Math 8210 – Topics

Topics in Harmonic Analysis and Geometric Measure Theory (GMT)

• Class notes
• Special Class on Tuesdays September 5 and 12, 3:30 – 4:30 pm, MW 154

 

 

• Presentations Set 2 (each paper is assigned to a group of two people, except that marked with a * , which is short and should be assigned to a group of one)
  • Chang, Csörnyei The Kakeya needle problem and the existence of Besicovitch and Nikodym sets for rectifiable sets
  • Chang, Csörnyei, Héra, Keleti Small unions of affine subspaces and skeletons via Baire category
  • Davies’ On accessibility of plane sets and differentiation of functions of two real variables
  • Falconer Sets with prescribed projections and Nikodým sets
  • Bongers Geometric Bounds for Favard Length* (to be assigned to only 1 person)
  • Ren, Wang Furstenberg sets estimate in the plane (this paper might be too hard and outside the scope of the class, but I include it in case someone is particularly interested in it. Alternatives are Mitsis or Oberlin)

 

List of Topics and Readings by week

• Stein & Shakarchi Chapter 8
• Distance Problems – Introduction
  1. Falconer’s distance problem What is Falconer’s Conjecture?
  2. Erdos distance problem: class notes (link above) and  Erdos’ original paper
  3. Eyvi’s notes;
  4. Tao’s blogs for discussion: genius, work-hard, learn and relearn your field
  5. Guth-Katz on Tao’s blog

 

• Geometric measure theory – What is it?
  1. Tatiana Toro’s Notices article
  2. reading: Introduction to Mattila’s book
  3. Advice to a Young Mathematician in the Princeton Companion – page 1000
  4. Watch Paths to Math: Svitlana Mayboroda (she is one of the great mathematicians on the planet today) and think about your own path to math
  5. Further reading: Orponen’s notes on GMT;  Brush up on Interpolation result

 

• Presentations Set 1 (Hausdorff dimension, Thickness, )
  • Eidelman’s, A Rare Plane Set with Hausdorff Dimension 2 (You will also be expected to be familiar with the papers in the introduction) (Tianyu Zhao & Saúl) 
  • Hunt-Kan-Yorke’s When Cantor Sets intersect thickly. (Also see this survey to put these papers into a larger context) (Hannah & Joseph) 
  • William’s How big is the Intersection of Two Thick Cantor sets  (Steve) 
  • Bieber’s A Complex Gap Lemma  (Also see this survey to put these papers into a larger context)
  • Ruelle’s  Repellers for Real Analytic maps   (Katelynn) 
  • Lafont’s On the Hausdorff Dimension of CAT(κ) Surfaces (Austin) 

 

• More reading:
  • Yavicoli’s Patterns in Thick Compact Sets
  •  McDonald-Taylor
    1. Introduction to thickness and the Newhouse gap lemma (Simon-Taylor, Yavicoli)
    2. patterns in thick sets (Yavicoli, Astel, McDonald-Taylor)
  • Astel’s Cantor Sets and Numbers with Restricted Partial Quotients

 

• Harmonic Analysis – What is it?
  1. The Princeton Companion pg. 448 –
  2. Introduction in Steins’ big book
  3. Tao’s overview on Harmonic Analysis

 

• Distance Problems – Further reading
  1. Falconer’s proof of (d+1)/2
  2. Wolff’s notes – Chapter 8
  3. Wolff’s 1999 paper (averaging the spherical operator to get a better exponent)
  4. Bennet-Iosevich-Taylor (trees and chains), Bochen- Iosevich (Cartesian product improvements), Greenleaf-Iosevich-Liu-Palsson (group actions); Iosevich-Magyar (any graph)

 

• Decoupling
  1. Guth’s talk on the topic
  2. Idea with Eyvi to use decoupling – see emails to Eyvi

 

• Projections
• discrete Favard (see emails with Singer)
• Kakeya and Davies’ theorem (see Orponen’s notes, Katz-Zahl, Mattila’s book, Falconer’s books)
• Kakeya Maximal (Wolff Chapter 10, Mattila_2015 (Chapter 22))

 

 

• The appearance of the Fourier transform in GMT: Mattila’s Introduction, Hausdorff measures, Minkowski dimension, weak convergence, energy integrals
• Differentiation of measures, Fourier transform, convolutions and measures, Vitali covering
• Projection theory: Marstrand’ projection theorem, transversality
• 1-sets, purely unrectifiability, Besicovitch’s projection theorem, Buffon needle problem, visibility and radial projections
• Bongers and Mattila1990 Theorem 4.1 for the decay of Buffon needle, Buffon circle; Connections: PDE, Probability
• Davies theorem and the Kakeya conjectures, slicing problem
• Arithmetic operations on fractals: Steinhaus, Erdős and Oxtoby, category analogue and generalizations
• Distance sets and Finite point configurations within fractal sets Big picture in Harmonic Analysis
• Davies theorem and Kakeya
• Oscillatory Integrals and Restriction (see Wolff notes section 7 starting on page 41)
• The big topics in harmonic analysis and lattice point counting; Connections:  Number theory
• The big topics in harmonic analysis continued…

Survey Articles:

• What is Falconer’s Conjecture?
• Lattice point counting
• Tao notes on harmonic analysis (7 pages)- he also wrote a more length Technical Survey here
• Connes Advice for the beginner

Research Articles

• Csornyei How To Make Davies’ Theorem Visible 
• 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:

• Grafakos’ Classical Fourier Analysis
• Mattila’s Fourier Analysis and Hausdorff Dimension
• Falconer’s The Geometry of Fractal Sets, 1986
• (Falconer’s Fractal Geometry_Mathematical Foundations and Applications 1990)
• (Chapter 3)  (Chapter 18)
• Wolff notes (Section 7 Restriction; Section 9 Fourier dimension; Section 10 Kakeya)

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.

 

Events, Recordings, and Talks related to this course: 

• Workshop in Analysis at Georgia Tech, Dec. 9-11, 2023, register here

 

• MAAM Mid Atlantic Analysis Seminar
• U. Edinburgh– Virtual Harmonic Analysis Seminar with recorded talks
• 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.
• MAAM: http://web.sas.upenn.edu/maam2/