Two mathematicians find order in the chaos to receive the Abel Prize

Hillel Furstenberg and Gregory Margulis have applied theories of probability, randomness and system dynamics to many areas of mathematics.

Israeli- American mathematician Hillel Furstenberg and Russian-American Gregory Margulis have just been honored to receive the 2020 Abel Prize - one of the most prestigious awards in the field of Mathematics.

This award is given for both 'pioneering the use of probability and dynamics methods in group theory, number theory and combinatorial mathematics " , published by the Norwegian Academy of Sciences and Literature. March 18, 2020. The two have narrowed the gap between various mathematical fields, solving problems that seem to be out of reach.

Hillel Furstenberg (left) and Gregory Margulis receive the Abel Awards 2020. (Image: Yosef Adest, Dan Renzetti).

Furstenberg said he was 'totally unbelievable' knowing he won the award. 'I've heard the Abel Prize and know previous winners. I see them as a separate union and I do not belong to that union, 'he said.

He also shared that from the beginning, he did not anticipate the impact that his ideas would bring: 'Like any mathematician, I followed my intuition and searched for things. sounds interesting ".

Margulis also feels extremely honored to have this recognition from the math community.

Chaotic systems

The main idea in both mathematicians' work was to use the methods of ergodic theory, a field of mathematics rooted in the study of physical problems such as the motion of billiard balls or the system. planet.

Ergodic theory studied systems that evolved over time, and eventually discovered almost all of their possible states. These systems are often chaotic, meaning we can only predict their future behavior by using probability.

Ergodic theory explores the behavior of matter even though they are in a seemingly disorderly chaotic system.

But that randomness can be a strength when applied to other math problems.'If you want to understand a big space, exploring it by chance is a way of doing it, ' explains Terence Tao, a mathematician at the University of California, Los Angeles.

In articles published in the 1960s and 1970s, Furstenberg used the ergodic idea to show that even the most random set of infinitely many integers hides structures - Alex Lubotzky, the Mathematics at Jewish University, and one of Furstenberg's students, explained.

'Even in chaos, if you look closely, you will find order in it. It's like stars in the sky - they look completely random, but the ancient Greeks could recognize the constellations , ' he said.

Furstenberg's ideas impact on areas that seem to be quite different from ergodic theory, including geometry and algebra. Based on part of Furstenberg's work, Tao and his colleague Ben Green - mathematician at Oxford University, UK - announced a breakthrough in number theory in 2004.

They show that the set of primes containing the addition terms - ranges that contain constant spaces - such as the constant space (difference constant) of 3, 5 and 7 is 2. This is one of the The most striking pattern ever discovered in the arrangement of seemingly random primes on integer sets.

Take on challenges early

Both Furstenberg and Margulis suffer discrimination because they are Jewish. Furstenberg was born in 1935 in Berlin, and his family escaped Nazi persecution when he was 4 years old. He settled in New York City, then moved to Israel and taught at the Jewish University in Jerusalem from 1965 until his retirement in 2003.

Margulis was born in Moscow in 1946 and is a victim of anti-Semitism in the Soviet system. This prevented him from going abroad in 1978 to receive the Fields Medal - another prestigious award in mathematics. Later, he emigrated to the United States and worked at Yale University in New Haven, Connecticut.

Hillel Furstenberg, a mathematician, teaches at Hebrew University in Jerusalem.(Photo: Yosef Adest).

Margulis achieved his most glorious accomplishments at an early age and his reputation of 'musk' naturally came to non-Soviet researchers - before they had a chance to meet him.'He is a legend'. Mathematician Isarel Lubotzky recalled.

Margulis won the Fields Medal at age 32 for her work on the theory of symmetry. It includes continuous transformations in geometry, such as the rigid motion of a plane or the rotation of a sphere.

The mathematician who introduced Margulis during the Fields Medal ceremony said: 'Many times, he shocked the experts by solving questions that seemed completely out of reach at the time. Some of his research results lay the groundwork for the work of at least the next three Fields Medal winners , '' said the Norwegian Academy.

Mathematician Gregory A. Margulis.(Photo: Dan Renzetti).

'This year's award connects many previous winners. It was great to see researchers crossing boundaries, ' said Hans Munthe-Kaas, a mathematician at the University of Bergen in Norway, chairman of the Abel Prize committee.

In particular, Yakov Sinai - the 2014 winner of the method - developed ergodic methods in chaotic theory (and he was the doctoral adviser of Margulis), and part of the work of Endre Szemerédi - who won the 2012 prize - linking ergodic theory with integer theory.

The Abel Prize is named after the Norwegian mathematician Niels Henrik Abel (1802 - 1829) and was established in 2003. The two winners will win 7.5 million Norwegian kroner, or about 19.4 billion copper.

Because of the ongoing Covid-19 pandemic, the Academy decided to postpone the award ceremony - which normally takes place in Oslo in June. Instead, the 2021 awards ceremony will honor the winners for both 2020 and 2021. 'This is an unusual time, so this year we have to do things a little differently' . Mr. Munthe-Kaas said.