Tessel­la­tions divide a space similarly to how the tiles of a mosaic divide a picture. Study­ing the individ­ual tiles and the patterns they form provides a way of analyz­ing both real-world phenom­ena like microstruc­tures and mathe­mat­i­cal objects like metric spaces. In this project, we study random tessel­la­tions on hyper­bolic space, with the goal of better under­stand­ing a surpris­ing connec­tion between proba­bil­ity, geome­try, and algebra.

The goal of this project to study a connec­tion between several fields of math through random tessel­la­tions. Recent work yielded insight about certain discrete subgroups of Lie groups related to hyper­bolic space by bring­ing together ideas from proba­bil­ity and geome­try. The Ideal Poisson Voronoi Tessel­la­tion (IPVT) has been used to prove signif­i­cant results, provid­ing motiva­tion to construct varia­tions of it. 

The tiles of a Voronoi tessel­la­tion are constructed as follows. Take a metric space and fix a set of points in it. Each point defines a tile made up of all the points of the space closer to it than any other fixed point. To get a random tessel­la­tion, we select the fixed points based on a random process. The IPVT uses this type of construc­tion with points chosen accord­ing to a Poisson point process on a suitable bound­ary of the metric space. There­fore, a natural exten­sion of this work is to consider other point processes. We are currently working with deter­mi­nan­tal point processes, which are closely related to the Poisson point process, but lead to more uniform tiles. 

Conversely, we can begin with the algebraic side. Mathe­mat­i­cal groups describe the symme­tries of a space, so it is often useful to study them in conjunc­tion with that space. The IPVT corre­sponds to a partic­u­lar type of group (lattices) acting on hyper­bolic space. We are also construct­ing point processes corre­spond­ing to smaller discrete subgroups of inter­est, such as Schot­tky groups.