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Given that this blog is currently being devoted to a rather intensive study of flows on manifolds, I thought that it might be apropos to highlight an amazing 22-minute video from 1994 on this general topic by the (unfortunately now closed) Geometry Center, entitled “Outside In“, which depicts Smale’s paradox (which asserts that an 2-sphere in three-dimensional space can be smoothly inverted without ever ceasing to be an immersion), following a construction of Thurston (who was credited with the concept for the video). I first saw this video at the 1998 International Congress of Mathematicians in Berlin, where it won the first prize at the VideoMath Festival held there. It did a remarkably effective job of explaining the paradox, its resolution in three dimensions, and the lack of a similar paradox in two dimensions, all in a clear and non-technical manner.

A (rather low resolution) copy of the first half of the video can be found here, and the second half can be found here. Some higher resolution short movies of just the inversion process can be found at this Geometry Center page. Finally, the video (and an accompanying booklet with more details and background) can still be obtained today from A K Peters, although I believe the video is only available in the increasingly archaic VHS format.

There are a few other similar such high-quality expository videos of advanced mathematics floating around the internet, but I do not know of any page devoted to collecting such videos. If any readers have their own favourites, you are welcome to post some links or pointers to them here.

In the previous lecture, we saw that Ricci flow with surgery ensures that the second homotopy group \pi_2(M) became extinct in finite time (assuming, as stated in the above erratum, that there is no embedded \Bbb{RP}^2 with trivial normal bundle). It turns out that the same assertion is true for the third homotopy group, at least in the simply connected case:

Theorem 1. (Finite time extinction of \pi_3(M)) Let t \mapsto (M(t),g(t)) be a Ricci flow with surgery on compact 3-manifolds with t \in [0,+\infty), with M(0) simply connected. Then for all sufficiently large t, \pi_3(M(t)) is trivial (or more precisely, every connected component of M(t) has trivial \pi_3).

[Aside: it seems to me that this theorem should also be true if one merely assumes that M(0) contains no embedded copy of \Bbb{RP}^2 with trivial bundle, as opposed to M(0) being simply connected, but I will be conservative and only state Theorem 1 with this stronger hypothesis, as this is all that is necessary for proving the Poincaré conjecture.]

Suppose we apply Ricci flow with surgery to a compact simply connected Riemannian 3-manifold (M,g) (which, by Lemma 1 from Lecture 2, has no embedded \Bbb {RP}^2 with trivial normal bundle). From the above theorem, as well as Theorem 1 from the previous lecture, we know that all components of M(t) eventually have trivial \pi_2 and \pi_3 for all sufficiently large t. Also, since M is initially simply connected, we see from Exercise 2 of Lecture 2, as well as Theorem 2.1 of Lecture 2, that all components of M(t) also have trivial \pi_1. The finite time extinction result (Theorem 4 from Lecture 2) then follows immediately from Theorem 1 and the following topological result, combined with the following topological observation:

Lemma 1. Let M be a compact non-empty connected 3-manifold. Then it is not possible for \pi_1(M), \pi_2(M), and \pi_3(M) to simultaneously be trivial.

This lemma follows immediately from the Hurewicz theorem, but for sake of self-containedness we give a proof of it here.

There are two known approaches to establishing Theorem 1; one due to Colding and Minicozzi, and one due to Perelman. The former is conceptually simpler, but requires a certain technical concentration-compactness type property for a min-max functional which has only been established recently. This approach will be the focus of this lecture, while the latter approach of Perelman, which has also been rigorously shown to imply finite time extinction, will be the focus of the next lecture.

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Returning (perhaps anticlimactically) to the subject of the Poincaré conjecture, recall from Lecture 2 that one of the key pillars of the proof of that conjecture is the finite time extinction result (see Theorem 4 from that lecture), which asserted that if a compact Riemannian 3-manifold (M,g) was initially simply connected, then after a finite amount of time evolving via Ricci flow with surgery, the manifold will be empty.

In this lecture and the next few, we will describe some of the key ideas used to prove this theorem. We will not be able to completely establish this theorem at present, because we do not have a full definition of “surgery”, but we will be able to establish some partial results, and indicate (in informal terms) how to cope with the additional technicalities caused by the surgery procedure. Hopefully, if time permits later in the class, once we have studied the surgery process, I will be able to revisit this material and flesh out these technicalities a bit more.

The proof of finite time extinction proceeds in several stages. The first stage, which was already accomplished in the previous lecture (in the absence of surgery, at least), is to establish lower bounds on the least scalar curvature R_{\min}. The next stage, which we discuss in this lecture, is to show that the second homotopy group \pi_2(M) of the manifold must become extinct in finite time, thus all immersed copies of the 2-sphere S^2 in M(t) for sufficiently large t must be contractible to a point. The third stage is to show that the third homotopy group \pi_3(M) also becomes extinct so that all immersed copies of the 3-sphere S^3 in M are similarly contractible. The final stage, which uses homology theory, is to show that a non-empty 3-manifold cannot have \pi_1(M), \pi_2(M), \pi_3(M) simultaneously trivial, thus yielding the desired claim (note that a simply connected manifold has trivial \pi_1(M) by definition; also, from Exercise 2 of Lecture 2 we see that all components of M remain simply connected even after surgery).

More precisely, in this lecture we will discuss (most of) the proof of

Theorem 1. (Finite time extinction of \pi_2(M)) Let t \mapsto (M(t),g(t)) be a Ricci flow with surgery on compact 3-manifolds with t \in [0,+\infty), with M(0) containing no embedded copy of \Bbb{RP}^2 with trivial normal bundle. Then for all sufficiently large t, \pi_2(M(t)) is trivial (or more precisely, every connected component of M(t) has trivial \pi_2).

The technical assumption about having no copy of \Bbb{RP}^2 with trivial normal bundle is needed solely in order to apply the known existence theory for Ricci flow with surgery (see Theorem 2 from Lecture 2).

The intuition for this result is as follows. From the Gauss-Bonnet theorem (and the fact that the Euler characteristic \chi(S^2)=V-E+F=2 of the sphere is positive), we know that 2-spheres tend to have positive (Gaussian) curvature on the average, which should make them shrink under Ricci flow. (Here I am conflating Gaussian curvature with Ricci curvature; however, by restricting to a special class of 2-spheres, namely minimal surfaces, one can connect the two notions of curvature to each other (and to scalar curvature) quite nicely.) On the other hand, the presence of negative scalar curvature can counteract this by expanding these spheres. But the lower bounds on scalar curvature tell us that the negativity of scalar curvature becomes weakened over time, and it turns out that the shrinkage caused by the Gauss-Bonnet theorem eventually dominates and sends the area of all minimal immersed 2-spheres into zero, at which point one can conclude the triviality of \pi_2(M) by the Sacks-Uhlenbeck theory of minimal 2-spheres.

The arguments here are drawn from the book of Morgan-Tian and from the paper of Colding-Minicozzi. The idea of using minimal surfaces to force disappearance of various topological structures under Ricci flow originates with Hamilton (who used 2-torii instead of 2-spheres, but the idea is broadly the same).

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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.)

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In this lecture, we move away from recurrence, and instead focus on the structure of topological dynamical systems. One remarkable feature of this subject is that starting from fairly “soft” notions of structure, such as topological structure, one can extract much more “hard” or “rigid” notions of structure, such as geometric or algebraic structure. The key concept needed to capture this structure is that of an isometric system, or more generally an isometric extension, which we shall discuss in this lecture. As an application of this theory we characterise the distribution of polynomial sequences in torii (a baby case of a variant of Ratner’s theorem due to (Leon) Green, which we will cover later in this course).

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