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We now turn to Perelman’s second scale-invariant monotone quantity for Ricci flow, now known as the Perelman reduced volume. We saw in the previous lecture that the monotonicity for Perelman entropy was ultimately derived (after some twists and turns) from the monotonicity of a potential under gradient flow. In this lecture, we will show (at a heuristic level only) how the monotonicity of Perelman’s reduced volume can also be “derived”, in a formal sense, from another source of monotonicity, namely the relative Bishop-Gromov inequality in comparison geometry (which has already been alluded to in previous lectures). Interestingly, in order to obtain this connection, one must first reinterpret parabolic flows such as Ricci flow as the limit of a certain high-dimensional Riemannian manifold as the dimension becomes infinite; this is part of a more general philosophy that parabolic theory is in some sense an infinite-dimensional limit of elliptic theory. Our treatment here is a (liberally reinterpreted) version of Section 6 of Perelman’s paper.
In the next few lectures we shall give a rigorous proof of this monotonicity, without using the infinite-dimensional limit and instead using results related to the Li-Yau-Hamilton Harnack inequality. (There are several other approaches to understanding Perelman’s reduced volume, such as Lott’s formulation based on optimal transport, but we will restrict attention in this course to the methods that are in Perelman’s original paper.)
Over a year ago, I had a brief post here pointing out Gene Weingarten‘s article in the Washington Post entitled “Pearls before breakfast“, in which the Post asked the question of what would happen if a world-class musician (in this case, Joshua Bell), were to perform incognito and out of context, in a Washington subway during the morning rush hour? If you haven’t yet read the article describing the experiment and the outcome, I recommend it to you.
Anyway, a few weeks ago, this article was awarded the 2008 Pulitzer Prize for Feature Writing. Congratulations to Gene, Joshua, and the other Washington Post staff!
[Actually, this article was highly atypical for Gene; he usually sticks to writing a weekly low-brow humour column entitled “Below the Beltway”. By a random coincidence, I, together with Curt McMullen, even have a very minor bit part in one of these columns (on page 2), thanks to a brief phone conversation we each had with Gene.]
It is well known that the heat equation
using the inner product associated to the volume measure . Indeed, if we evolve f in time at some arbitrary rate , a simple application of integration by parts (equation (29) from Lecture 1) gives
from which we see that (1) is indeed the gradient flow for (3) with respect to the inner product. In particular, if f solves the heat equation (1), we see that the Dirichlet energy is decreasing in time:
Thus we see that by representing the PDE (1) as a gradient flow, we automatically gain a controlled quantity of the evolution, namely the energy functional that is generating the gradient flow. This representation also strongly suggests (though does not quite prove) that solutions of (1) should eventually converge to stationary points of the Dirichlet energy (2), which by (3) are just the harmonic functions (i.e. the functions f with ).
As one very quick application of the gradient flow interpretation, we can assert that the only periodic (or “breather”) solutions to the heat equation (1) are the harmonic functions (which, in fact, must be constant if M is compact, thanks to the maximum principle). Indeed, if a solution f was periodic, then the monotone functional E must be constant, which by (4) implies that f is harmonic as claimed.
It would therefore be desirable to represent Ricci flow as a gradient flow also, in order to gain a new controlled quantity, and also to gain some hints as to what the asymptotic behaviour of Ricci flows should be. It turns out that one cannot quite do this directly (there is an obstruction caused by gradient steady solitons, of which we shall say more later); but Perelman nevertheless observed that one can interpret Ricci flow as gradient flow if one first quotients out the diffeomorphism invariance of the flow. In fact, there are infinitely many such gradient flow interpretations available. This fact already allows one to rule out “breather” solutions to Ricci flow, and also reveals some information about how Poincaré’s inequality deforms under this flow.
The energy functionals associated to the above interpretations are subcritical (in fact, they are much like ) but they are not coercive; Poincaré’s inequality holds both in collapsed and non-collapsed geometries, and so these functionals are not excluding the former. However, Perelman discovered a perturbation of these functionals associated to a deeper inequality, the log-Sobolev inequality (first introduced by Gross in Euclidean space). This inequality is sensitive to volume collapsing at a given scale. Furthermore, by optimising over the scale parameter, the controlled quantity (now known as the Perelman entropy) becomes scale-invariant and prevents collapsing at any scale – precisely what is needed to carry out the first phase of the strategy outlined in the previous lecture to establish global existence of Ricci flow with surgery.
