Terry Tao has just completed detailed summaries of another series of lectures, this time by Shing-Tung Yau:
- What is a geometric structure?
- The Basic Tools to Discuss Geometric Structures
- Application of the Geometric Structures to Solve Problems in Algebraic Geometry and Topology
By geometric structure, Yau means additional structure that one imposes on a topological manifold. For example, a complex-analytic structure on a manifold is the requirement that the manifold be given by glueing together copies of Cn, such that the map which glues two copies together is required to be complex analytic. The main question is generally given a topological manifold, what sorts of structures can be imposed on them? The answer is frequently hard, and occasionally bizarre: there are uncountably infinite smooth structures on Rn for n = 4, while for any other value of n, there is only one. (Something the previous link mentioned was new to me: some exotic R4s can be smoothly embedded as open sets in the usual R4.)