"What is a random surface?" - A plenary talk at ICM 2022

Introduction: "What is a random surface?" was a plenary talk given by Scott Sheffield (Massachusetts Institute of Technology, USA) at International Congress of Mathematicians (ICM) 2022. This post will discuss about certain minor details that are simple and interesting regarding this talk. 

1. The construction of the problem

   Consider k identical regular triangles. We want to glue them in a way to make into a closed surface (i.e. a polyhedron whose faces are regular triangles). By double counting, the number of triangles times three equals the number of edges of the polyhedron times two, thus k must be even or k = 2n. If we consider the planar version of this problem (i.e. all triangles lie on a plane), the positions of the triangles are clear and you may view these triangles as triangular bricks on the floor of your house. However, when these triangles are at arbitrary positions in the 3-dimensional space, the angle between two adjacent triangles might vary. This yields various configurations of the 2n triangles. A random surface might be viewed as a random way to generate a surface by gluing these 2n triangles. In other words, as mentioned in Scott Sheffield's paper in ICM 2022 Proceedings, "We obtain a random sphere-homeomorphic surface by sampling uniformly from the gluings that produce a topological sphere.". 

   The problem of discovering different configurations of the 2n triangles is interesting. You might try the following problem, which might be classified as enumerative combinatorics: for a fixed positive integer n, how many surfaces with 2n regular triangles can we obtain? When n equals two, the answer is one: the regular tetrahedron. Indeed, since there are 4 triangles, the number of edges is (4 x 3)/2 = 6. When two triangles are glued, they have exactly five edges in total. Since only one more edge can be added, the third triangle must have a common edge with each of the previously mentioned two edges. As the three triangles are glued together, a regular tetrahedron with no bottom is formed, and we only need to add the bottom in order to create a complete polyhedron. 

2. Convergence

   As n tends to infinity, "these random surfaces (appropriately scaled) converge in law. The limit is a 'canonical' sphere-homeomorphic random surface.". Convergence in law is a concept that can be learned from higher undergraduate course in probability and can be found in, for example, the book "Probability Essentials" written by Jean Jacod and Philip Protter, so I won't recall it here. This amazing proposition might remind us of the Law of Large Numbers, since both have the same mathematical spirit: there are a wide variety of results that we can obtain, but it is likely that the overall picture is close to a particular, fixed "expectation" as soon as we are not limited to too few observations. However, when it comes to random surfaces, there are certain problems behind. 

   A problem is that when n tends to infinity, the volume of the set bounded by this surface might be very large (or possibly very small). Different values of the volume lead to completely different versions of the surface. In other to "uniformize" or "normalize", we should assume, for example, that each surface will be (re)scaled so that its volume is exactly one. This is perhaps the idea behind the word "appropriately scale" written in the paper. Moreover, there should be a formal geometric definition about the "gap" or "distance" between different shapes so that the convergence is well-defined.