This week I visited Washington University in St. Louis to give a colloquium, and caught up with a couple of my old foliations friends, namely Rachel Roberts and Larry Conlon. Actually, I had caught up with Rachel (to some extent) the weekend before, when we both spoke at the Texas Geometry and Topology Conference in Austin, where Rachel gave a talk about her recent proof (joint with Will Kazez) that every taut foliation on a 3-manifold (other than ) can be approximated by both positive and negative contact structures; it follows that admits a symplectic structure with pseudoconvex boundary, and one deduces nontriviality of various invariants associated to (Seiberg-Witten, Heegaard Floer Homology, etc.). This theorem was known for sufficiently smooth (at least ) foliations by Eliashberg-Thurston, as exposed in their confoliations monograph, and it is one of the cornerstones of -dimensional symplectic geometry; unfortunately (fortunately?) many natural constructions of foliations on 3-manifolds can be done only in the or world. So the theorem of Rachel and Will is a big deal.
If we denote the foliation by which is the kernel of a 1-form and suppose the approximating positive and negative contact structures are given by the kernels of 1-forms where and pointwise, the symplectic form on is given by the formula
for some small , where is any closed 2-form on which is (strictly) positive on (and therefore also positive on the kernel of if the contact structures approximate the foliation sufficiently closely). The existence of such a closed 2-form is one of the well-known characterizations of tautness for foliations of 3-manifolds. I know several proofs, and at one point considered myself an expert in the theory of taut foliations. But when Cliff Taubes happened to ask me a few months ago which cohomology classes in are represented by such forms , and in particular whether the Euler class of could be represented by such a form, I was embarrassed to discover that I had never considered the question before.
The answer is actually well-known and quite easy to state, and is one of the applications of Sullivan’s theory of foliation cycles. One can also give a more hands-on topological proof which is special to codimension 1 foliations of 3-manifolds. Since the theory of taut foliations of 3-manifolds is a somewhat lost art, I thought it would be worthwhile to write a blog post giving the answer, and explaining the proofs.
Dennis Sullivan tells the story of attending a dynamics seminar at Berkeley in 1971, in which the speaker ended the seminar with the solution of (what Dennis calls) a “thorny problem”: the speaker explained how, if you have N pairs of points in the plane (all distinct), where each pair is distance at most apart, the pairs can be joined by a family of N disjoint paths, each of diameter at most (where depends only on , not on N, and goes to zero with ). This fact led (by a known technique) to an important application which had hitherto been known only in dimensions 3 and greater (where the construction is obvious by general position).
Sullivan goes on:
A heavily bearded long haired graduate student in the back of the room stood up and said he thought the algorithm of the proof didn’t work. He went shyly to the blackboard and drew two configurations of about seven points each and started applying to these the method of the end of the lecture. Little paths started emerging and getting in the way of other emerging paths which to avoid collision had to get longer and longer. The algorithm didn’t work at all for this quite involved diagrammatic reason.
The graduate student in question was Bill Thurston.
The purpose of this blog post is to give a short, constructive, computation-free proof of the following theorem:
Theorem: Every compact subset of the Riemann sphere can be arbitrarily closely approximated (in the Hausdorff metric) by the Julia set of a rational map.
A rational map is just a ratio of (complex) polynomials. Every holomorphic map from the Riemann sphere to itself is of this form. The Julia set of a rational map is the closure of the set of repelling periodic points; it is both forward and backward invariant. The complement of the Julia set is called the Fatou set.
Kathryn Lindsey gave a nice constructive proof that any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Her proof depends on an interpolation result by Curtiss. Kathryn is a postdoc at Chicago, and talked about her proof in our dynamics seminar a few weeks ago. It was a nice proof, and a nice talk, but I wondered if there was an elementary argument that one could see without doing any computation, and today I came up with the following.
Nothing stands still except in our memory.
- Phillipa Pearce, Tom’s Midnight Garden
In mathematics we are always putting new wine in old bottles. No mathematical object, no matter how simple or familiar, does not have some surprises in store. My office-mate in graduate school, Jason Horowitz, described the experience this way. He said learning to use a mathematical object was like learning to play a musical instrument (let’s say the piano). Over years of painstaking study, you familiarize yourself with the instrument, its strengths and capabilities; you hone your craft, your knowledge and sensitivity deepens. Then one day you discover a little button on the side, and you realize that there is a whole new row of green keys under the black and white ones.
In this blog post I would like to talk a bit about a beautiful new paper by Juliette Bavard which opens up a dramatic new range of applications of ideas from the classical theory of mapping class groups to 2-dimensional dynamics, geometric group theory, and other subjects.
