Finding Order in Apparent Randomness

Simons Junior Fellow Morris Ang develops and proves theorems that help explain the hidden mathematical patterns in random surfaces or objects.

When grains of pollen are seen suspended in water, or when bits of dust float through the air, we are seeing Brownian motion in action. This phenomenon, long observed in the natural world but first formally described by botanist Robert Brown in 1827, is a way to explain how atoms and small molecules — far too small to be seen with the naked eye — impact the movements of tiny particles in seemingly random ways.

As an undergraduate at Stanford University, Morris Ang learned something amazing about Brownian motion: If particles are moving on a surface, stretching and shrinking that surface does not change the behavior of the particle — thus revealing a hidden source of order within an otherwise random process.

This unexpected discovery would inspire Ang’s passion for understanding the underlying mathematics of the physical world.

Ang, whose work focuses on probability theory and mathematical physics, completed his doctorate at MIT. A first-year Junior Fellow with the Simons Society of Fellows, he is now completing a three-year postdoctoral fellowship in Columbia University’s Department of Mathematics, working with Julian Dubedat.

We recently sat down to discuss Ang’s work. Our conversation has been edited for clarity.

One mathematical concept you study is called a conformal transformation. What is that?

In simple terms, a conformal transformation is a special way of changing one shape into another — for instance, a disk into a square. At small scales, conformal transformations preserve shapes and angles, but not lengths or areas. Brownian motion, a model that describes a randomly moving particle, has conformal symmetry; if you run Brownian motion on a disk, and then use a conformal transformation to map it into a square, what you get is Brownian motion in the square. We call that the conformal invariance of Brownian motion.

I understand that your work centers on two other conformally invariant random objects as well.

Yes! The first is a random curve known as Schramm-Loewner evolution, or SLE. SLE describes interfaces between two objects in statistical physics. Take, for example, the Ising model, a well-known model of magnetism. In the Ising model, each site of a 2D lattice has either a plus spin or a minus spin, and each site wants to be similar to its neighbors: If my neighbors all have plus spins, then I probably have a plus spin too. This model also has a temperature parameter. At high temperatures, the system is entirely disordered; it just looks like static, with no clumps of the same spin. At low temperatures, most of the sites have the same spin. At just the right temperature, however, you get a beautiful fractal geometry that looks the same at all scales. SLE describes the interfaces between the plus and minus spins at that critical temperature.

The other conformally invariant random object I focus on is called Liouville quantum gravity, or LQG.

To explain LQG, I’ll start with a combinatorial example. Let’s say I take a thousand triangles made of cardboard, randomly pair up the edges of the triangles with each other, and glue them all together. The result could look something like a topological sphere. What, then, does this sphere’s random geometry look like? Well, when you use a large number of small triangles, you get this very rough, fractal kind of surface — that’s Liouville quantum gravity.

Now, let me explore LQG from a physical perspective. In quantum mechanics, when you consider the trajectory of a particle over time, you calculate the sum of all its possible trajectories. Informally, quantum field theory is the extension of quantum mechanics in which the objects of study are not particles but fields. In quantum field theory, you instead want to take a sum over all possible surfaces. LQG is a random surface which gives a mathematically rigorous way to make sense of such a sum for certain quantum field theories.

At MIT, my research focused on the interplay between random curves (SLE) and random surfaces (LQG). In our work, my collaborators and I extended the LQG/SLE framework pioneered by my doctoral adviser, Scott Sheffield, and others, with the goal of making connections between the works of different mathematical communities. For instance, one group of mathematicians explores the quantum field theory underlying LQG, while another studies geometric properties such as distances on random surfaces. Things that are easy to solve from one perspective might be intractable from another; the strength of the LQG/SLE approach is the ability to synthesize inputs coming from these different approaches to prove results inaccessible with any single approach.

To demonstrate what I mean, let me describe a favorite theorem that I proved at MIT with collaborators Guillaume Remy and Xin Sun. It focuses on what is known as an annulus. This is what mathematicians call the region between two concentric circles. Continuing with our cardboard triangle analogy, if you glue together a large number of triangles in a random way to make an annulus, you get what is known as a Brownian annulus. This is a special type of LQG surface that has a geometric property called modulus which, roughly speaking, describes an annulus’s thickness. There is a conjectural description of the Brownian annulus modulus obtained via a brilliant but mathematically non-rigorous argument in string theory. In our paper, we successfully ascertained the law of the Brownian annulus modulus, giving a firm mathematical grounding to that conjecture. That paper is available on arXiv, a service supported by the Simons Foundation that many mathematicians use to share their work.

What is your focus now at Columbia?

I am still working on the interaction between SLE and LQG with two different projects. First, I am working to understand the gluing of general LQG surfaces. When you glue LQG surfaces together, you get a bigger LQG surface, and the gluing interface turns out to be an SLE curve. This is well understood in specific cases, such as when the LQG surfaces involved are disks or spheres, but not for surfaces in general.

In addition, I am also exploring a more general relationship between LQG and SLE. The current LQG and SLE tools work well when they are matched in some sense. For example, LQG and SLE each have a parameter that describes the roughness of the geometry, and when these parameters are matched in a special way, you get miraculous interactions between them. But what happens when these parameters are mismatched? How does this change the interactions? This is another focus of my work.