In the physical world, objects tend to have an expected shape and form. But if you move into the abstract realm of the truly random, things can start to seem a little strange.
Prof Wendelin Werner, who will deliver the Royal Irish Academy Hamilton Lecture on October 16th at the Royal Irish Academy in Dublin, is no stranger to seeing the larger picture and beauty in randomness among possibilities.
His talk on “How surprisingly intricate are random structures?” is on unusual and unexpected properties thrown up in the abstract world of randomness and uncountable options.
“Let’s say you choose something at random in a natural way among infinitely and uncountably many possibilities,” he explains. “Then in many circumstances, the object you typically get is an object that has properties that none of what the known objects would actually have.”
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Mountains and coastlines
Werner, who is Rouse Ball Professor of Mathematics at the University of Cambridge, knows the concept needs concrete examples to be more relatable, and landscapes offer a good canvas for explanation.
We tend to have an expected image what mountains look like, with a range of peaks, valleys and jagged outcrops of rock. But if we were to be able to choose one truly at random in some natural mathematical abstract way, things would be much wilder, he notes.
“As a topographic landscape it would be a complete mess, with infinite peaks and valleys all over the place,” says Werner. “In this very large realm of possibilities, a typical object would not always be a standard one you would tend to look at and know as familiar.”
These rough objects are not unrelated to rough coastlines, he adds: “If you look at the Irish coastline on a map, it is a very irregular curve that doesn’t intersect itself. A horizontal slice through the mountain range we just talked about will in fact consist of coastline-like curves.”
Werner likes to communicate about these kinds of concepts as a way to stimulate the imagination. “I hope that in some way by lecturing about these mathematical surprises I can encourage people to start thinking about new ideas, that it opens the windows of the mind and lets in some new air,” he says.
Hamiltonian creativity
The annual RIA Hamilton Lecture celebrates the 19th-century Irish mathematician Sir William Rowan Hamilton, who was struck by inspiration while walking beside the Royal Canal in Dublin 180 years ago on October 16th, 1843.
Hamilton captured his insight by using a penknife to etch a new form of equation into a stone at Broome Bridge in Cabra. This cerebral graffiti expressed a new structure in mathematics, a “quaternion”, that represents spatial orientations and rotations of elements in three-dimensional space, and would seal Hamilton’s reputation as the “liberator of algebra”.
While Werner does not use quaternions in his exploration of mathematics, he appreciates the creativity and beauty of Hamilton’s vision.
“The way I see his discovery of quaternions is like finding a wonderful perfect gem, like a diamond,” he says.
And while quaternions have had some applications in modern life — including landing objects on the moon and creating realistic effects in video games and movies — Werner believes they may offer more over time.
“Quaternions are not a central thing that many people are working with in mathematics today, but they are like a wonderful item in the cupboard of curiosities that you can learn from and be inspired by,” he says. “And it is likely that we haven’t yet discovered all what their uses could be — there may be more instances in physics, or in the world generally, where this diamond will shine again in the future.”
Where Werner leans on Hamilton more frequently in his own work is in Hamilton’s interpretation of the equations of motion.
“His rewriting of Newtonian mechanics, his phrasing of a more energetic approach, is relevant for my work in understanding dynamics and how particles interact randomly,” he says.
Be drawn by your interests
Hamilton’s broad interests and open mind shine through to Werner as a perspective to be admired. “He was interested in many things, including astronomy, and he had an original mind, he asked his own questions and explored the directions that interested him.”
This enjoyment of learning is key to finding your direction as a mathematician, he says, and he would advise students with an interest in maths to move towards the areas that draw them in naturally. “If you like a subject and it resonates with you, that is a potential signal that you may have something to contribute to this area.”
Werner, who was born in Germany but spent most of his youth in France, recalls that few people in his family were surprised when he decided to become a mathematician.
“It’s a very personal decision that you make,” he says. “Then if you enjoy it, you can figure out what interests you over time, take it step by step and follow what interests you. If you keep learning about a topic and then get to the point where your professor says they don’t know what comes next, that is very exciting, that is where you think about whether you would like to try to start your own research.”
Random vibrations in space
Werner’s own research on random processes seeks to understand how particles interact at distance and how random motions affect those interactions. In 2006, he was awarded a prestigious Fields Medal “for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion and conformal theory.”
What does that mean in practice? He offers an example:
“To come back to our coastlines, the stochastic Loewner evolutions are the natural mathematical random curves that show up there,” he says. “In a slightly different set-up, suppose there are two points in the plane or in space. We tend to say that we know the distance between the two points. But maybe there are some sort of vibrations of space, that randomly perturbs the distance between these two points. What is the natural way in which this happens? Just as the random mountains, some fracture lines and discontinuities do then show up.”
These kinds of questions have long been studied in physics, he adds. “These random distances link in with what physicists have been calling quantum gravity back in the 1980s.”
Prof Wendelin Werner will deliver the RIA Hamilton Lecture 2023 on Monday, October 16th, from 18.30 at the Royal Irish Academy, 19 Dawson Street, Dublin 2. The talk is free to attend but places must be reserved. See ria.ie for details.
The 2023 RIA Hamilton Lecture is sponsored by Ibec.
Time to act on climate is yesterday
Maths can help us to understand global issues, such as the climate emergency, but we need more than just maths to solve them. That’s according to the mathematician behind this year’s Royal Irish Academy Hamilton Lecture.
“Of course, maths has a role to play in understanding what is going on with climate, but it is not going to be the solution,” says Prof Wendelin Werner. “As far as I can tell, the main problem now is that people need to listen and really understand they have to do something substantial, and the time to do that is not tomorrow, the time to do something was actually yesterday.”
The evidence around climate change is not pleasant to hear, notes Werner, but countries must not deprioritise the problem and just assume that technology will come to save us.
“Of course, scientists and mathematicians should do whatever they can to help. But it cannot be a government policy to push this priority down the road and rely on scientists or mathematicians to come up with a wonderful miracle theorem and that will solve all our problems. The energy pumped into our heated atmosphere is monstrous, and it is very unlikely that one can come up with a miraculous way to dissolve it, in the same way as vaccines can just eradicate diseases. Society as a whole needs to act now.”
