## From The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH): “Solving problems with intuition”

3.14.22

Barbara Vonarburg

Sarah Zerbes is the first ETH professor of theoretical mathematics. She finds her best ideas through intuition. Photo: Alessandro Della Bella/ETH Zürich.

In her research as a number theorist, Sarah Zerbes focuses on one of the oldest – but also most topical – branches of mathematics. Her work is closely tied to one of the great open mathematical problems, the solution for which carries a prize of one million dollars.

There are two kinds of researchers in mathematics, Zerbes says: “One kind builds theories and sees the big picture.” The other focuses on a particular problem that needs to be solved. “I’m a problem solver,” says the German-born 43-year-old, who last autumn was appointed Professor of Mathematics at ETH Zurich. The problems she deals with relate to one of the most famous and mysterious conjectures in mathematics. It was proposed by two British mathematicians, Bryan Birch and Peter Swinnerton-Dyer, in 1965, after they had spent many nights conducting numerical experiments on what at the time was the sole computer at the University of Cambridge. “These days,” Zerbes says, “anyone could do those calculations on a laptop.”

Birch and Swinnerton-Dyer (BSD for short) were unable to prove their conjecture. In 2000, a foundation listed it as one of seven major mathematical problems whose solution would each be rewarded with one million dollars. “It has to do with a class of equations that are very important in mathematics, as well as for some cryptography applications,” Zerbes says: “They’re called elliptic curves.” The challenge is to find certain solutions for these curves. “The BSD conjecture states that the solutions to these equations are determined by an object that stems, surprisingly, from a completely different area of mathematics,” Zerbes says, “namely functions.” This object is known as a complex analytic L-function.

**Huge network of new conjectures**

The BSD conjecture is one of the most important open problems in the field of number theory, but it has also opened up a new area of research. There is now an extensive network of other conjectures generalising the BSD conjecture. “Together with my husband, I have proved several new sub-problems in this network,” Zerbes says. She has been collaborating with her husband, David Loeffler, for many years. He is currently a visiting professor at ETH Zürich, alongside his full professorship at the University of Warwick, UK, and works in the same office in the ETH Zürich Main Building as his wife. “Sharing an office isn’t always easy, as it’s very difficult to separate our personal lives from our work. We have the occasional heated discussion,” Zerbes admits, “but we complement each other very well.”

Unlike herself, her husband is a theory builder who is interested in the big picture. “He has an enormous library in his head and he can understand and categorise things directly.” She’s less adept at this, she says: “My strength is intuition.” Her best ideas come to her when she simply sits and drinks coffee. “I concentrate, contemplate and wait for inspiration,” she says: “I don’t even need a sheet of paper for it.” Only later does she write down her idea in her notebook or on the board in her office, accompanied by much discussion, erasing and rewriting. “First, you always have to see the overarching structure. Only then can you start working out the details, which often takes years,” she says. That’s also what Zerbes and Loeffler experienced in their work in connection with the BSD conjecture.

**Breakthrough after eight years**

“We’ve spent the past eight years developing new examples of Euler systems,” Zerbes says. Named after the Swiss mathematician Leonhard Euler, these systems are very complicated mathematical structures that can be used to prove new cases of this conjecture. Once the fundamental idea was born, the couple was able to finish the first part of their programme within a few years. “But then we were stuck,” Zerbes says. For years, they made no progress, until they flew to a conference in Princeton, US. “There, a mathematician from Lyon gave a lecture in which he presented a tool that he had developed for something else entirely,” she says, “but it was exactly what we were missing.” Although the two mathematicians realized within minutes that they would now succeed, it still took another four years with a lot of work on details. “We achieved the breakthrough last year,” says Zerbes, before summing up by saying, “We were very lucky.”

But the million-dollar prize is still out of reach. While it can be shown that the BSD conjecture does indeed hold under certain conditions, there are some cases that no one currently knows how to solve. “We don’t know either,” Zerbes says. “Also, what we’ve proven isn’t parts of the original conjecture, but parts of a generalization; there are other parts that would require a completely new idea.” So the prize isn’t what motivates their research. “It’s the problem itself that’s so fascinating,” Zerbes says: “how deep it is, how complicated the arguments are that might lead to progress, and how lucky one has to be to make any progress.”

