Geometric Measure Theory

2020-10-16
2 min read

This course is intended to provide a graduate level introduction to geometric measure theory. Topics include:

  • Covering Lemmas
  • Hausdorff Measure and Dimension
  • Construction of sets using Self Similarity, Venetian Blinds, and Duality
  • The behavior of dimension under projections.
  • Lipschitz Functions
  • Tangent Measures
  • Rectifiable Sets
  • Harmonic Measure

The second half is more specialized than the first, in that we will focus more on tools related to rectifiability and tangent measures. This essay by Tatiana Toro is a good overview of these topics and their applications.

Sources

The primary text that this course is structured around is Pertti Mattila’s Geometry of Sets and Measures in Euclidean Space. However, I have also drawn from other great sources, including

  • Kenneth Falconer’s classic Geometry of Fractal Sets
  • Juha Heinonen’s Lectures on Analysis on Metric spaces
  • Evans and Gariepy’s Measure theory and fine properties of functions
  • Mattila’s Fourier Analysis and Hausdorff Dimension
  • David’s Wavelets and Singular Integrals
  • De Lellis' Rectifiable Sets, Densities and Tangent Measures

There is no need to own any of these textbooks since notes are provided that isolate the parts of these texts I aim to cover. However, these authors are far better writers than I am, so if you are able to get a hold of the books, I will also refer to the parts of the texts we are using.

Materials

There are three main resources I provide on this webpage.

  • Lecture notes: these will typically be more detailed from the sources I derive them from.
  • Video Lectures: These videos discuss important results and examples an provide a high-level view of the material, but I will also sometimes work through some details when I think they are important to emphasize.
  • Worksheets: For each week I have collected some exercises to try your hand at using the methods covered in the lectures.

Prerequisites

This class only assumes you are comfortable with abstract measure theory (e.g. Egoroff’s theorem shouldn?t scare you), however having had courses in topology, harmonic analysis, functional analysis, and complex analysis will improve your experience. If you don?t know Measure Theory and are self-learning, here is a youtube class on the subject.

For those looking for an advanced undergraduate level introduction to this topic, I recommend Kenneth Falconer’s Fractal Geometry: Mathematical Foundations and Applications.