From The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH): “At the interface of physics and mathematics”

From The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH)

Barbara Vonarburg

Sylvain Lacroix is a theoretical physicist who conducts research into fundamental concepts of physics – an exciting but intellectually challenging field of science. As an Advanced Fellow at ETH Zürich’s Institute for Theoretical Studies (ITS), he works on complex equations that can be solved exactly only thanks to their large number of symmetries.

“It was fascinating to learn abstract mathematical concepts and see them neatly applied in the realm of physics,” says Sylvain Lacroix, Advanced Fellow at the Institute for Theoretical Studies. Photo: Nicola Pitaro/ETH Zürich.

“I got hooked on the interplay of physics and mathematics while I was still at secondary 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 University of Lyon [Université Claude Bernard Lyon 1] (FR), 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 The European Organization for Nuclear Research [Organización Europea para la Investigación Nuclear][Organisation européenne pour la recherche nucléaire] [Europäische Organisation für Kernforschung](CH) [CERN] – 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, “but it ultimately comes down to 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 Zürich 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 movement 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 have any practical applications of that kind,” Lacroix says. “I don’t examine concrete physical models; instead, I study mathematical structures and attempt to find general approaches 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 [Universität Hamburg](DE), before finally moving to Zürich 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 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 feel very much at home here. Living in Zürich is wonderful, it’s such a great feeling being here.” In his spare time, he likes watching movies, reading books and socialising. “I love meeting up with friends in restaurants or cafés,” he says. He also feels fortunate that he didn’t start working in Zürich 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 in the Flumserberg mountain resort for the first time: “I love hiking and it’s incredible 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.”

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!”

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ETH Zurich campus

The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH) is a public research university in the city of Zürich, Switzerland. Founded by the Swiss Federal Government in 1854 with the stated mission to educate engineers and scientists, the school focuses exclusively on science, technology, engineering and mathematics. Like its sister institution The Swiss Federal Institute of Technology in Lausanne [EPFL-École Polytechnique Fédérale de Lausanne](CH) , it is part of The Swiss Federal Institutes of Technology Domain (ETH Domain)) , part of the The Swiss Federal Department of Economic Affairs, Education and Research [EAER][Eidgenössisches Departement für Wirtschaft, Bildung und Forschung] [Département fédéral de l’économie, de la formation et de la recherche] (CH).

The university is an attractive destination for international students thanks to low tuition fees of 809 CHF per semester, PhD and graduate salaries that are amongst the world’s highest, and a world-class reputation in academia and industry. There are currently 22,200 students from over 120 countries, of which 4,180 are pursuing doctoral degrees. In the 2021 edition of the QS World University Rankings ETH Zürich is ranked 6th in the world and 8th by the Times Higher Education World Rankings 2020. In the 2020 QS World University Rankings by subject it is ranked 4th in the world for engineering and technology (2nd in Europe) and 1st for earth & marine science.

As of November 2019, 21 Nobel laureates, 2 Fields Medalists, 2 Pritzker Prize winners, and 1 Turing Award winner have been affiliated with the Institute, including Albert Einstein. Other notable alumni include John von Neumann and Santiago Calatrava. It is a founding member of the IDEA League and the International Alliance of Research Universities (IARU) and a member of the CESAER network.

ETH Zürich was founded on 7 February 1854 by the Swiss Confederation and began giving its first lectures on 16 October 1855 as a polytechnic institute (eidgenössische polytechnische Schule) at various sites throughout the city of Zurich. It was initially composed of six faculties: architecture, civil engineering, mechanical engineering, chemistry, forestry, and an integrated department for the fields of mathematics, natural sciences, literature, and social and political sciences.

It is locally still known as Polytechnikum, or simply as Poly, derived from the original name eidgenössische polytechnische Schule, which translates to “federal polytechnic school”.

ETH Zürich is a federal institute (i.e., under direct administration by the Swiss government), whereas The University of Zürich [Universität Zürich ] (CH) is a cantonal institution. The decision for a new federal university was heavily disputed at the time; the liberals pressed for a “federal university”, while the conservative forces wanted all universities to remain under cantonal control, worried that the liberals would gain more political power than they already had. In the beginning, both universities were co-located in the buildings of the University of Zürich.

From 1905 to 1908, under the presidency of Jérôme Franel, the course program of ETH Zürich was restructured to that of a real university and ETH Zürich was granted the right to award doctorates. In 1909 the first doctorates were awarded. In 1911, it was given its current name, Eidgenössische Technische Hochschule. In 1924, another reorganization structured the university in 12 departments. However, it now has 16 departments.

ETH Zürich, EPFL (Swiss Federal Institute of Technology in Lausanne) [École polytechnique fédérale de Lausanne](CH), and four associated research institutes form The Domain of the Swiss Federal Institutes of Technology (ETH Domain) [ETH-Bereich; Domaine des Écoles polytechniques fédérales] (CH) with the aim of collaborating on scientific projects.

Reputation and ranking

ETH Zürich is ranked among the top universities in the world. Typically, popular rankings place the institution as the best university in continental Europe and ETH Zürich is consistently ranked among the top 1-5 universities in Europe, and among the top 3-10 best universities of the world.

Historically, ETH Zürich has achieved its reputation particularly in the fields of chemistry, mathematics and physics. There are 32 Nobel laureates who are associated with ETH Zürich, the most recent of whom is Richard F. Heck, awarded the Nobel Prize in chemistry in 2010. Albert Einstein is perhaps its most famous alumnus.

In 2018, the QS World University Rankings placed ETH Zürich at 7th overall in the world. In 2015, ETH Zürich was ranked 5th in the world in Engineering, Science and Technology, just behind the Massachusetts Institute of Technology(US), Stanford University(US) and University of Cambridge(UK). In 2015, ETH Zürich also ranked 6th in the world in Natural Sciences, and in 2016 ranked 1st in the world for Earth & Marine Sciences for the second consecutive year.

In 2016, Times Higher Education World University Rankings ranked ETH Zürich 9th overall in the world and 8th in the world in the field of Engineering & Technology, just behind the Massachusetts Institute of Technology(US), Stanford University(US), California Institute of Technology(US), Princeton University(US), University of Cambridge(UK), Imperial College London(UK) and University of Oxford(UK) .

In a comparison of Swiss universities by swissUP Ranking and in rankings published by CHE comparing the universities of German-speaking countries, ETH Zürich traditionally is ranked first in natural sciences, computer science and engineering sciences.

In the survey CHE ExcellenceRanking on the quality of Western European graduate school programs in the fields of biology, chemistry, physics and mathematics, ETH Zürich was assessed as one of the three institutions to have excellent programs in all the considered fields, the other two being Imperial College London(UK) and The University of Cambridge(UK), respectively.