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Perhaps Thurston’s best known achievement is the proof of the hyperbolisation theorem for Haken manifolds, which showed that 3-manifolds which obeyed a certain number of topological conditions, could always be given a hyperbolic geometry (i.e. a Riemannian metric that made the manifold isometric to a quotient of the hyperbolic 3-space ). This difficult theorem connecting the topological and geometric structure of 3-manifolds led Thurston to give his influential geometrisation conjecture, which (in principle, at least) completely classifies the topology of an arbitrary compact 3-manifold as a combination of eight model geometries (now known as Thurston model geometries). This conjecture has many consequences, including Thurston’s hyperbolisation theorem and (most famously) the Poincaré conjecture. Indeed, by placing that conjecture in the context of a conceptually appealing general framework, of which many other cases could already be verified, Thurston provided one of the strongest pieces of evidence towards the truth of the Poincaré conjecture, until the work of Grisha Perelman in 2002-2003 proved both the Poincaré conjecture and the geometrisation conjecture by developing Hamilton’s Ricci flow methods. (There are now several variants of Perelman’s proof of both conjectures; in the proof of geometrisation by Bessieres, Besson, Boileau, Maillot, and Porti, Thurston’s hyperbolisation theorem is a crucial ingredient, allowing one to bypass the need for the theory of Alexandrov spaces in a key step in Perelman’s argument.)
One of my favourite results of Thurston’s is his elegant method for everting the sphere (smoothly turning a sphere in inside out without any folds or singularities). The fact that sphere eversion can be achieved at all is highly unintuitive, and is often referred to as Smale’s paradox, as Stephen Smale was the first to give a proof that such an eversion exists. However, prior to Thurston’s method, the known constructions for sphere eversion were quite complicated. Thurston’s method, relying on corrugating and then twisting the sphere, is sufficiently conceptual and geometric that it can in fact be explained quite effectively in non-technical terms, as was done in the following excellent video entitled “Outside In“, and produced by the Geometry Center:
In addition to his direct mathematical research contributions, Thurston was also an amazing mathematical expositor, having the rare knack of being able to describe the process of mathematical thinking in addition to the results of that process and the intuition underlying it. His wonderful essay “On proof and progress in mathematics“, which I highly recommend, is the quintessential instance of this; more recent examples include his many insightful questions and answers on MathOverflow.
I unfortunately never had the opportunity to meet Thurston in person (although we did correspond a few times online), but I know many mathematicians who have been profoundly influenced by him and his work. His death is a great loss for mathematics.
Here is a nice version of the periodic table (produced jointly by the Association for the British Pharmaceutical Industry, British Petroleum, the Chemical Industry Education Centre, and the Royal Society for Chemistry) that focuses on the applications of each of the elements, rather than their chemical properties. A simple idea, but remarkably effective in bringing the table to life.
It might be amusing to attempt something similar for mathematics, for instance creating a poster that takes each of the top-level categories in the AMS 2010 Mathematics Subject Classification scheme (or perhaps the arXiv math subject classification), and listing four or five applications of each, one of which would be illustrated by some simple artwork. (Except, of course, for those subfields that are “seldom found in nature”. :-) )
A project like this, which would need expertise both in mathematics and in graphic design, and which could be decomposed into several loosely interacting subprojects, seems amenable to a polymath-type approach; it seems to me that popularisation of mathematics is as valid an application of this paradigm as research mathematics. (Admittedly, there is a danger of “design by committee“, but a polymath project is not quite the same thing as a committee, and it would be an interesting experiment to see the relative strengths and weaknesses of this design method.) I’d be curious to see what readers would think of such an experiment.
Now that the quarter is nearing an end, I’m returning to the half of the polymath1 project hosted here, which focussed on computing density Hales-Jewett numbers and related quantities. The purpose of this thread is to try to organise the task of actually writing up the results that we already have; as this is a metathread, I don’t think we need to number the comments as in the research threads.
To start the ball rolling, I have put up a proposed outline of the paper on the wiki. At present, everything in there is negotiable: title, abstract, introduction, and choice and ordering of sections. I suppose we could start by trying to get some consensus as to what should or should not go into this paper, how to organise it, what notational conventions to use, whether the paper is too big or too small, and so forth. Once there is some reasonable consensus, I will try creating some TeX files for the individual sections (much as is already being done with the first polymath1 paper) and get different contributors working on different sections (presumably we will be able to coordinate all this through this thread). This, like everything else in the polymath1 project, will be an experiment, with the rules made up as we go along; presumably once we get started it will become clearer what kind of collaborative writing frameworks work well, and which ones do not.
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.
I’m continuing my series of articles for the Princeton Companion to Mathematics ahead of the winter quarter here at UCLA (during which I expect this blog to become dominated by ergodic theory posts) with my article on generalised solutions to PDE. (I have three more PCM articles to release here, but they will have to wait until spring break.) This article ties in to some extent with my previous PCM article on distributions, because distributional solutions are one good example of a “generalised solution” or “weak solution” to a PDE. They are not the only such notion though; one also has variational and stationary solutions, viscosity solutions, penalised solutions, solutions outside of a singular set, and so forth. These notions of generalised solution are necessary when dealing with PDE that can exhibit singularities, shocks, oscillations, or other non-smooth behaviour. Also, in the foundational existence theory for many PDE, it has often been profitable to first construct a fairly weak solution and then use additional arguments to upgrade that solution to a stronger solution (e.g. a “classical” or “smooth” solution), rather than attempt to construct the stronger solution directly. On the other hand, there is a tradeoff between how easy it is to construct a weak solution, and how easy it is to upgrade that solution; solution concepts which are so weak that they cannot be upgraded at all seem to be significantly less useful in the subject, even if (or especially if) existence of such solutions is a near-triviality. [This is one manifestation of the somewhat whimsical “law of conservation of difficulty”: in order to prove any genuinely non-trivial result, some hard work has to be done somewhere. In particular, it is often the case that the behaviour of PDE depends quite sensitively on the exact structure of that PDE (e.g. on the sign of various key terms), and so any result that captures such behaviour must, at some point, exploit that structure in a non-trivial manner; one usually cannot get very far in PDE by relying just on general-purpose theorems that apply to all PDE, regardless of structure.]
