Professor Jason Miller of the Department of Pure Mathematics and Mathematical Statistics (DPMMS) has been elected Fellow of the Royal Society. He joins over 90 outstanding researchers from across the world who have been elected to the UK's national academy this year and treads in the footsteps of Stephen Hawking and Isaac Newton. The honour recognises Miller's work on probability, in particular the study of random geometry.
Escaping the straight line
The Austrian artist and architect Friedensreich Hundertwasser once wrote about the "tyranny of the straight line". It's a shape that doesn't exist in nature, which we humans draw with a ruler and impose on the world around us. Indeed, the geometry we learn at school, the geometry of the ancient mathematician Euclid, is all about regular shapes: polygons and circles, planes and spheres. That's not surprising since these shapes are easy to describe.
What Miller's work has helped to show, however, is that mathematics is far from powerless in the face of shapes that are much wilder. One source of geometric unruliness is randomness. A dust particle floating around a room is unlikely to follow a straight line path. Random collisions with molecules and other particles cause it to veer off course so its path becomes jagged and complex. Such behaviour is known as Brownian motion after the botanist Robert Brown, who first observed it in pollen grains floating in water in 1827.
Brownian motion as modelled by the Wiener process. Image: Shiyu Ji, CC BY-SA 4.0
While entirely unpredictable, it's still possible to get a mathematical grip on Brownian motion. A so-called random walk is a sequence of discrete steps each taken in a random direction in space. Each individual step can be drawn as a straight line segment, but many steps viewed together will form an irregular zig-zaggy path.
There are different ways of mathematically defining the details of such a random walk, but many of them have something in common: if you let the number of steps tend to infinity while letting their size tend to zero (so that the path still fits into a finite region of space — this is known as taking a scaling limit) then you end up with something called the Wiener process, named after the American mathematician Norbert Wiener. This process, no longer discrete in time and space but continuous, provides a good mathematical model for Brownian motion. The model is useful, not only in understanding dust and pollen, but also in modelling stock prices and in medical imaging techniques such as MRI scans.
Random surfaces
Some of Jason Miller's most celebrated contributions to mathematics involve, not random paths, but random surfaces. One type of random surface is the so-called Brownian map. It can be viewed as a two-dimensional analogue of Brownian motion in that it arises as a scaling limit of discrete random structures made from straight-line pieces. An example are random triangulations which cover a surface such as the sphere in a network of triangles.
The Brownian map is hard to draw, but the image below illustrates a triangulation with many tirangles.
This picture represents a uniform triangulation of the sphere with over 30,000 vertices. Image by Thomas Budzinski, used by permission.
Random surfaces are interesting in their own right, from a purely mathematical point of view. But there's also another motivation for studying them, coming from a surprising direction: they arise in physicists' quest to find a theory of everything, which describes all the physics of the Universe in one unifying mathematical framework.
One contender for such a framework is string theory, which holds that the fundamental particles of nature don't behave like tiny little balls, but like tiny little vibrating strings. While the random path of a tiny ball might look like Brownian motion, the random wriggling of a tiny string would sweep out a random surface. In the 1980s the physicist Alexander Polyakov developed a way of describing random surfaces inspired by string theory, and by conformal field theories, known as Liouville quantum gravity (LQG).
Both the Brownian map and LQG were studied extensively by mathematicians and physicists. Over time people came to believe that the Brownian map and a particular surface known as the √(8/3)-LQG sphere should be equivalent in some sense. Actually proving that they were, however, was a different matter. The two objects came equipped with very different mathematical structures and it was far from obvious how to endow one object with the structure of the other. Without doing so, it was hard to even find the mathematical words in which to talk about the presumed equivalence.
Natural ideas
Miller, together with Scott Sheffield of the Massachusetts Institute of Technology, broke the deadlock with a series of three papers, published in 2020 and 2021, which showed the equivalence of the two objects and also provided a robust unification of the corresponding theories. Central to their work was the so-called Schramm-Loewner evolution (SLE). Similar to the Wiener process, this arises as scaling limits of random paths in the plane. Using the way in which SLE paths explore a random surface, Miller and Sheffield came up with a distance measure that works both on the Brownian map and on the √(8/3)- LQG sphere. This provided the key to seeing the two objects in the same mathematical light. You can find out more in this beautiful article by Kevin Hartnett in Quanta.
"Scott and Jason have been able to implement natural ideas and not be rolled over by technical details," Wendelin Werner, one of Miller's colleagues here at DPMMS and winner of a Fields Medal in 2006 for work on the SLE, told Quanta. "They have been basically able to push for results that looked out of reach using other approaches." Apart from being elected FRS, Miller has won a range of other awards, including a Whitehead Prize in 2016, a Clay Research Award in 2017, and the Eisenbud and Fermat Prizes in 2023. He was also an invited speaker at the 2018 International Congress of Mathematicians.
As far as Friedensreich Hundertwasser is concerned, we do not know what he thought of wild mathematical shapes like those studied by Miller, or if he was even aware of them. Perhaps he would have revised his opinion of the straight line as "tyrannical" and even "immoral". With mathematical imagination straight-line shapes can turn into structures much more wonderful than first impressions might suggest. That these can also unveil the secrets of the Universe is a marvellous bonus.