Low-dimensional topologists study 3- and 4-dimensional surfaces or manifolds. Here a manifold is a topological space M that locally looks like Euclidean space R^k for some positive integer k. We call k the dimension of the manifold, and we call M a k-manifold. People like to say that if you were a myopic insect living on a 2-dimensional manifold then you would think you were living in the plane. The residents of Earth in the fifteenth century fit that description: everyone in those days thought that the Earth was flat. Figure 1 illustrates a 2-dimensional manifold (namely, a torus).

Figure 1: The torus: an example of a 2-manifold.

Thanks to work of Möbius and others in the mid-nineteenth century, the structure of 2-manifolds is well understood. In fact a closed, orientable 2-manifold is, up to topological equivalence, a sphere with handles attached. Topologists would like to have a similar description of 3-manifolds and 4-manifolds. Particularly because our universe is a 3-dimensional manifold and space-time is a 4-dimensional manifold, there is considerable motivation for these questions.

It took mathematicians quite some time to come to grips with the concept of manifold (from the mid-nineteenth century to nearly the mid-twentieth century). The development of the idea of Riemann surface aided in the process. Special new tools are required for the analysis of 3- and 4-dimensional manifolds. Among the first investigators in this more difficult realm were Poul Heegaard and Henri Poincaré. One of Poincaré's important contributions was the seminal Poincaré Conjecture---that a 3-manifold with the geometric/topological properties of a sphere must in fact be (homeomorphic to) a sphere. Poincaré first formulated his conjecture in the language of homology, and it was Heegaard who proved that that formulation was false. To do so, he constructed a 3-manifold by gluing together two handle-bodies (see below) along their boundaries. The idea is illustrated in Figure 2. A more general version of this construction that can be used to describe any 3-manifold later came to be known as the Heegaard splitting (again see the discussion below).

Figure 2: Gluing together two handle-bodies.

Furthermore, Heegaard and Dehn's article Analysis Situs (1907, in the Enz. der Math. Wissenschaften, edited by Poincaré himself) marks the foundation of combinatorial topology. One of the thrusts of the present workshop is to create some communication between the combinatorial point of view in the subject and the geometric point of view in the subject.

The combinatorial point of view is illustrated by the basic concept of triangulation of a surface or manifold. Figures 3 and 4 illustrate triangulations of planar regions.

Figure 3: Triangulation of a planar region.

Figure 4: Another triangulation of a planar region.

The idea is to represent the given surface or region as a network of triangles. In such a configuration, the edges of the triangles form a graph, and the surface may be profitably studied through analysis of that graph. Figure 5 shows a triangulation of the 2-torus.

Figure 5: Triangulation of a 2-torus.

In the 1950s, Edwin Moise and R H Bing established the existence of triangulations for topological 3-manifolds. It is not hard to see that a triangulated 3-manifold possesses a Heegaard splitting. If T is a triangulation then (i) a fattening of the graph induced by the edges of T will give a handle-body and (ii) a fattening of the dual graph to that in part (i) will give another handle-body. Here, by a handle-body we mean (roughly speaking) a ball with handles attached. Heegaard's main result is that the union of the two indicated handle-bodies is the entire original manifold M. This results in the so-called Heegaard splitting of M. Figure 6 gives the idea of the construction of the Heegaard splitting as described here.

Figure 6: The Heegaard splitting.

The geometric point of view in the subject of 3- and 4-dimensional manifolds is due---at least in modern times---primarily to William P. Thurston. One key ingredient of the geometric approach is hyperbolic geometry. This is a non-Euclidean geometry of the sort developed by Bolyai and Lobachevsky to show that Euclid's parallel postulate is independent of the other geometric axioms. In the early 1980s Thurston postulated that any 3-manifold can be decomposed into fundamental pieces, each of which supports a basic model geometry---either spherical, Euclidean, hyperbolic, S² × R, H² × R (this is hyperbolic geometry), the universal cover of SL₂R, nil-geometry, or solv-geometry. This idea became known as the Thurston geometrization program.

There is hope that the idea of triangulations (and the more recent idea, due to Jaco and Rubinstein, of efficient triangulations), along with Heegaard splittings, can be used to understand the geometrization program and the Poincaré conjecture.

Certainly a big event in the past few years is that Grigori Perelman of St. Petersburg used deep methods of partial differential equations (particularly the Ricci flow of Richard Hamilton) to prove the Poincaré conjecture. Perelman's write-up of his result was rather sketchy, but John Morgan and Gang Tian have recently published a 450-page book that provides all the details of the argument. Perelman's methods also apply to the geometrization program, but there is not yet a consensus as to whether they constitute a bona fide mathematical proof. Perelman's important results leave many questions about the structure of 3-manifolds unanswered, and one purpose of this workshop is to explore those avenues.

Clearly the ideas being discussed in this workshop are at the cutting edge of modern ideas in geometry and topology. Many of the world's leading experts are here at AIM to focus on the latest ideas--particularly a new result of Namazi for constructing hyperbolic metrics--and put them into a broad context.