The Companion also has a section on history of mathematics; for instance, here is Leo Corry‘s PCM article “The development of the idea of proof“, covering the period from Euclid to Frege. We take for granted nowadays that we have precise, rigorous, and standard frameworks for proving things in set theory, number theory, geometry, analysis, probability, etc., but it is worth remembering that for the majority of the history of mathematics, this was not completely the case; even Euclid’s axiomatic approach to geometry contained some implicit assumptions about topology, order, and sets which were not fully formalised until the work of Hilbert in the modern era. (Even nowadays, there are still a few parts of mathematics, such as mathematical quantum field theory, which still do not have a completely satisfactory formalisation, though hopefully the situation will improve in the future.)
[Update, Jan 4: bad link fixed.]
My colleague Ricardo Pérez-Marco showed me a very cute proof of Pythagoras’ theorem, which I thought I would share here; it’s not particularly earth-shattering, but it is perhaps the most intuitive proof of the theorem that I have seen yet.
In the above diagram, a, b, c are the lengths BC, CA, and AB of the right-angled triangle ACB, while x and y are the areas of the right-angled triangles CDB and ADC respectively. Thus the whole triangle ACB has area x+y.
Now observe that the right-angled triangles CDB, ADC, and ACB are all similar (because of all the common angles), and thus their areas are proportional to the square of their respective hypotenuses. In other words, (x,y,x+y) is proportional to . Pythagoras’ theorem follows.
For the last year or so, I’ve maintained two advice pages on my web site: one for career advice to students in mathematics, and another for authors wishing to write and submit papers (for instance, to one of the journals I am editor of). It occurred to me, though, that an advice page is particularly well suited to the blog medium, due to the opportunities for feedback that this medium affords (and especially given that many of my readers have more mathematical experience than I do).
I have thus moved these pages to my blog; my career advice page is now here, and my advice for writing and submitting papers is now here. I also took the opportunity to split up these rather lengthy pages into lots of individual subpages, which allowed for easier hyperlinking, and also to expand each separate topic somewhat (for instance, each topic is now framed by an appropriate quotation). Each subpage is also open to comments, as I am hoping to get some feedback to improve each of them.
[Update, June 10: Of course, the comments page for this post, and for the pages mentioned above, are also a good place to post your own tips on mathematical writing or careers. :-) ]
I gave a non-technical talk today to the local chapter of the Pi Mu Epsilon society here at UCLA. I chose to talk on the cosmic distance ladder – the hierarchy of rather clever (yet surprisingly elementary) mathematical methods that astronomers use to indirectly measure very large distances, such as the distance to planets, nearby stars, or distant stars. This ladder was really started by the ancient Greeks, who used it to measure the size and relative locations of the Earth, Sun and Moon to reasonable accuracy, and then continued by Copernicus, Brahe and Kepler who then measured distances to the planets, and in the modern era to stars, galaxies, and (very recently) to the scale of the universe itself. It’s a great testament to the power of indirect measurement, and to the use of mathematics to cleverly augment observation.
For this (rather graphics-intensive) talk, I used Powerpoint for the first time; the slides (which are rather large – 3 megabytes) – can be downloaded here. [I gave an earlier version of this talk in Australia last year in a plainer PDF format, and had to get someone to convert it for me.]
[Update, May 31: In case the powerpoint file is too large or unreadable, I also have my older PDF version of the talk, which omits all the graphics.]
I’ve received quite a lot of inquiries regarding a recent article in the New York Times, so I am borrowing some space on this blog to respond to some of the more common of these, and also to initiate a discussion on maths education, which was briefly touched upon in the article.
Firstly, some links:
- The video for the talk “Structure and randomness in the prime numbers” mentioned in the article can be found here (requires RealPlayer). The slides can be found here. My other expository and research material on number theory can be found here.
- I don’t have any specific advice regarding gifted education, though some articles on my own experiences can be found here. I do however have some thoughts on career advice at the undergraduate level and beyond.
- I have some responses to several other common queries (e.g. regarding books, interviews, invitations, etc.) at my contact information page.
Most of the feedback I received, though, concerned the issue of maths education. I mentioned in the article that I feel that the skill of thinking in a mathematical and rigorous way is one which can be taught to virtually anyone, and I would in the future hope to be involved in some project aimed towards this goal. I received a surprising number of inquiries on this, particularly from parents of school-age children. Unfortunately, my maths teaching experience is almost completely restricted to the undergraduate and graduate levels – and my own school experience was perhaps somewhat unusual – so I currently have close to zero expertise in K-12 maths education. (This may change though as my son gets older…) Still, I think it is a worthy topic of discussion as to what the mathematical academic community can do to promote interest in mathematics, and to encourage mathematical ways of thinking and of looking at the world, so I am opening the discussion to others who may have something of interest to say on these matters.
(Update, March 13: A bad link has been repaired. Also, I can’t resist a somewhat political plug: for Californian readers, there is an open letter in support of California’s K-12 education standards, together with some background information.)