Abel Prize 2020

Congratulations to Gregory (Grisha) Margulis and Hillel (Harry) Furstenberg on being awarded the 2020 Abel Prize. The prize is one of the most prestigious in mathematics and is presented annually by the Norwegian Academy of Science and Letters.

The official announcement states that Margulis and Furstenberg were awarded the prize “for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics” and their work is described by Hans Munthe-Kaas, chair of the Abel committee, as “bringing down the traditional wall between pure and applied mathematics”. So, who are they?

Gregory Margulis

Born in Moscow in 1946, Margulis gained international recognition aged only 16 when he received a silver medal at the International Mathematical Olympiad. He began his academic career at Moscow State University and began working towards his PhD under the supervision of 2014 Abel Prize Laureate Yakov Sinai. At the age of 32, he was awarded the 1978 Fields Medal for his work on the ‘arithmeticity and superrigidity theorems’, but was unable to travel to Finland to receive the medal as the soviet authorities refused to provide him with a visa.

Another major result followed in 1984 with his proof of the Oppenheimer Conjecture – a problem in Number Theory first stated in 1929. The ideas he introduced here centred on what is known as ‘ergodic theory’ (more on this later), and have since been used by three recent Fields Medallists: Elon Lindenstrauss, Maryam Mirzakhani and Akshay Venkatesh. In 2008, Pure and Applied Mathematics Quarterly ran an article listing Margulis’s major results which ran to more than 50 pages.

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Hillel Furstenberg

Originally thought to be a pseudonym for a group of mathematicians due to the vast range of ideas published in his early work, Furstenberg is a mathematician with a deep technical knowledge of countless areas of mathematics. He published his first papers as an undergraduate in 1953 and 1955, with the latter giving a topological proof of Euclid’s famous theorem that there are infinitely many prime numbers.

One of his key results came in 1977 when he used methods from ergodic theory to prove a celebrated result by 2012 Abel Prize Laureate Endre Szemerédi on arithmetic progressions of integers. The insights that came from his proof have led to numerous important results, including the recent proof by Ben Green and Terence Tao that the sequence of prime numbers includes arbitrarily large arithmetic progressions.

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So, what is ergodic theory?

Ergodic theory relates to probability and what we call ‘random walks’, best explained by thinking about a dog trying to find some treats buried in a garden…

If you hide some treats in your garden and let your dog try to to find them, it will most likely start sniffing in what seems to be an apparently random pattern. However, after a short period of time, the dog will more often than not successfully find the treats. This method of search might not seem to be systematic, but yet the dog is following its instinct telling it to randomly change its direction at regular intervals to maximise its chance of success. You can think of it as moving one step forwards, then flipping a coin to decide whether you next go left or right for one step, and repeating this indefinitely.

In maths, the dog’s behaviour is encoded in the concept of a random walk. A random walk is a mathematical object that describes a path consisting of a succession of random steps in some mathematical space. There are numerous examples of physical systems that are modelled by random walks: the behaviour of gas molecules, stock markets, the statistical properties of neurons firing in the brain… But, random walks can also be seen as a tool to explore a mathematical object, in the same way that the dog tries to understand the garden. Of course, Hillel Furstenberg and Gregory Margulis are not using random walks to find treats in a garden, they do random walks on graphs or on groups in order to reveal the secrets of these objects.

If the trajectory of the dog is ergodic, this means that the dog will eventually get close to the treat in the long term. In fact, if we were to draw a circle around the treat, of any size (even as small as you can possibly imagine), after some finite amount of time the dog will be sniffing inside the circle, and therefore will probably discover the treat. This is ergodic theory in a nutshell.

More information on the Abel Prize announcement can be found on the website of the Norwegian Academy of Sciences and Letters here or in the official citation here.

Alex Bellos Interviews Abel Prize Winner Robert Langlands

Author and broadcaster Alex Bellos interviews 2018 Abel Prize Laureate Robert Langlands after he receives the award from King Harald V of Norway. Langlands discusses his early childhood in Canada, his choice of maths at university because it was ‘easy’, his meeting with Norwegian mathematician Atle Selberg at Princeton, and finally his advice for young mathematicians looking to make their mark on the subject.

Produced by Tom Crawford with support from the Norwegian Academy of Science and Letters. The third in a series of videos documenting my experience at the 2018 Abel Prize week in Oslo.

Abel Prize Laureate 2019: Karen Uhlenbeck

Karen Uhlenbeck was selected by a committee of five mathematicians nominated by the European Mathematical Society and the International Mathematics Union. Her work involves the study of partial differential equations, calculus of variations, gauge theory, topological quantum field theory, and integrable systems. The full citation from the announcement can be found here and a short biography by Jim Al-Khalili here.

“Karen Uhlenbeck receives the Abel Prize 2019 for her fundamental work in geometric analysis and gauge theory, which has dramatically changed the mathematical landscape. Her theories have revolutionised our understanding of minimal surfaces, such as those formed by soap bubbles, and more general minimisation problems in higher dimensions.” – Hans Munthe-Kaas, Chair of the Abel Committee.

Karen’s work covers minimisation problems, such as solving for the shape of a soap bubble acting to minimise its energy under gravity. Here’s a fantastic slow-motion experiment from Ray Goldstein at the University of Cambridge demonstrating the change in the shape of a soap bubble as the two supporting wires are pulled apart.

Karen also works in topological quantum field theory which has very important consequences for physicists, not least in relation to the Yang-Mills Mass Gap Hypothesis – one of the 7 million-dollar Millennium Problems. You can read more about the problem here.

“If I really understand something, I’m bored.” Karen Uhlenbeck

Throughout her career Karen has been very active in the area of mentorship and furthering the cause of women in mathematics. She is the founder of the Institute of Advanced Study Women’s Program, now entering its 25th year, and the Park City Mathematics Institute Summer Session, which places a huge emphasis on interdisciplinary research and collaboration between mathematicians from all areas.

The Abel Prize was established on 1 January 2002 – 200 years after the birth of Niels Henrik Abel. The purpose is to award the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 6 million NOK (about 750,000 Euro) and was awarded for the first time on 3 June 2003.

You can read the official announcement from the Norwegian Academy of Science and Letters here.

2018 Abel Laureate Robert Langlands

The Norwegian Academy of Science and Letters kindly provided me with a scholarship to attend the Abel Prize week in Oslo earlier this year where I interviewed the 2018 Abel Laureate Robert Langlands.

In the first of a series of videos documenting my experience, Robert describes how he came to do Mathematics at university…

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