I spent a few weeks on the USTC campus, in Hefei, China, in the summer of 2011, where I was invited to give some lectures in the annual summer school there. I gave 10 lectures on the topics related to F-structures and collapsing Riemiann manifolds.

Lecture 1 Riemannian geometry. Examples of the 3-sphere and warped products. Connections and curvature. Why study collapsing?
Lecture 2 Differential geometry, the Lie derivative, and Lie groups.
Lecture 3 Interaction between Lie group structure and Riemannian geometry. Example: SU(2). The Hopf action and the Hopffibration. The Heisenberg group and its quotients.
Lecture 4 The fundamental examples of collapse: The Berger spheres, collapsing warped products, collapse along a torus action, collapse of nilmaifolds.
Lecture 5 The Gromov-Hausdorff distance and its properties. Covergence of Riemannian manifolds.
Lecture 6 Basic definitions surrounding F-structures.
Lecture 7 Additional F-structure definitions. Examples of F-structures. Basic Theorems.
Lecture 8 Non-polarizable F-structures, complete flat manifolds, and elementary F-strucutures
Lecture 9 How various types of F-structures imply collapse with bounded curvature.
Lecture 10 How collapse with bounded curvature implies collapse

 Addendum: References and additional concepts.