Characteristic classes of foliations

I recently learned from Jim Carlson’s blog of the passing of Harsh Pittie on January 16 this year. I never met Harsh, but he is very familiar to me through his work, in particular for his classic book Characteristic classes of foliations. I first encountered this book as a graduate student, late last millenium. The book has several features which make it very attractive. First of all, it is short — only 105 pages. My strong instinct is always to read books cover-to-cover from start to finish, and the motivated reader can go through Harsh’s book in its entirety in maybe 4 hours. Second of all, the exposition is quite lovely: an excellent balance is struck between providing enough details to meet the demands of rigor, and sustaining the arc of an idea so that the reader can get the point. Third of all, the book achieves a difficult synthesis of two “opposing” points of view, namely the (differential-)geometric and the algebraic. In fact, this achievement is noted and praised in the MathSciNet review of the book (linked above) by Dmitri Fuks.

Filling geodesics and hyperbolic complements

Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3-manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flow-invariant way into stable, unstable and flow directions) and contact (i.e. they preserve a contact form — that is, a 1-form for which is a volume form). Their preprint gives some very interesting new constructions of such flows, obtained by surgery along a Legendrian knot (one tangent to the kernel of the contact form) which is transverse to the stable/unstable foliations of the Anosov flow.

Quasigeodesic flows on hyperbolic 3-manifolds

My student Steven Frankel has just posted his paper Quasigeodesic flows and Mobius-like groups on the arXiv. This heartbreaking work of staggering genius interesting paper makes a deep connection between dynamics, hyperbolic geometry, and group theory, and represents the first significant progress that I know of on a conjectural program I formulated a few years ago.

One of the main results of the paper is to show that every quasigeodesic flow on a closed hyperbolic 3-manifold either has a closed orbit, or the fundamental group of the manifold admits an action on a circle with some very peculiar properties, namely that it is Mobius-like but not Mobius. The problem of giving necessary and sufficient conditions on a vector field on a 3-manifold to guarantee the existence of a closed orbit is a long and interesting one, and the introduction to the paper gives a brief sketch of this history as follows:

Laying train tracks

This morning I was playing trains with my son Felix. At the moment he is much more interested in laying the tracks than putting the trains on and moving them around, but he doesn’t tend to get concerned about whether the track closes up to make a loop. The pieces of track are all roughly the following shape:

The Hall-Witt identity

The purpose of this blog post is to try to give some insight into the “meaning” of the Hall-Witt identity in group theory. This identity can look quite mysterious in its algebraic form, but there are several ways of describing it geometrically which are more natural and easier to understand.

If is a group, and are elements of , the commutator of and (denoted ) is the expression (note: algebraists tend to use the convention that instead). Commutators (as their name suggests) measure the failure of a pair of elements to commute, in the sense that . Since , the property of being a commutator is invariant under conjugation (here the superscript means conjugation by ; i.e. ; again, the algebraists use the opposite convention).

Ziggurats and the Slippery Conjecture

A couple of months ago I discussed a method to reduce a dynamical problem (computing the maximal rotation number of a prescribed element  in a free group given the rotation numbers of the generators) to a purely combinatorial one. Now Alden Walker and I have uploaded our paper, entitled “Ziggurats and rotation numbers”, to the arXiv.

The purpose of this blog post (aside from continuing the trend of posts titles containing the letter “Z”) is to discuss a very interesting conjecture that arose in the course of writing this paper. The conjecture does not need many prerequisites to appreciate or to attack, and it is my hope that some smart undergrad somewhere will crack it. The context is as follows.

Zonohedra and the Sylvester-Gallai theorem

When I was in Melbourne recently, I spent some time browsing through a copy of “Twelve Geometric Essays” by Harold Coxeter in the (small) library at AMSI. One of these essays was entitled “The classification of zonohedra by means of projective diagrams”, and it contained a very cute proof of the Sylvester-Gallai theorem, which I thought would make a nice (short!) blog post.

The Sylvester-Gallai theorem says that a finite collection of points in a projective plane are either all on a line, or else there is some line that contains exactly two of the points. Coxeter’s proof of this theorem falls out incidentally from an apparently unrelated study of certain polyhedra known as zonohedra.

For subsets and of a vector space , the Minkowski sum  is the set of points of the form for and . If and are polyhedra, so is , and the vertices of are sums of vertices of and . One natural way to think of is that it is the projection of the product under the affine map .