Last week while in Tel Aviv I had an interesting conversation over lunch with Leonid Polterovich and Yaron Ostrover. I happened to mention the following gem from the remarkable book A=B by Wilf-Zeilberger. The book contains the following Theorem and “proof”:
Theorem 1.4.2. For every triangle ABC, the angle bisectors intersect at one point
Proof. Verify this for the 64 triangles for which the angle at A and B are one of 10, 20, 30, , 80. Since the theorem is true in these cases it is always true.
We are asked the provocative question: is this proof acceptable? The philosophy of the W-Z method is illustrated by pointing out that this proof is acceptable if one adds for clarity the remark that the coordinates of the intersections of the pairs of angle bisectors are rational functions of degree at most 7 in the tangents of A/2 and B/2; hence if they agree at 64 points they agree everywhere.
Leonid countered with a personal anecdote. Recall that an altitude in a triangle is a line through one vertex which is perpendicular to the opposite edge. Leonid related that one day his geometry class (I forget the precise context) were given the problem of showing that the altitudes in a hyperbolic triangle (i.e. a triangle in the hyperbolic plane) meet at a single point — the orthocenter of the triangle. After the class had struggled with this for some time, the professor laconically informed them that the result obviously followed immediately from the corresponding fact for Euclidean triangles “by analytic continuation”. Philosophically speaking, this is not too far from the W-Z example, although the details are slightly more shaky — in particular, the class of Euclidean triangles are not Zariski dense in the class of triangles in constant curvature spaces, so a little more remains to be done.
Actually, one might even go back and rethink the W-Z example — how exactly are we to verify that the angular bisectors intersect at a point for the triangles in question without doing a calculation no less complicated that the general case? Let’s raise the stakes further. After some thought, we see that not only will the intersections of pairs of angle bisectors be given by rational functions of the tangents of A/2 and B/2, but the (algebraic) heights of the coefficients of these rational functions can be easily estimated, and one can therefore compute an effective lower bound on how far apart the intersections of the angle bisectors would be if they were not equal. We can then literally draw the triangles on a piece of physical paper using a protractor, and verify by eyesight that the angle bisectors appear to coincide to within the necessary accuracy. After rigorously estimating the experimental errors, we can write qed.
The other day by chance I happened to look at Richard Kenyon’s web page, and was struck by a very beautiful animated image there. The image is of a region tiled by colored squares, which are slowly rotating. As the squares rotate, they change size in such a way that the new (skewed, resized) squares still tile the same region. I thought it might be fun to try to guess how the image was constructed, and to produce my own version of his image.
In winter and spring of 2001, Nathan Dunfield and I ran a seminar at Harvard whose purpose was to go through Thurston’s proof of the geometrization theorem for Haken manifolds. This was a very useful and productive exercise, and there was wide participation from faculty and students. As well as talks by Nathan and myself, there were talks by David Dumas, Laura de Marco, Maryam Mirzakhani, Curt McMullen, Dylan Thurston, and John Holt. At the conclusion of the semester, Bill Thurston agreed to come out and lead a discussion on geometrization, in which he ended up talking a bit about what had led him to formulate the conjecture in the first place, what ideas had played into it, how and when he had gone about proving it, his ideas about exposition, and so on.
I had recently bought a video camera, and decided to tape Bill’s talk. I never did anything with it until now (in fact, I don’t think I ever re-watched anything that I taped), but it turned out to be not too difficult to transfer the file from tape to computer. Since this seems like an interesting fragment of intellectual history, I thought it might be worthwhile to post the result to YouTube — the video link is here.
My eldest daughter Lisa recently brought home a note from her school from her computer class teacher. Apparently, the 5th grade kids have been learning to program in Logo, in the MicroWorlds programming environment. I have very pleasant memories of learning to program in Logo back when I was in middle school. If you’re not familiar with Logo, it’s a simple variant of Lisp designed by Seymour Papert, whereby the programmer directs a turtle cursor to move about the screen, moving forward some distance, turning left or right, etc. The turtle can also be directed to raise or lower a pen, and one can draw very pretty pictures in Logo as the track of the turtle’s motion.
