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 **C**^{n}, 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 **R**^{n} 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 **R**^{4}s can be smoothly embedded as open sets in the usual **R**^{4}.)