MATH 757: Topics in Topology (Fall 2007)

Instructor: William Thurston

Meeting Time & Room

There are several nice theories concerning enumeration and probabilistic properties of triangulations and other cell-divisions of surface. We will develop some of this theory, with the main goal of developing a good intuitive and geometric understanding of typical and arbitary triangulations of surfaces.

  1. There is an amazingly simple and efficient code due to Poulalhon and Schaeffer that gives a bijective enumerations of triangulations and in terms of trees, which in turn are encoded by binary sequences.
  2. The Poulalhon-Schaeffer construction is in turn based on a construction of Walter Schnyder that in effect constructs a canonical curvature-transporting graph-flow on a planar graph. How does this generalize to surfaces of more complicated topology? Are there comparable constructions for smooth surfaces?
  3. There are other constructions that work, sometimes more easily, other cell-divisions. In particular, quadrangulations and other types of bipartite planar graphs have another nice system for enumeration. Using these, Le Gall and Paulin showed if you take a uniformly-random sequence of quadrangulations of S², with metrics defined as an assemblage of squares scaled so that the diameter is constant, then this sequence of metric spaces is almost surely precompact in the Gromov-Hausdorff topology, with limit set consisting of metric spaces with the topology of S² that have Hausdorff dimension a.e. 4.
  4. A triangulated surface has a conformal structure. I would like to know whether there are limit theorems for the conformal structures on uniformly-random triangulated surfaces as there are for the metric structure. For instance, associated to any triangulation of the sphere is a circle-packing, unique up to Moebius transformation. The packing can be adjusted on the sphere so the center of mass of the centers of circles is at the center of the sphere. What do these circle-packings look like in the limit? Do the circles go to 0 in size? Does the uniform measure on centers of circles typically go to a measure on S² that is topologically equivalent to Lebesgue measure (0 on points, positive on open sets)?