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On Friday, Yau concluded his lecture series by discussing the PDE approach to constructing geometric structures, particularly Einstein metrics, and their applications to many questions in low-dimensional topology (yes, this includes the Poincaré conjecture). Yau also discussed the situation in high-dimensional topology, which appears to be completely different (and much less well understood).
Yau’s slides for this talk are available here.
On Thursday, Yau continued his lecture series on geometric structures, focusing a bit more on the tools and philosophy that goes into actually building these structures. Much of the philosophy, in its full generality, is still rather vague and not properly formalised, but is nevertheless supported by a large number of rigorously worked out examples and results in special cases. A dominant theme in this talk was the interaction between geometry and physics, in particular general relativity and string theory.
As usual, there are likely to be some inaccuracies in my presentation of Yau’s talk (I am not really an expert in this subject), and corrections are welcome. Yau’s slides for this talk are available here.
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