At the interface of physics and mathematics

“I got hooked on the interplay of physics and mathematics while I was still at high school,” says 30-year-old Sylvain Lacroix, who was born and grew up near Paris. “It was fascinating to learn abstract mathematical concepts and see them neatly applied in the realm of physics.” During his studies at the École Normale Supérieure de Lyon, he devoted much of his energy and enthusiasm to physics problems that had highly complex underlying mathematical structures. So when it came to selecting a topic for his doctoral thesis, this area of research seemed like the obvious choice. He decided to explore the theory of what are known as integrable models – a subject he has continued to pursue up to the present day.

Lacroix readily acknowledges that most people outside his line of work find the term “integrable models” completely incomprehensible: “I have to admit that it’s probably not the simplest or most accessible field of physics,” he says, almost apologetically. That’s why he takes pains to explain it in layman’s terms: “We define a model as a body of laws, a set of equations that describe the behaviour of certain quantities, for example how the position of an object changes over time.” An integrable model is characterised by equations that can be solved exactly, which is by no means a given.

Symmetry is the key

Many of the equations used in modern physics – such as that practised at CERN, the European laboratory for particle physics in Geneva – are so complex that they can be solved only approximately. These approximation methods often serve their purpose well, for instance if there is only a weak interaction between two particles. However, other cases require exact calculations – and that’s where integrable models come in. But what makes them so exact? “That’s another aspect that is tricky to explain,” Lacroix says, “the key word is symmetry.” Take, for example, the symmetry of time or space: a physics experiment will produce the same results whether you perform it today or – under identical conditions – ten days from now, and whether it takes place in Zurich or New York. Consequently, the equation that describes the experiment must remain invariant even if the time or location changes. This is reflected in the mathematical structure of the equation, which contains the corresponding constraints. “If we have enough symmetries, this results in so many constraints that we can simplify the equation to the point where we get exact results,” says the physicist.

Integrable models and their exact solutions are actually very rare in mathematics. “If I chose a random equation, it would be extremely unlikely to have this property of exact solvability,” Lacroix says. “But equations of this kind really do exist in nature.” Some describe the motion of waves propagating in a channel, for example, while others describe the behaviour of a hydrogen atom. “But it’s important to note that my work doesn’t necessarily have practical applications of that kind,” Lacroix says. “I don’t examine concrete physical models; instead, I study mathematical structures and attempt to find general ways that will allow us to construct new exactly solvable equations.” Although some of these formulas may eventually find a real-world application, others probably won’t.

After completing his doctoral thesis, Lacroix spent three years working as a postdoc at the University of Hamburg, before finally moving to Zurich in September 2021. “I don’t have a family, so I had no problem making the switch,” he says. He is relieved that he can now spend five years at the ITS as an Advanced Fellow and focus entirely on his research without already having to worry about the future. He admits it was a pleasure getting to know different countries as a postdoc and that he enjoyed moving from place to place. “But it makes it very hard to have any kind of stability in your life.”

A beautiful setting

Lacroix spends most of his time working in his office at the ITS, which is located in a stately building dating from 1882 not far from the ETH Main Building. “It’s a lovely place,” he says, glancing out the window at the green surroundings and the city beyond. “I enjoy living in Zurich. I really like the atmosphere here so far.” In his spare time, he likes watching movies, reading books and socialising. “I love meeting up with friends going in restaurants or cafés,” he says. He also feels fortunate that he didn’t start working in Zurich until after the Covid measures had been relaxed.

“I’m vaccinated and everyone’s very careful at ETH. We still have restrictions in place, but life is slowly getting back to normal – and that made it much easier to get to know my colleagues from day one,” he says. One of the greatest privileges of working at the ITS, Lacroix says, is that it offers an international environment that brings together researchers from all over the world. As well as offering a space for experts to exchange ideas and holding seminars where Fellows can present their work, the Institute also has a tradition of organising joint excursions. In the autumn of 2021, Lacroix joined his colleagues on a hike at Flumserberg for the first time: “I love hiking and it’s wonderful to have the mountains so close.”

Normally, however, he can be found sitting at his desk jotting down a series of mostly abstract equations on a sheet of paper. Occasionally his computer comes in handy, he says, because it has become so much more than just a calculating device; today’s computers can also handle abstract mathematical concepts, which can be very useful. Most people don’t really understand much of what Lacroix puts down on paper, but that doesn’t bother him: “I’ve learned to live with that,” he says; “I don’t feel isolated in my research at all – at least not in the academic sphere.”

Achieving a better understanding of quantum field theory

Integrable models are extremely symmetrical models, Lacroix explains. The basic principle of symmetry plays an important role in modern physics, for example in quantum field theory – the theoretical basis of particle physics – as well as in string theory, which scientists hope will eventually provide a unified description of particle physics and gravity. So could such an all-encompassing unified field theory turn out to be an integrable model? “That would obviously be great, especially for me!” Lacroix says with a wry smile. “But it’s a bit optimistic to believe that whatever unified theory of physics finally emerges will have enough symmetries to make it completely exact.”

Even if the equations he studies don’t explain the world directly, he still believes they can help us achieve a better understanding of theoretical physics. For example, we can take advantage of so-called “toy models”, which have a particularly large number of symmetries, to simplify extremely complex equations in quantum field theory. “This gives us a better understanding of how the theory works, even if these models are too simplistic for the real world,” Lacroix says. Yet his primary interest lies in the purely mathematical questions that integrable models pose, and he admits that the equations they involve sometimes even appear in his dreams: “It’s hard to shake off what I’ve been thinking about the entire day. But I’ve never managed to solve a mathematical problem in my dreams – at least not so far!”