Just to make sure it does not sound intimidating, let me say: the level of the content will be modified to fit the level of the participants.
We will start by defining basic notions like Borel measures, Radon measures. We will define
Hausdorff measure, Hausdorff content etc.
We will talk about some special measures and how they are constructed. In this part, we will learn to think of
measures as martingales. Will explain what a Frostman measure is and use it.
We will talk about various covering theorems (5r, Vitali, Besicovitch, 1/3 trick) and how they are used for Maximal inequalities and differentiation.
We will talk about basic density theorems.
We will talk about Lipschitz maps
We will talk about various ways harmonic analysis can be brought into this game, and prove a quantitative version of Sard's theorem (P. Jones). In particular, we will need to say something about Haar wavelets (and martingales, again).
We will talk about Rectifiability.
We will talk about how singular integrals (SI) come into the picture. How curvature is related. In particular we will discuss some work of David-Semmes, Mattila-Melnikov-Verdera, Mattila-Verdera, Coifman-Jones-Semmes etc. Uniform rectifiability and SI connection, curvature and TSP. (Pajot's book and a survey by Verdera are good references here). For ease of presentation, the focus will be 1-dimensional sets.
We will talk about Christ-cubes
We will talk about the Besicovitch 1/2 conjecture and prove the 3/4 theorem.
If we have time left (there probably won't be), we will keep going...