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Order Amid Chaos: Who Are The Two Mathematicians Sharing This Year’s Abel Prize?

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Two mathematicians who showed how an underappreciated branch of the field could be employed to solve important problems share this year’s Abel Prize, the mathematics equivalent of a Nobel. The Abel Prize for 2020 goes to Hillel Furstenberg, 84, of the Hebrew University of Jerusalem, and Gregory Margulis, 74, of Yale University. Both are retired professors.

The citation for the prize, awarded by the Norwegian Academy of Science and Letters, lauds the two mathematicians “for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.” Dr. Furstenberg and Dr. Margulis will split the award money of 7.5 million Norwegian kroner, or more than $700,000.

There is no Nobel Prize in mathematics, and for decades, the most prestigious awards in math were the Fields Medals, awarded in small batches every four years to the most accomplished mathematicians who are 40 or younger. The Abel Prize, granted annually for research in mathematics, is named in commemoration of the brilliant 19th-century Norwegian mathematician Niels Henrik Abel. A pioneer in the development of several branches of modern mathematics, Abel studied the mathematical works of the 17th-century Englishman Sir Isaac Newton, the 18th-century German Leonhard Euler and his contemporaries the Frenchman Joseph-Louis Lagrange and the German Carl Friedrich Gauss in preparation for his own research.

The Niels Henrik Abel Memorial fund was established on the 1st of January, 2002, and it is administered by the Norwegian Ministry of Education and Research. The main purpose of the fund is to award an international prize for “outstanding scientific work in the field of mathematics.” The prize is also intended to help raise the status of mathematics in society and to stimulate the interest of young people in mathematics. Responsibility for the Abel Prize and for other uses of the funds lies with the Norwegian Academy of Science and Letters. The fund also supports one or two Abel Symposia per year on various branches of mathematics, and in 2005 the fund created the Bernt Michael Holmboe Memorial Prize for the promotion of excellence in teaching mathematics, in honour of Abel’s own mathematics teacher. The prize, which is worth about $1 million, was first awarded in 2003 to the French mathematician Jean-Pierre Serre.

Hillel Furstenberg was born in Berlin in 1935. His family was Jewish and they managed to flee from Nazi Germany to the U.S. in 1939. Sadly, his father did not survive the journey, and Furstenberg grew up with his mother and sister in an orthodox community in New York. Following a career in mathematics at several universities in the U.S., he left the country in 1965 for the Hebrew University of Jerusalem, where he stayed until his retirement in 2003. Spending most of his career in Israel, he helped establish the country as a world center for mathematics.
Furstenberg has won the Israel Prize and the Wolf Prize.

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When he published one of his early papers, a rumour circulated that he was not an individual but instead a pseudonym for a group of mathematicians. The paper contained ideas from so many different areas: surely it could not possibly be the work of one man?
Furstenberg said that he reacted with “total disbelief” when he learnt he had won. “I had known about the prestige of the Abel Prize and knew the list of former laureates,” he told an interviewer during the announcement. “I simply felt that these are people of a certain league, and I was not in that league.” He added that early on, he did not foresee the impact that his ideas were going to have. “Like any mathematician, I follow my nose and look for what seems to be very interesting.”

Gregory Margulis was born in Moscow in 1946. In 1978, he won the Fields Medal at only 32 years of age but was unable to receive the medal in Helsinki since Soviet authorities refused him a visa. He was one of the top young mathematicians in the Soviet Union but was unable to find a job at Moscow University as he faced discrimination for being of Jewish origin. Instead, he found work at the Institute for Problems in Information Transmission. During the 1980s he visited academic institutions in Europe and the U.S. before settling at Yale in 1991, where he has been ever since. Margulis is a winner of the Lobachevsky Prize and the Wolf Prize.

From early on, Margulis, showed a unique talent in mathematics, but he was only allowed to travel abroad in 1979 when Soviet academics were given more personal freedoms.

Due to the ten-year age difference and the travel restrictions of the Soviet authorities, the laureates did not formally collaborate. However, they influenced each other’s work.

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Margulis said that he, too, felt greatly honoured to receive such recognition from the mathematical community.

The two have studied chaotic systems. A common thread in the work of both mathematicians has been the use of techniques from ergodic theory, a field of mathematics that originated in the study of physics problems such as the motion of billiard balls or planetary systems. Ergodic theory studies systems that evolve in time, eventually exploring almost all their possible configurations. These systems are typically chaotic, meaning that their future behaviour can only be guessed using probability.

But that randomness can be a strength when applied to other mathematical problems. “If you want to understand one big space, one way to do it is to explore it randomly,” explains Terence Tao, a mathematician at the University of California, Los Angeles.

In seminal papers in the 1960s and 1970s, Furstenberg used ergodic ideas to show how even the most random sets of infinitely many whole numbers had to conceal some kind of regular structure, explains Alex Lubotzky, a mathematician at the Hebrew University of Jerusalem who was one of Furstenberg’s students. “Even if you have chaos, if you look carefully you will find order in it,” he says. “It’s like the stars in the sky they look completely random, but the ancient Greeks could see the constellations.”
Furstenberg’s ideas had an impact across fields seemingly distant from ergodic theory – including geometry and algebra. Building in part on his work, Tao and his collaborator Ben Green, a mathematician now at the University of Oxford, UK, announced a breakthrough in number theory in 2004. They showed that the set of prime numbers contains arithmetic progressions — sequences that contain constant intervals, such as the two steps between each of 3, 5 and 7 — of arbitrary length. These sequences form one of the most striking patterns ever discovered in the seemingly random arrangement of primes across the set of all whole numbers.

 

The writer Abid Amin Naeem is research scholar at the department of Mathematics at COMSATS University Islamabad. He can be reached at [email protected]

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