Math 8210 – Topics

Topics in Harmonic Analysis and Geometric Measure Theory (GMT)

Upcoming Zoom Talks related to this course: 

  • MAAM Mid Atlantic Analysis Seminar
  • U. Edinburgh– Virtual Harmonic Analysis Seminar
  • Friday, August 28, 12:00 (noon) Eastern time, Montreal Zoom Seminar, video
    Malabika Pramanik (UBC), Restriction of eigenfunctions to sparse sets on manifolds, abstract and zoom info.
  • Thursday, August 27, 4:15 pm Eastern time, OSU Zoom Colloquium, Kenneth Falconer, Symmetry and Enumeration of Fractals
  • MAAM: http://web.sas.upenn.edu/maam2/

List of Topics and Readings by week:

  1. Falconer’s distance problem and special colloquium talk by Falconer; reading: What is Falconer’s Conjecture?
  2. The appearance of the Fourier transform in GMT: Mattila’s Introduction, Hausdorff measures, Minkowski dimension, weak convergence, energy integrals; reading: Introduction to Mattila’s book, Advice to a Young Mathematician.
  3. Differentiation of measures, Fourier transform, convolutions and measures, Vitali covering
  4. Projection theory: Marstrand’ projection theorem, transversality
  5. 1-sets, purely unrectifiability, Besicovitch’s projection theorem, Buffon needle problem, visibility and radial projections
  6. Bonger’s and Mattila1990 Mattila1990Theorem 4.1 for the decay of Buffon needle, Buffon circle; Connections: PDE, Probability
  7. Davies theorem and the Kakeya conjectures, slicing problems
  8. Presentations, Reading, and Discussion
  9. Arithmetic operations on fractals: Steinhaus, Erdős and Oxtoby, category analogue and generalizations
  10. Distance sets and Finite point configurations within fractal sets
  11. Presentation and Discussion; Big picture in Harmonic Analysis
  12. Davies theorem and Kakeya
  13. Oscillatory Integrals and Restriction (see Wolff notes section 7 starting on page 41)
  14. The big topics in harmonic analysis and lattice point counting; Connections:  Number theory
  15. The big topics in harmonic analysis continued…

Survey Articles:

Research Articles

  • Csornyei How To Make Davies’ Theorem Visible (Nik)
  • Csornyei The Kakeya Needle problem and the existence of Besicovitch and Nikodym sets for rectifiable sets (Nik?)
  • Erdos_Oxtoby (Steinhaus)
  • IMT Intersections of sets and Fourier analysis
  • IMT Fourier integral operators, fractal sets and the regular value theorem (Mathew)
  • Jiang A nonlinear version of the Newhouse thickness theorem
  • ST Interior of Sums of Planar Sets and Curves (Alexander)
  • Guth, Netz On the Erdős distinct distance problem in the plane, also see notes and lectures on the polynomial method
  • Laba, Survey Recent progress on Favard length estimates for planar Cantor sets
  • Shmerkin On the Packing dimension of Furstenberg sets
  • Tao A quantitative version of the Besicovitch projection theorem via multiscale analysis
  • Toro paper 1, paper 2  (GMT and PDEs): Doubling and Flatness: Geometry of Measures  & GMT and Introduction
  • NVP The power law for the Buffon needle probability of the four-corner Cantor set
  • BV An estimate from below for the Buffon needle probability of the four-corner Cantor set
  • BLZ Quantitative visibility estimates for unrectifiable sets in the plane
  • Bongers Geometric Bounds for Favard Length
  • CDT Upper and lower bounds on the rate of decay of the Favard curve length for the four-corner Cantor set (Adam_ also see BV and BT)
  • HT On the Fourier Dimension of Sums and Product sets.
  • MIT On the Mattila Sjolin Theorem for Distance Sets

Books:

Blogs and Notes:

  • Tao’s Blog on Harmonic Analysis
  • Gower’s Blog
  • Orponen’s notes on GMT
  • Zahl’s notes and lectures
  • Shmerkin’s slides on Bourgain’s projection theorem.

Class Meetings (Videos and Notes):

  • Week 1: Tour of course website and content, Hello and why are you taking this course, discussion of topics, Erdos’ distance problem and Falconer’s distance problem.  Reading assignment: Erdos distance paper from 1946 and the survey given above on the Falconer distance problem.  We also announced that class Friday will be replaced by the colloquium, and you are also encouraged to attend the talk listed above on Friday.
  • Week 2: Energy integrals and Frostman’s Lemma, See lecture notes above. Monday; Wednesday; Friday
  • Week 3, 4: Differentiation of measures, Fourier Transforms, energy, Falconer’s distance proof
  • Week 5: Projection theory