Random Hyperbolic Lattices

Lattices are fundamental objects in mathematics, and their study has significant applications in modeling different types of symmetries in statistical physics, chemistry (especially in crystallography), and art. The tiles on the floor and in the shower, a cobbled street, a chessboard – anywhere you see a grid pattern – the underlying mathematical structure is a lattice. The preceding examples are all Euclidean lattices, which model symmetries in classical Euclidean geometry.

In previous collaborative work with Professor Gregory Margulis (Yale University), Professor Athreya studied the properties of a model of a random Euclidean lattice, proving that the probability that a random lattice does not intersect a large subset of Euclidean space is inversely proportional to the size of the set.

A natural and important generalization of this question is to develop and understand a model of a random hyperbolic lattice, that is, the set of symmetries of a grid in hyperbolic geometry. During his Center appointment Professor Athreya will collaborate with Professors Margulis and Yair Minsky (Yale University) to do just that.

In order to construct a random hyperbolic lattice, they will develop a way to select geometric representatives of fundamental elements of the lattice known as generators. They are planning to use the technology of representation varieties to formulate and attack the problem. In technical language, the goal will be to estimate and understand the mean and variance of the number of lattice points in a fixed set, which, in the hyperbolic setting, will require new mathematical technology. This is a long-term, foundational project that will connect with several areas of mathematics, including geometry, topology, and dynamical systems.