Geodesic Currents and Counting Problems

Imagine drawing a circle in the plane, centered at the origin and of radius R, and you want to count the number of points with integer coefficients enclosed by the circle. For example, if R=3 there are 13 such points, if R=10 there are 253, and if R=20 there are 1129 points. Clearly, the larger R is the more points there are, but exactly how are these numbers related? One can prove, using elementary mathematics, that the number of integer points inside a circle of radius R grows like pi*R^2, i.e. the area of the region it encloses. More precisely, the number of such points is asymptotic to the area, meaning that the ratio of the two quantities tends to 1 as R goes to infinity.

The simple problem described above is closely related to counting curves on surfaces. To a topologist, a surface is a 2-dimensional object which can be obtained by cutting out a polygon in the plane and then gluing sides together in pairs. For example, if the polygon is a square and we glue two opposite sides together we get a tube. If we glue the two boundary circles of the tube together, we get a donut, which we call a torus. The torus comes with a metric, a way to measure distances, given by its identification with a square in the plane which has the usual flat (Euclidean) metric. A curve on the torus is a closed loop (think of a string wrapped around the surface where you tie the two endpoints together) which we "pull tight" so it becomes as short as possible. As it turns out, the number of curves on the torus of length at most R is exactly the same as number of integer points in the plane inside a circle of radius R.

If we use another polygon instead of a square in the construction above we get a more complicated surface. In fact, in general we get a surface that looks like several tori glued together. The number of tori is called the genus g of the surface. However, to get a nice (constant curvature) metric on the surface, we need to cut the polygon out of the hyperbolic plane (which is negatively curved, like the inside of a bowl) instead of the usual Euclidean plane (which is flat). This drastically changes the growth of the number of curves: it was shown in the 60s by Huber that the asymptotic growth is exponential in the length when g>1. However, if we look instead only at curves that do not self-intersect there are much fewer curves and we again get a polynomial growth rate (this was first observed by Birman-Series in the 80s and proved in more detail by Rivin in 2001). Finding the exact asymptotic growth of these curves is a hard problem and was solved by a deep theorem by Mirzakhani in 2008. She proved that the number of simple curves of length at most R on a surface of genus g>1 is asymptotic to a constant times R^{6g-6}.

Mirzakhani's result became instantly famous since it was a part of her triad of results on curve counting, volume growth, and the Witten conjecture (an important problem in physics) breaking ground in both the world of geometry and dynamics and having important implications to physics. In this project we use new methods to approach the problem of counting curves which allows us to generalize her result. In fact, we also get a new, and very different, proof of Mirzakhani's result. The original proof requires expert understanding of several fields of mathematics and is hard to grasp in full detail even for experts in the fields; the new approach has potential to open up the field to researchers from a wider field of expertise. The new proof also gives a new way to compute important constants related to Mirzakhani's theorem. The novelty of these methods is the use of so called geodesic currents, a space that unifies the study of curves, measured laminations, and hyperbolic metrics, all integral notions to curve counting.

The goal of this project is to advance our understanding of the geometry and dynamics of surfaces and their associated group of symmetries. As a project in pure mathematics the life cycle of impact on society is long and its immediate effects will be felt within the mathematical community and related fields.

The most significant impact of the project will be due to the methods it develops to give a new proof of Mirzakhani's famous curve counting theorem, which will be presented in the form of a book. This book, although containing original research, will be written for non-experts. It will explain all necessary background and techniques in detail, making these tools available to interested researchers in other fields. No such text is currently available; existing surveys of Mirzakhani's work require substantial background in the field. Opening up the field to other researchers and presenting new tools to approach the counting theorems have the potential for long lasting effects. In particular, the book will present new ways to compute certain constants in Mirzakhani's theorems which have important connections to other fields, including mathematical physics.

The project uses methods from several fields of mathematics (geometry, group theory, dynamics) and has applications to still wider fields (number theory, mathematical physics, probability). As such, the PI will seek to work with and build connections with experts across these fields.

In pure mathematics the economic benefits are often not seen until long after the breakthroughs have been made, and often they are hard to predict. However, it is clear that the advancement of important aspects of modern life, from economics to technology, has its base in mathematics. To give just one example, methods from geometry and topology has recently found applications in topological data analysis. This in turn has found applications in a wide range of areas, from image analysis to evolution of viruses. Hence, strengthening UK research in pure mathematics is an important long-term investment.