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 .
When I started in graduate school, I was very interested in 3-manifolds, especially Thurston’s geometrization conjecture. Somehow in dimension 3, there is a marvelous marriage of flexibility and rigidity: generic 3-manifolds are flexible enough to admit hyperbolic structures — i.e. Riemannian metrics of constant curvature -1, modeled on hyperbolic space. But these structures are so rigid that they are determined up to isometry (!) entirely by the fundamental group of the manifold, and provide a bridge from topology to the rigid world of number fields and arithmetic. 3-manifolds, especially the hyperbolic ones, display an astonishing range of interesting phenomena, so that even though the individual manifolds are discrete and rigid, they come in infinite families parameterized by Dehn surgery. When Perelman proved Thurston’s conjecture, I gradually moved away from 3-manifold topology into some neighboring fields such as dynamics and geometric group theory; subjectively this move felt to me like a transition from a baroque world of highly intricate, finely tuned and beautiful objects to more rough and disordered domains in which the rule was chaos and disorder, and where one had to restrict attention and focus to find the kinds of structured objects that one can say something about mathematically. In these new domains my familiarity with 3-manifold topology was always extremely useful to me, but almost always as a source of inspiration or analogy or example, rather than that some specific theorem about 3-manifolds could be used to say something about groups in general, or dynamical systems in general, or whatever. Many important recent developments in geometric group theory are generalizations of geometric ideas which were first identified or studied in the world of 3-manifolds; but there was not much connection at the deepest level, at least as far as I could see.
This impression was dramatically shaken by Agol’s proof of the virtual Haken conjecture and virtual fibration conjectures in 3-manifold topology by an argument which depends for one of its key ingredients on the theory of non-positively curved cube complexes — a subject in geometric and combinatorial group theory which, while inspired by key examples in low-dimensions (especially surfaces in the hands of Scott, and graphs in the hands of Stallings), is definitely a high-dimensional theory with no obvious relations to manifolds at all. Even so, the transfer of information in this case is still from the “broad” world of group theory to the “special” world of 3-manifolds. It shows that 3-manifold topology is even richer than hitherto suspected, but it does not contradict the idea that the beautiful edifice of 3-manifold topology is an exceptional corner in the vast unstructured world of geometry.
I have just posted a paper to the arXiv, coauthored with Henry Wilton, and building on prior work I did with Alden Walker, that aims to challenge this idea. Let me quote the first couple of paragraphs of the introduction:
Geometric group theory was born in low-dimensional topology, in the collective visions of Klein, Poincaré and Dehn. Stallings used key ideas from 3-manifold topology (Dehn’s lemma, the sphere theorem) to prove theorems about free groups, and as a model for how to think about groups geometrically in general. The pillars of modern geometric group theory — (relatively) hyperbolic groups and hyperbolic Dehn filling, NPC cube complexes and their relations to LERF, the theory of JSJ splittings of groups and the structure of limit groups — all have their origins in the geometric and topological theory of 2- and 3-manifolds.
Despite these substantial and deep connections, the role of 3-manifolds in the larger world of group theory has been mainly to serve as a source of examples — of specific groups, and of rich and important phenomena and structure. Surfaces (especially Riemann surfaces) arise naturally throughout all of mathematics (and throughout science more generally), and are as ubiquitous as the complex numbers. But the conventional view is surely that 3-manifolds per se do not spontaneously arise in other areas of geometry (or mathematics more broadly) amongst the generic objects of study. We challenge this conventional view: 3-manifolds are everywhere.