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Topics in Geometric Measure Theory (TCC course)

Time

Wednesdays 11-1. From 16 January to 6 March.

Lecture notes

Typed notes (latex) (updated 8 March), containing problems for credit too.

Whiteboard notes: Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8

Tentative List of Main Topics

Baire category theorem and its applications in fractal geometry

  • Kakeya (Besicovitch) sets. Proof of existence using the Baire category theorem.
  • Typical Nikodym sets.
  • Dimension of Kakeya sets. Cordoba's L^2 argument.

Mapping planar sets of positive Lebesgue measure onto balls

  • Covering planar sets by Lipschitz strips. The Matousek--Preiss proof using Dilworth's theorem on chains/antichains
  • Uy's proof using complex & harmonic analysis

Prerequisites

Solid background in measure theory. Previous exposure to Hausdorff dimension and Baire category can be helpful.