We now set aside our discussion of the finite time extinction results for Ricci flow with surgery (Theorem 4 from Lecture 2), and turn instead to the main portion of Perelman’s argument, which is to establish the global existence result for Ricci flow with surgery (Theorem 2 from Lecture 2), as well as the discreteness of the surgery times (Theorem 3 from Lecture 2).
As mentioned in Lecture 1, local existence of the Ricci flow is a fairly standard application of nonlinear parabolic theory, once one uses de Turck’s trick to transform Ricci flow into an explicitly parabolic equation. The trouble is, of course, that Ricci flow can and does develop singularities (indeed, we have just spent several lectures showing that singularities must inevitably develop when certain topological hypotheses (e.g. simple connectedness) or geometric hypotheses (e.g. positive scalar curvature) occur). In principle, one can use surgery to remove the most singular parts of the manifold at every singularity time and then restart the Ricci flow, but in order to do this one needs some rather precise control on the geometry and topology of these singular regions. (In particular, there are some hypothetical bad singularity scenarios which cannot be easily removed by surgery, due to topological obstructions; a major difficulty in the Perelman program is to show that such scenarios in fact cannot occur in a Ricci flow.)
In order to analyse these singularities, Hamilton and then Perelman employed the standard nonlinear PDE technique of “blowing up” the singularity using the scaling symmetry, and then exploiting as much “compactness” as is available in order to extract an “asymptotic profile” of that singularity from a sequence of such blowups, which had better properties than the original Ricci flow. [The PDE notion of a blowing up a solution around a singularity, by the way, is vaguely analogous to the algebraic geometry notion of blowing up a variety around a singularity, though the two notions are certainly not identical.] A sufficiently good classification of all the possible asymptotic profiles will, in principle, lead to enough structural properties on general singularities to Ricci flow that one can see how to perform surgery in a manner which controls both the geometry and the topology.
However, in order to carry out this program it is necessary to obtain geometric control on the Ricci flow which does not deteriorate when one blows up the solution; in the jargon of nonlinear PDE, we need to obtain bounds on some quantity which is both coercive (it bounds the geometry) and either critical (it is essentially invariant under rescaling) or subcritical (it becomes more powerful when one blows up the solution) with respect to the scaling symmetry. The discovery of controlled quantities for Ricci flow which were simultaneously coercive and critical was Perelman’s first major breakthrough in the subject (previously known controlled quantities were either supercritical or only partially coercive); it made it possible, at least in principle, to analyse general singularities of Ricci flow and thus to begin the surgery program discussed above. (In contrast, the main reason why questions such as Navier-Stokes global regularity are so difficult is that no controlled quantity which is both coercive and critical or subcritical is known.) The mere existence of such a quantity does not by any means establish global existence of Ricci flow with surgery immediately, but it does give one a non-trivial starting point from which one can hope to make progress.
In this lecture we discuss Perelman’s original approach to finite time extinction of the third homotopy group (Theorem 1 from the previous lecture), which, as previously discussed, can be combined with the finite time extinction of the second homotopy group to imply finite time extinction of the entire Ricci flow with surgery for any compact simply connected Riemannian 3-manifold, i.e. Theorem 4 from Lecture 2.
Van Vu and I have just uploaded to the arXiv our preprint “On the permanent of random Bernoulli matrices“, submitted to Adv. Math. This paper establishes analogues of some recent results on the determinant of random Bernoulli matrices (matrices in which all entries are either +1 or -1, with equal probability of each), in which the determinant is replaced by the permanent.
More precisely, let M be a random Bernoulli matrix, with n large. Since every row of this matrix has magnitude , it is easy to see (by interpreting the determinant as the signed volume of a parallelopiped) that is at most , with equality being satisfied exactly when M is a Hadamard matrix. In fact, it is known that the determinant has magnitude with probability ; for a more precise result, see my earlier paper with Van. (There is in fact believed to be a central limit theorem for ; see this paper of Girko for details.) These results are based upon the elementary “base times height” formula for the volume of a parallelopiped; the main difficulty is to understand what the distance is from one row of M to a subspace spanned by several other rows of M.
