Description

This is the page for a course at Princeton taught by the lovely Prof. Camillo de Lellis (and truthfully entitled “Topics in Geometric Analysis: Rectifiable sets and rectifiable measures”) that I took during Fall 2023. Below is the course description and some resources about the topic.

“This is a monographic course on the topic of rectifiability, a concept introduced first by Besicovitch in the particular case of 1-dimensional sets in the plane. After introducing the Hausdorff dimension, the Hausdorff measures, the course covers Besicovitch’s theory and touches upon the famous Besicovitch 1/2 conjecture, which is almost a century old and still unsolved. It then defines rectifiable sets and rectifiable measures in general dimension and codimension and covers the most basic and widely used rectifiability criteria..”

Everything we learned was very elegant and beautiful despite its technicality. Since the course was not meant for undergrads we moved extremely fast and so I was exposed to a lot of new concepts very quickly. We focused a lot on proving the Besicovitch-Marstrand-Preiss Theorem and some criteria for rectifiability and pure unrectifiability, developing analyses using rectifiable and tangent measures.

Reading List

• Frank Morgan’s “Geometric Measure Theory: A Beginner’s Guide” (in the direction of currents and varifolds).
• Pertti Matilla’s “Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability” (in the direction of rectifiablity and tangent structure).
• Prof. De Lellis’ lecture notes on rectifiability and density.
• Leon Simon’s textbook (builds toward varifolds and integration).
• Evans & Gareipy Measure Theory and Fine Properties of Functions.

Notes & Problem Sets

• My notes from lecture
• My problem sets! I don’t think any are really awfully wrong, but some are incorrect for sure. Email me if there’s anything to discuss:
  • PSet 1
  • PSet 2
• Background notes for my final presentation