Let’s restrict our turtle’s movements to alternating between taking a step of a fixed size S, and turning either left or right through some fixed angle A. Then a (compiled) “program” is just a finite string in the two letter alphabet L and R, indicating the direction of turning at each step. A “random turtle” is one for which the choice of L or R at each step is made randomly, say with equal probability, and choices made independently at each step. The motion of a Euclidean random turtle on a small scale is determined by its turning angle A, but on a large scale “looks like” Brownian motion. Here are two examples of Euclidean random turtles for A=45 degrees and A=60 degrees respectively.
The purpose of this blog post is to describe the behavior of a random turtle in the hyperbolic plane, and the appearance of an interesting phase transition at . This example illustrates nicely some themes in probability and group dynamics, and lends itself easily to visualization.
Let be the free group on two generators, and let be the endomorphism defined on generators by and . We define Sapir’s group to be the ascending HNN extension
This group was studied by Crisp-Sageev-Sapir in the context of their work on right-angled Artin groups, and independently by Feighn (according to Mark Sapir); both sought (unsuccessfully) to determine whether contains a subgroup isomorphic to the fundamental group of a closed, oriented surface of genus at least 2. Sapir has conjectured in personal communication that does not contain a surface subgroup, and explicitly posed this question as Problem 8.1 in his problem list.
After three years of thinking about this question on and off, Alden Walker and I have recently succeeded in finding a surface subgroup of , and it is the purpose of this blog post to describe this surface, how it was found, and some related observations. By pushing the technique further, Alden and I managed to prove that for a fixed free group of finite rank, and for a random endomorphism of length (i.e. one taking the generators to random words of length ), the associated HNN extension contains a closed surface subgroup with probability going to 1 as . This result is part of a larger project which we expect to post to the arXiv soon.
I am currently teaching a class at the University of Chicago on hyperbolic groups, and I have just introduced the concept of -hyperbolic (geodesic) metric spaces. A geodesic metrix space is -hyperbolic if for any geodesic triangle , and any there is some with . The quintessential -hyperbolic space is the hyperbolic plane, the unique (up to isometry) simply-connected complete Riemannian 2-manifold of constant curvature . It follows that any simply-connected complete Riemannian manifold of constant curvature is -hyperbolic for some depending on ; roughly one can take .
What gives this condition some power is the rich class of examples of spaces which are -hyperbolic for some . One very important class of examples are simply-connected complete Riemannian manifolds with upper curvature bounds. Such spaces enjoy a very strong comparison property with simply-connected spaces of constant curvature, and are therefore the prime examples of what are known as CAT(K) spaces.
Definition: A geodesic metric space is said to be , if the following holds. If is a geodesic triangle in , let be a comparison triangle in a simply connected complete Riemannian manifold of constant curvature . Being a comparison triangle means just that the length of is equal to the length of and so on. For any there is a corresponding point in the comparison edge which is the same distance from and as is from and respectively. The condition says, for all as above, and all , there is an inequality .
The term CAT here (coined by Gromov) is an acronym for Cartan-Alexandrov-Toponogov, who all proved significant theorems in Riemannian comparison geometry. From the definition it follows immediately that any space with is -hyperbolic for some depending only on . The point of this post is to give a short proof of the following fundamental fact:
CAT(K) Theorem: Let be a complete simply-connected Riemannian manifold with sectional curvature everywhere. Then with its induced Riemannian (path) metric is .
This morning I heard the awful news that Bill Thurston died last night. Many of us knew that Bill was very ill, but we all hoped (or imagined?) that he would still be with us for a while yet, and the suddenness of this is very harsh. As Sarah Koch put it in an email to me, “Although this was not unexpected, it is still shocking.” On the other hand, I am glad to hear that he was surrounded by family, and died peacefully.
I counted Bill as my friend, as well as my mentor, and I have many vivid and happy memories of time I spent with him. I hope that writing down a few of these reminiscences will be cathartic for me, and for others who are coping with this loss.