In early July 2013, I gave 4 lectures at the ICMS summer school in Ricci curvature, the focus of which was ideas in geometric analysis revolving around epsilon-regularity.

At the moment there is some overlap in the lecture notes, owing to the fact that the lectures themselves proceeded either more or less quickly according to the plan. At some point I will clean up these notes and make then into a more unified presentation.

Lecture 1 Spacs Diff and Mod, the L2 metric, canonical metrics,  Riemannian functionals, Euler Lagrange equations, elliptic systems
Lecture 2 The role of analysis, Sobolev constants, and isoperimetric constants
Lecture 3 Distance functions on moduli spaces, notions of compactness and pre-compactness, and the proof of the epsilon-regularity theorem for Einstein manifolds
Lecture 4 Weak convergence, orbifold points, bubbles, and topology