As a number theorist, she also feels connected with generations of mathematicians. “The ancient Greeks of 2,000 years ago were already studying some of the problems that my colleagues and I are working on now,” Zerbes says. Number theory is one of the oldest branches of mathematics. It mostly deals with such equations as the famous Pythagorean theorem: x2+y2=z2. It asks whether integer or rational number solutions can be found for these equations. In the case of Pythagoras, it is known that there are infinitely many rational numbers that solve the equation and that they describe right triangles having sides of length x, y and z. More complicated equations have been keeping mathematicians busy for centuries, and have led to the development of other topics, such as the BSD conjecture.

**Learning Latin as a living language**

In school, Zerbes wasn’t initially interested in mathematics; she preferred Latin. “This language is incredibly analytical and logical,” she says. This is something that still fascinates her today. “I’m now learning Latin as a modern, spoken language,” she says. It bothered her that they only ever translated word for word in school, and that even after six years of lessons she was still incapable of reading a text fluently. Now she has found an instructor who teaches Latin as a living language. “The lessons are conducted exclusively in Latin, and we have discussions and read the ancient texts, which is really interesting,” she says. Only now does she notice how sarcastic, but also funny, Cicero’s writing was.

As a schoolgirl, she didn’t have any interest in mathematics until, at age 14, she had an outstanding teacher for half a year. “Before that, I didn’t understand maths at all because everything was always packaged in word problems,” Zerbes says. The new teacher was excellent at explaining mathematical concepts. “He was clear, abstract and precise,” she recalls. Now quite interested, when that teacher was replaced again, she took it upon herself to get some mathematics books from the library. After completing her school-leaving exams, she applied to study at the world-famous Cambridge University in England and was accepted. She also obtained her doctorate there. When she was later appointed professor at University College London, she invited the teacher from her time at school to attend her introductory lecture. “He actually came, which made me extremely happy,” Zerbes says. “After all, it was his teaching that made all the difference, because that’s when I really started to enjoy maths.”

Zerbes has since received multiple awards and is one of the world’s leading experts in number theory. She herself has never had any trouble asserting herself as a woman in a male-dominated environment, but she knows some women in the field who have been bullied because of their sex. “I generally haven’t had any bad experiences,” she says, adding, “I’ve had to develop a thick skin on account of suffering from loss of hair for 35 years, which probably hasn’t hurt, either.” Or maybe, she says, she has just been lucky.

**Mountaineering and ice climbing**

Moving from England to Switzerland was easy for Zerbes. “ETH Zürich is one of the best universities in the world,” she says proudly. “The working conditions and the students are outstanding.” In addition, some of her family lives in southern Germany, and she and her husband are keen mountaineers. “I’m particularly fond of ice climbing,” Zerbes says, “which I recently did in Scuol, in Lower Engadine.” The couple spends most weekends in the mountains, skiing in winter, “to gain another perspective out in nature,” she says, “because otherwise you do dig yourself into quite a deep hole of mathematical problems.” She works out nearly every day, especially swimming and climbing a lot. “Exercise is important to me, as a counterbalance to research,” she says.

She also finds reading relaxing. Her website features a long list of books she has enjoyed, including such works as Thomas Mann’s *Buddenbrooks* and Kazuo Ishiguro’s *The Remains of the Day*. “There are few good books about mathematics,” Zerbes says. There is only one she recommends: *Regarding Roderer* by Guillermo Martinez, an Argentine mathematician and novelist. Zerbes isn’t bothered by the fact that mathematics is hardly accessible to the general public. She is also happy to overcome the many difficulties that come with the field. She mentions the very first lecture she attended at Cambridge, in which a professor said that mathematics research is bitter and frustrating most of the time. You’re always struggling against the same problems, which can be very draining emotionally. But then, when something works, the feeling is indescribable. “I think of that often,” she says, “because that’s really how it is.”

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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 Excellence Ranking* 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.

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