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In order to motivate the lengthy and detailed analysis of Ricci flow that will occupy the rest of this course, I will spend this lecture giving a high-level overview of Perelman’s Ricci flow-based proof of the Poincaré conjecture, and in particular how that conjecture is reduced to verifying a number of (highly non-trivial) facts about Ricci flow.

At the risk of belaboring the obvious, here is the statement of that conjecture:

Theorem 1. (Poincaré conjecture) Let M be a compact 3-manifold which is simply connected (i.e. it is connected, and every loop is contractible to a point). Then M is homeomorphic to a 3-sphere $S^3$.

[Unless otherwise stated, all manifolds are assumed to be without boundary.]

I will take it for granted that this result is of interest, but you can read the Notices article of Milnor, the Bulletin article of Morgan, or the Clay Mathematical Institute description of the problem (also by Milnor) for background and motivation for this conjecture. Perelman’s methods also extend to establish further generalisations of the Poincaré conjecture, most notably Thurston’s geometrisation conjecture, but I will focus this course just on the Poincaré conjecture. (On the other hand, the geometrisation conjecture will be rather visibly lurking beneath the surface in the discussion of this lecture.)

I’m continuing my series of articles for the Princeton Companion to Mathematics through the holiday season with my article on “Differential forms and integration“. This is my attempt to explain the concept of a differential form in differential geometry and several variable calculus; which I view as an extension of the concept of the signed integral in single variable calculus. I briefly touch on the important concept of de Rham cohomology, but mostly I stick to fundamentals.

I would also like to highlight Doron Zeilberger‘s PCM article “Enumerative and Algebraic combinatorics“. This article describes the art of how to usefully count the number of objects of a given type exactly; this subject has a rather algebraic flavour to it, in contrast with asymptotic combinatorics, which is more concerned with computing the order of magnitude of number of objects in a class. The two subjects complement each other; for instance, in my own work, I have found enumerative and other algebraic methods tend to be useful for controlling “main terms” in a given expression, while asymptotic and other analytic methods tend to be good at controlling “error terms”.

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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