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: x
. 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.”