The permanent looks formally very similar to the determinant, but does not have a geometric interpretation as a signed volume of a parallelopiped and so can only be analysed combinatorially; the main difficulty is to understand the cancellation that can arise from the various signs in the matrix. It can be somewhat larger than the determinant; for instance, the maximum value of for a Bernoulli matrix M is , attaned when M consists entirely of +1’s. Nevertheless, it is not hard to see that has the same mean and standard deviation as , namely 0 and respectively, which shows that is at most with probability 1-o(1). Our main result is to show that one also has that is at least with probability 1-o(1), thus obtaining the analogue of the previously mentioned result for the determinant (though our o(1) bounds are significantly weaker).
In particular, this shows that the probability that the permanent vanishes completely is o(1) (in fact, we get a bound of for some absolute constant ). This result appears to be new (although there is a cute observation of Alon (see e.g. this paper of Wanless for a proof) that if is one less than a power of 2, then every Bernoulli matrix has non-zero permanent). In contrast, the probability that the determinant vanishes completely is conjectured to equal (which is easily seen to be a lower bound), but the best known upper bound for this probability is , due to Bourgain, Vu, and Wood.
In the previous lecture, we saw that Ricci flow with surgery ensures that the second homotopy group became extinct in finite time (assuming, as stated in the above erratum, that there is no embedded 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 ) Let be a Ricci flow with surgery on compact 3-manifolds with , with M(0) simply connected. Then for all sufficiently large t, is trivial (or more precisely, every connected component of M(t) has trivial ).
[Aside: it seems to me that this theorem should also be true if one merely assumes that M(0) contains no embedded copy of 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 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 and 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 . 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 , , and 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.
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 . The next stage, which we discuss in this lecture, is to show that the second homotopy group of the manifold must become extinct in finite time, thus all immersed copies of the 2-sphere in M(t) for sufficiently large t must be contractible to a point. The third stage is to show that the third homotopy group also becomes extinct so that all immersed copies of the 3-sphere in M are similarly contractible. The final stage, which uses homology theory, is to show that a non-empty 3-manifold cannot have simultaneously trivial, thus yielding the desired claim (note that a simply connected manifold has trivial 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 ) Let be a Ricci flow with surgery on compact 3-manifolds with , with M(0) containing no embedded copy of with trivial normal bundle. Then for all sufficiently large t, is trivial (or more precisely, every connected component of M(t) has trivial ).
The technical assumption about having no copy of 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 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 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).
I usually try to keep political issues out of this blog, and I certainly try to avoid asking friends and readers of this blog for favours, but there is an urgent situation developing in mathematics (and related disciplines) in my home country of Australia, and I need to ask all of you for assistance to prevent an impending disaster.
When I was an undergraduate at Flinders University in South Australia from 1989 to 1992, the level of mathematics education in Australia was comparable to that of world-class institutions overseas. Even in a small and little-known university such as Flinders, I received a first-rate honours undergraduate education in mathematics, computer science, and physics which I continue to use daily in my career. (Examples of topics I learned as an undergraduate include wavelets; information theory; Lie algebras; differential geometry; nonlinear PDE; quantum mechanics; statistical mechanics; and harmonic analysis. I rely on my knowledge of all of these topics today, for instance many of them are are helpful for me in teaching my current class on the Poincaré conjecture.) In addition, several of the faculty (including the chair and my undergraduate advisor, Garth Gaudry) had the time to spare an hour a week with me to discuss mathematics, as they were not overloaded with large teaching loads and other duties. I honestly think that I would not be where I am today without the high-quality undergraduate education that I received (in particular, I would definitely have floundered in graduate school at Princeton, if I were admitted at all).
The situation for mathematics education in Australia began however to deteriorate in later years, due to a combination of factors including government neglect (the federal government is the most significant source of funding for most universities in Australia) and the low priority of basic education in mathematics and sciences among university administrators. In particular, at Flinders University, the School of Mathematics suffered severe attrition due to lack of support and was eventually folded into the School of Informatics and Engineering. In fact the number of mathematicians on the faculty at Flinders has dwindled down to just three (in my day it was close to 20).
The decline of mathematics departments across the country, particularly in a time in which mathematics skills are desperately needed in the workforce, has been documented thoroughly in the 2006 national strategic review of the mathematical sciences. In response to that report, the federal government in 2007 announced an increase in the funding allocation to universities based on their student enrollments in key majors including mathematics and the sciences. The newly elected federal government is also likely to continue and extend this support in its upcoming budget in May of this year.
Unfortunately, it appears that at many universities, the additional funding was diverted away from the schools that it was intended to support, for the administrator’s own priorities. (See also the letter by the international authors of the above mentioned review, Jean-Pierre Bourguignon, Brenda Dietrich, and Iain Johnstone, condemning this diversion.) As a consequence, many mathematics departments are in fact in worse shape than before.