Posted in Dynamics, Groups, Hyperbolic geometry, Surfaces
Tagged curve complex, Juliette Bavard, mapping class groups, quasimorphism, ray complex, Surfaces, surfaces of infinite type, weakly properly discontinuous
I was saddened to hear the news that Geoff Mess recently passed away, just a few days short of his 54th birthday. I first met Geoff as a beginning graduate student at Berkeley, in 1995; in fact, I believe he gave the first topology seminar I ever attended at Berkeley, on closed 3-manifolds which non-trivially cover themselves (the punchline is that there aren’t very many of them, and they could be classified without assuming the geometrization theorem, which was just a conjecture at the time). Geoff was very fast, whip-smart, with a daunting command of theory; and the impression he made on me in that seminar is still fresh in my mind. The next time I saw him might have been May 2004, at the N+1st Southern California Topology Conference, where Michael Handel was giving a talk on distortion elements in groups of diffeomorphisms of surfaces, and Geoff (who was in the audience), explained in an instant how to exhibit certain translations on a (flat) torus as exponentially distorted elements. Geoff was not well even at that stage — he had many physical problems, with his joints and his teeth; and some mental problems too. But he was perfectly pleasant and friendly, and happy to talk math with anyone. I saw him again a couple of years later when I gave a colloquium at UCLA, and his physical condition was a bit worse. But again, mentally he was razor-sharp, answering in an instant a question about (punctured) surface subgroups of free groups that I had been puzzling about for some time (and which became an ingredient in a paper I later wrote with Alden Walker).
Geoff Mess in 1996 at Kevin Scannell’s graduation (photo courtesy of Kevin Scannell)
Geoff published very few papers — maybe only one or two after finishing his PhD thesis; but one of his best and most important results is a key step in the proof of the Seifert Fibered Theorem in 3-manifold topology. Mess’s paper on this result was written but never published; it’s hard to get hold of the preprint, and harder still to digest it once you’ve got hold of it. So I thought it would be worthwhile to explain the statement of the Theorem, the state of knowledge at the time Mess wrote his paper, some of the details of Mess’s argument, and some subsequent developments (another account of the history of the Seifert Fibered Theorem by Jean-Philippe Préaux is available here).
The title of this post is a nod to the excellent and well-known Div, grad, curl and all that by Harry Schey (and perhaps also to the lesser-known sequel to one of the more consoling histories of Great Britain), and the purpose is to explain how to generalize these differential operators (familiar to electrical engineers and undergraduates taking vector calculus) and a few other ones from Euclidean 3-space to arbitrary Riemannian manifolds. I have a complicated relationship with the subject of Riemannian geometry; when I reviewed Dominic Joyce’s book Riemannian holonomy groups and calibrated geometry for SIAM reviews a few years ago, I began my review with the following sentence:
Riemannian manifolds are not primitive mathematical objects, like numbers, or functions, or graphs. They represent a compromise between local Euclidean geometry and global smooth topology, and another sort of compromise between precognitive geometric intuition and precise mathematical formalism.
Don’t ask me precisely what I meant by that; rather observe the repeated use of the key word compromise. The study of Riemannian geometry is — at least to me — fraught with compromise, a compromise which begins with language and notation. On the one hand, one would like a language and a formalism which treats Riemannian manifolds on their own terms, without introducing superfluous extra structure, and in which the fundamental objects and their properties are highlighted; on the other hand, in order to actually compute or to use the all-important tools of vector calculus and analysis one must introduce coordinates, indices, and cryptic notation which trips up beginners and experts alike.
For almost 50 years, Paul Sally was a towering figure in mathematics education at the University of Chicago. Although he was 80 years old, and had two prosthetic legs and an eyepatch (associated with the Type 1 diabetes he had his whole life), it was nevertheless a complete shock to our department when he passed away last December, and we struggled just to cover his undergraduate teaching load this winter and spring. As my contribution, I have been teaching an upper-division undergraduate class on “topics in geometry”, which I have appropriated and repurposed as an introduction to the classical geometry and topology of surfaces.
I have tried to include at least one problem in each homework assignment which builds a connection between classical geometry and some other part of mathematics, frequently elementary number theory. For last week’s assignment I thought I would include a problem on the well-known connection between Pythagorean triples and the modular group, perhaps touching on the Euclidean algorithm, continued fractions, etc. But I have introduced the hyperbolic plane in my class mainly in the hyperboloid model, in order to stress an analogy with spherical geometry, and in order to make it easy to derive the identities for hyperbolic triangles (i.e. hyperbolic laws of sines and cosines) from linear algebra, so it made sense to try to set up the problem in the language of the orthogonal group , and the subgroup preserving the integral lattice in .