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The Abel Prize in Mathematics was shared by two pioneers in the fields of probability and dynamics

Hillel Farstenberg, 84 years old, and Grigory Margulis, 74 years old, retired professors, shared the mathematical equivalent of the Nobel Prize



Hillel Farstenberg

Two mathematicians who demonstrated how an underestimated branch of the field of research can be used to solve important problems shared the Abelian Prize this year - the mathematical equivalent of the Nobel Prize.

It was received by Hillel Farstenberg , 84 years old, from the Hebrew University of Jerusalem, and Grigory Margulis , 74 years old, a Soviet and American mathematician from Yale University. Both are retired professors.

The prize, awarded by the Norwegian Academy of Sciences and Literature, was awarded "for an innovative approach to using methods from probability theory and dynamics in group theory, number theory and combinatorics."

Farstenberg and Margulis will share a cash prize of NOK 7.5 million, or about $ 700,000.


Grigory Margulis is

not awarded the Nobel Prize for mathematics, and for several decades the Field Prize was the most prestigious award in this field , medals for which were given in small groups every four years the most outstanding mathematicians under the age of 40 years.

The Abel Prize, named after Niels Henrik Abel, a Norwegian mathematician, is more like a Nobel Prize. Since 2003, it has been awarded annually to highlight important breakthroughs in mathematics. Former laureates include Andrew Wiles, who proved Fermat's Great Theorem and works at Oxford University; John F. Nash Jr., whose biography was filmed in the movie "Mind Games"; Karen Uhlenbeck, professor emeritus at the University of Texas at Austin, became the first woman to receive this award last year.

This year, pioneers of new ideas and techniques became laureates of the Abel Prize.

Francois Labourier, a mathematician from the University of the Cote d'Azur in France, who sat on the prize committee, said that most mathematicians in the 20th century did not really like the probability theory, which was at the very bottom of the mathematical hierarchy, under number theory, algebra, and differential geometry.

“Probability theory was just applied mathematics,” said Labourier. But Farstenberg and Margulis found ways to show how probability methods can solve abstract problems.

“At that time it was truly revolutionary,” said Labourier. “They were one of the first to show that probabilistic methods lie at the very center of mathematics.” Now it’s almost obvious. ”

Farstenberg said he had a phone call Monday night to announce the award. “I didn’t hear what they said on the phone,” he told us during a telephone interview. - I heard the words “Norwegian Academy” and “prize”, and thought: Are they talking about the Abel Prize? It was hard to believe. I called my wife to the phone. And so it was. ”



Margulis said he was also called on Monday. “Of course, I was very happy and proud,” he said. “It is a great honor.”

Here is an example of how randomness can be used in theoretical mathematics.

Imagine how a drunkard hesitantly moves around the room, pushing himself against the walls. Noting how often it passes at a certain point on the floor, we can conclude about the shape and size of the room. The general idea of ​​using the trajectory of an object to obtain information about the space in which it moves is called an ergodic theory .

Farstenberg used this approach in his doctoral dissertation at Princeton University to examine whether the complete history of measuring a sequence of numbers could provide useful information about what would happen to her next. “Is it possible to say exactly what will happen next, or is it possible to at least talk about the likelihood of what will happen next?” - he said.

Farstenberg showed that a dynamic system, the periodic images of which reproduce a sequence of numbers, can give a similar forecast.

Many years later, Farstenberg used a similar approach for an alternative proof of the theorem on numbers, which had already been proved by another mathematician, Endre Szemeredi.. In a sufficiently large subset of integers - which mathematicians describe as a set with positive density - you can find arithmetic progressions of arbitrary length (sequences of numbers like 3, 7, 11, 15 - where the numbers are at the same distances from each other).

However, Szemeredi's proof was long and complicated.

“Farstenberg provided an excellent brief proof,” said Terence Tao, a mathematician at the University of California, Los Angeles.

In 2004, Tao and Ben Green, mathematicians from the University of Oxford, citing Farstenberg's work and using ergodic theory, proved another important theorem: that sequences of arbitrary length also exist among primes - such integers that are divisible by only 1 and by yourself.

Some of the noteworthy works of Margulis, another Abelian laureate, relate to communication networks similar to the Internet, where computers constantly send messages to each other. To achieve the highest communication speed, it would be necessary to connect each pair of computers directly. However, this would require the use of a huge amount of cable that goes beyond a practical range.

“These are the networks that you are trying to design so that they are as thin as possible on the one hand,” said Peter Sarnak, a mathematician at the Advanced Research Institute at Princeton, “and on the other hand, they would have such a property that you you can quickly go from one point to another along a short path. "

Margulis was the first to describe a step-by-step procedure for creating such networks, known as expanders .

Margulis with the help of ergodic theory reformulated the problem, but often this approach does not facilitate their solution. Sarnak said that if a student came to him and showed the first steps of those that Margulis did, he would say to him: “So what? What have you achieved? You just reformulated it, and now it looks even more complicated. "

However, ergodic theory has helped to uncover universal truth, allowing Margulis to quickly advance in solving problems that were previously considered too intractable. “He reached a solution from scratch in just a couple of articles, and he did it in an amazingly original way,” Sarnak said.

Expanders have practical applications not only in the design of computer networks, but also in applications such as error correction algorithms, random number generators, and cryptography.

Farstenberg was born in Berlin in 1935. His Jewish family was able to leave Germany shortly before the outbreak of World War II, and reached the United States, settling in New York in the Manhattan quarter of Washington Heights. While still a student at Yeshiva University, he had already published scientific papers.

After defending his doctorate at Princeton, he taught for a year there, and then moved to the Massachusetts Institute of Technology, to then get a job at the University of Minnesota. In 1965, he moved to the Hebrew University of Jerusalem, where he worked until his retirement in 2003.

Grigory Alexandrovich Margulis was born in Moscow in 1946, and defended his doctorate at Moscow State University in 1970. He received the Fields Prize in 1978, at the age of 32, but was not allowed to leave the USSR for the award ceremony held in Helsinki, Finland.

As a Jew, he was unable to get a job in any of the prestigious institutions. He worked at the A. A. Kharkevich Institute of Information Transmission Problems, but his practical decisions led him to discoveries related to expanders.

“I somehow ended up in the right place at the right time,” Margulis said. “If I hadn’t got there, then I probably would not have dealt with this issue.”

In the 1980s, he was able to travel to universities in other countries, and in 1991 settled at Yale.

The Abel Prize ceremony, scheduled to be held in Oslo on May 19, 2020, was temporarily postponed due to the coronavirus pandemic.

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