Classes

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.
We review some basic concepts from Measure Theory, highlighting the bits that will be important in future chapters.
How well can we partition a set if we can only partition it using a specified collection of balls? We introduce several ways of approaching this, particularly the Besicovitch Covering Theorem, then use this to study Radon-Nikodym derivatives more in depth.
We define Hausdorff measure, which allows us to extend the notion of length and area to sets other than curves and surfaces, and generalizes these notions for non-integral dimensional sets. We also introduce Frostman measures, which are key for estimating dimension. Finally, we introduce self-similar sets, a convenient way of constructing sets of any specified dimension.