There is a particular crisis unfolding at the University of Southern Queensland. On March 17, the university announced a rationalisation and restructuring proposal that would cut the number of mathematics faculty from 14 to 6, eliminate the majors in mathematics, chemistry, physics, and statistics, and phase out all non-service courses (for instance, any of the types of courses I mentioned above at Flinders would be lost). Similar cuts were also proposed in statistics, computer science, and physics, although other schools retained their funding and some even obtained increases. This is despite the increases in funding from the federal government for mathematics and statistics students (enrollments in these areas at USQ has held steady so far, though of course with the proposed cuts this is unlikely to last). Already as a consequence of these proposals, initiatives of the department such as an education program for high school mathematics teachers have had to be scrapped. Somewhat ironically, the Dean of Sciences at USQ, Janet Verbyla, who has been heavily involved in proposing the cuts, had also presided over similar reductions in the school of mathematics at Flinders.
If the proposed cuts at USQ go ahead, it is likely that other small universities in Australia will be tempted to similarly ignore concerns about mathematics and science education and perform similar cuts, even while receiving government support for these disciplines. (The University of New England, which currently shares some statistics courses at USQ, would for instance be particularly vulnerable.) So the crisis here is not purely localised to USQ, but could be very damaging for mathematics and sciences in Australia as a whole.
The consultation period for these cuts ends very soon, on April 14, and the vice-chancellor of USQ, Bill Lovegrove, plans to announce the specific cuts on April 18 at an unspecified future date. While there has been some media attention in Australia given to this issue, it has not yet had much effect in reversing the decisions of these administrators. Because of this, I am reluctantly turning to my friends and readers of this blog to ask for your urgent assistance in saving the school of mathematics and computing at USQ. In collaboration with several good friends and colleagues in Australia, I have begun a web page on this blog,
that is documenting the situation and outlining ways to help, including an online petition
that you can sign to show support, and people to contact in the university administration and in the Australian government to express your concerns, or to express support for mathematics and its role in the sciences. Please also inform others, especially those in Australia and who may have influence in media, political, or administrative circles, of the current crisis. There is still time, especially in view of the expected increase in support for mathematics and sciences in the upcoming federal budget, to reverse the situation before the damage becomes permanent, and to show that the political support for mathematics education is not so negligible as to be easily ignored.
Thank you all in advance for any help you can give – and I promise that I will keep the remainder of my blog on topic and focus primarily on mathematics. :-)
[Update, April 9: See my editorial at the Funneled Web, “Mathematics in Today’s world“, for a more detailed discussion of the USQ crisis, and also the broader context of the importance of higher mathematics education, and the pivotal role universities have to play in providing it.]
[Update, April 12: The Toowoomba Chronicle has a two-page article by Merryl Miller focusing on the crisis, and in particular focusing on its impact on a 10-year old child prodigy, Adam Walsh, currently taking maths classes at USQ. (Reprinted with permission.)]
[Update, April 14: The petition has been formally sent to the USQ administration. Apparently, the previously planned announcement of the cuts on April
18 has been delayed to some unspecified later date, but no further details are currently available.]
[Update, April 17: In response to the concerns of constituents, Hon. Mike Horan MP, the state member for Toowoomba South, spoke in the Queensland parliament urging the University of Southern Queensland to reconsider its cutbacks to mathematics and statistics. (The full and official transcript of the day’s session in Parliament can be found here; the speech above is on page 1198.)]
[Update, April 29: The USQ administration released a revised draft proposal on April 22, but the details are largely unchanged (e.g. 11 staff cuts to the department of mathematics and computing instead of 12, and a “review” of the maths major and its courses rather than automatic elimination). The revised plan has already attracted criticism from the National Tertiary Education Union, and we are continuing to organise further opposition to the proposals. (For instance, László Lovász, President of the International Mathematical Union, wrote a letter of support on April 25.]
[Update, May 1. A second revised draft proposal has been released, which uses some new (but possibly non-permanent) sources of funding to add some specialised positions to partially offset the cuts (e.g. there will be 2-3 such positions in mathematics and statistics, although the 11 staff cuts are still in effect). The USQ administration has apparently also agreed to recheck the student load and financial data that is being used to underlie these proposals, as there appears to be some irregularities with this data in previous rationales.]