In order to understand how I, a Master student in mathematics, ended up working with real trees (not just the mathematical objects, we have to go back in time to my very first lecture at ETH Zurich. The subject was “Mathematics of Data Science”, and I attended it by coincidence, just because I was told it should be amazing – and it was! But not only was the lecture great, but I also discovered that this topic had exactly the flavour of mathematics I loved.

However, much to the dismay of the computer science students, the lecture stuck strictly to its title, meaning we only learned about the MATHEMATICS behind Data Science and never applied the techniques to real data. When I saw an ad for a short data science internship at the SLF (Institute for Snow and Avalanche Research), I got immediately excited. I wanted to use my newly acquired knowledge on real-world data, and, to my even greater excitement, the data would be in the field of environmental science, a field that I’ve always been interested in. So, just a few mails, a short video call and six months later I was sitting in a train that climbed its way up to the mountain town of Davos for the start of my internship.

The project I would spend the next five weeks working on, was about detecting slow movements of mountain slopes by using tree ring data. You might wonder why we are not using a more straightforward approach, like remote sensing techniques, and instead work with trees? These are wonderful questions!

 The first thing one has to know about slope movements is that they are processes that happen over hundreds to thousands of years, whereas remote sensing data is often only available for the last decade.

Therefore, the available data covers a too short period of time to understand these very long-term processes of slope movements well. And that is exactly where trees come in! Trees are widely available on slopes in Europe and can be centuries old. So, if we could use trees as sensors for slope movements, this would extend our data from just a decade to potentially centuries.

And how can we use trees as sensors for slope movements?

The idea behind this approach is that naturally trees always try to grow straight upwards. However, if the slope beneath the trees moves, this tilts the trees and to stand straight again, the trees grow asymmetrically in the following years. This asymmetry can be measured by comparing the differences in tree ring width from two different directions of the trees (don’t worry, there were no trees cut down to collect the data, only two small cores were collected per tree. Sounds like a crazy idea. It is! But it was successfully used by other research groups to detect rapid slope movements. Which leads us to the question: Can we do the same for slow slope movements?

During my five-week journey to answer this question, I tried to apply many of the techniques learned during my course of Data Science. I learned more than I ever thought I would about bark beetles and polar expeditions from chats during lunch. Unfortunately, regardless of what I tried, I could not find a clear relationship between the tree ring data and slow slope movements. However, unexpectedly, I discovered a more rapid event in the data that turned out to coincide perfectly with a documented mudflow in the corresponding area. So, even though I could not answer the initial research question, we could prove again that the method works in the setting of more rapid events.

In addition to my project itself, I had a wonderful summer time in Davos with many after-work hikes and bike tours, I got to know so many nice people from different fields and talking with them really broadened my horizon of what research apart from mathematics can be like. Great thanks go to my supervisors Dr. Alessandra Bottero and Dr. Andrea Manconi who gave me the opportunity to be part of their team and ETH Studio for enabling me this enriching experience! 

And if you got curious, then give it a try and apply as well😊!

ETH Studios

ETH Studios are global innovation outposts where ETH students tackle real-world challenges alongside companies, researchers, and startups. Whether in New York, Heilbronn, Davos, or at the experimental StudioX, students gain entrepreneurial and interdisciplinary experience while contributing to sustainable and tech-driven solutions.

Contact Francesca Caggiula at eth-studios@ethz.ch if you are interested in more information on and about the ETH Studios.

Duminil-​Copin is the guest speaker at the ETH Global Lecture Series From Coffee to Mathematics, and he’s determined to make sure all of us – from ETH students to the general public in attendance – understand what he’s talking about.

Duminil-​Copin is introduced by Catherine Robert, from the French Embassy in Bern. Professor of mathematics at the Institut des Hautes Etudes Scientifiques in Bures-​Sur-Yvette, he is the winner of the Fields Medal, the top prize for mathematics that is handed out at the International Congress of the International Mathematical Union every four years. We are reminded of this fact later in his presentation when he uses images of his Fields Medal to illustrate flipping a coin.

The professor is here to tell us about his work on phase transitioning. His method for catching and keeping our attention lies in using simple comparisons – like slowly turning a game of Hex into a mathematical Ising model of magnetism – served with large helpings of humour.

Duminil-​Copin tells us that the job of a mathematician in layman’s terms is finding links that don’t seem connected. Such as: How is the game of Hex connected to coffee, and how are magnets connected to melting polar ice caps? He begins with a game called Hex, which is very popular with mathematicians.

If you are also not familiar with Hex, here are the basics: it is a two-player strategic game, played on a board in the shape of a rhombus made up of 11×11 hexagonal spaces, for example. Each player gets assigned a colour and opposite sides of the board – one plays horizontally, the other vertically. They alternate trying to connect spaces by placing their colour into them until one player creates a chain of adjacent spaces, successfully connecting their sides together. Because the board is arranged symmetrically, there cannot be a draw, there can only be one winner.

Duminil-​Copin explains that the more symmetry an object has, the easier it is to manipulate. The symmetry of conformal maps, where the right angles of latitude and longitude remain in place even when the map is expanded, translated or rotated are is found on globes or Mercator maps. If you can change the Hex board into a conformal map, then the probability of winning is the same.

This image of one side winning is an illustration of the theory of percolation, in which connections create a network of clusters for a type of phase transition that is geometric. “Conformal symmetry allows you to understand a lot of things,” says Duminil-​Copin, which is why mathematicians like symmetry.

So the Hex game is an example of percolation. Especially older generations, Duminil-​Copin says, will be familiar with the coffee percolator, which solves the first puzzle linking Hex to coffee – with the hot water pushing up through the coffee to the top.

Our second puzzle is the connection between magnets and melting polar ice caps. Duminil-​Copin takes us to the Arctic, where percolation, or the porosity of material, is having an effect on the planet. When the sun melts the ice on the surface, the resulting water percolates down, making small holes in the ice. The water in those holes then refreezes and expands, taking up more space as ice. This creates more holes, continuing the process until the ice breaks up.

How does this connect to the magnets? During his studies, Duminil-​Copin explains, he looked at how magnets react to heat. And created a model proving how at a certain temperature – the Curie point, named after Pierre Curie – the steel loses its magnetism. This type of phase transition – a brutal change of behaviour in a system as parameters vary – is also reflected in the melding ice caps.

Duminil-​Copin tries to understand this type of phase transition through mathematical modelling like the Ising model of magnetism, which looks a lot like the Hex game. He explains that by studying this model, you can do a lot of things, like using the percolation theory to prove that the phase transition of a 3D Ising model is continuous.

As he shows the stages from game to model, I suddenly found myself understanding things that would have given me a headache just 45 minutes before.

The importance of knowing how mathematicians like Duminil-​Copin work is made clear: “We live in a world where we want to understand things immediately. And this is very far from what we do in math and what sciences do in general. And (…) one of the reasons (…) this discussion between scientists and the public (was) difficult during the pandemic is that the people actually discovered research. They were only (faced with) science before, the result of the process. The process of research (results in) understanding something difficult through mistakes and (that’s) something that takes time.”

After Duminil-​Copin’s lecture, the questions from the audience come mainly from ETH students, so things get complicated really quickly. But one stands out:

“How do we know that percolation is a good model for porosity of material?”

“First, I’m going to give you the non-politically correct answer, which is that I don’t care so much,” Duminil-​Copin jokes. “It’s (just) a beautiful mathematical model, so that is my main motivation,” he adds. And laughs.

Watch the full recording:

Find out more about our ETH Global Lecture Series here.