's solution of the finite field Kakeya problem
- Schlag's geometric proof of Bourgain's Kakeya estimates in 3d (an earlier instance of using algebraic geometry
in harmonic analysis).
- The multilinear Kakeya problem: Bennett-Carbery-Tao and Guth papers.
- Distance set problem: Elekes-Sharir on incidences, Guth-Katz on the joints problem, Guth-Katz on distance sets.