Solving problems with intuition

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

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 Zurich, alongside his full professorship at the University of Warwick, UK, and works in the same office in the ETH Zurich 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.

«First, you always have to see the overarching structure. Only then can you start working out the details, which often takes years.»      Sarah Zerbes

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 realised 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 generalisation; 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 Zurich 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.”