What comes to mind when you think of the most prestigious awards in mathematics? Chances are, it’s either the Abel prize or the Fields medal. I think most of us have heard about Abel at one point or another, but I became increasingly curious about Fields. Who is behind the so-called Nobel prize of mathematics?
John Charles Fields was born in 1863, in Ontario, Canada and studied at the University of Toronto before moving to the United States in 1887 to study for a PhD at Johns Hopkins University in Baltimore. Fields was involved in several mathematical societies—the Royal Society of Canada and the Royal Canadian Institute among others—and he spent most of his life lecturing at the University of Toronto. Though a great mathematician, he excelled at organising mathematical events and promoting research.
One of his greatest achievements was working with the International Mathematical Union to hold the 1924 International Congress of Mathematicians (ICM) in Toronto. This was only the second congress held by the union after the First World War. During the first congress, Germany, Austria–Hungary, Bulgaria and Turkey were excluded from the union and many were worried this might escalate. But Fields was determined to make the 1924 congress work. He spent months in Europe fundraising tirelessly. Some say personal acquaintances with rulers of Europe aided his efforts—attendance of a dinner with the king of Sweden and a later meeting with Mussolini in Bologna certainly support this claim—but no direct sources remain. After organising the conference, with money to spare, he paid for European mathematicians’ travel costs across the Atlantic. And this is where the idea of the Fields medal was born.
It is proposed to found two gold medals to be awarded at successive International Mathematical Congress for outstanding achievements in mathematics […]
The idea of the award was first presented on 24 February 1931. Because of Fields’ talent for acquiring funds, the medals were to be funded by the remaining finances from the 1924 ICM. Fields wrote, “It is proposed to found two gold medals to be awarded at successive International Mathematical Congress for outstanding achievements in mathematics […]” The awards would be open to the whole world and would be made by an international committee.”
He proposed the medals would be handed out at every congress for work already done, but also to encourage further achievement, starting with the next congress in 1936. Unfortunately, Fields’ health started to decline in 1932. Just days before his death, he noted in his will to leave an additional 47,000 Canadian dollars to fund the medal. As he had intended, the very first medals were awarded in the 1936 ICM in Oslo, Norway.
Today, Fields medals are awarded every four years to between two and four brilliant mathematicians under the age of 40 with a prize of C$15,000. The age limit was not directly due to Fields himself; it was added in 1966 to promote diversity, although recently this choice has been criticised as there is anecdotal evidence suggesting female mathematicians fare better later in their careers.
The medal itself was designed by Robert Tait McKenzie and features Fields’ monogram alongside a profile of Archimedes and the date of the medal’s founding, incorrectly written in Roman numerals: MCNXXXIII (1933). The reverse features an illustration of one of Archimedes’ most famous achievements: deducing the surface areas and volumes of the cylinder and the sphere which can be inscribed within the cylinder. Each face of the medal bears a Latin phrase:
(To transcend one’s human limitations and master the universe)
on the obverse, and the reverse reads:
(Mathematicians gathered from the whole world to honour noteworthy contributions to knowledge)
The date of the prize being awarded and the recipient’s name is engraved on the rim of the medal.
In 1936, Jesse Douglas and Lars Ahlfors were the first to receive Fields medals. What was so special about these two mathematicians, that they stood out from across the globe to become the first winners of the (arguably) highest honour in mathematics?
Jesse Douglas’s love for mathematics started in high school. While studying at the City College of New York, he became the youngest recipient of the college’s Belden medal for excellence in mathematics in his first year. After his undergraduate degree, Douglas began his doctoral study at Columbia University under the supervision of Edward Kasner, who introduced Douglas to the problem that became his most noteworthy achievement—Plateau’s problem.
Plateau’s problem (also known as the soap bubble problem) is about showing the existence of a minimal surface for a given boundary, and possibly with other constraints. This has a fascinating physical application in the form of soap films. A frame filled with a thin soap bubble, due to the action of surface tension, will always take the shape of minimum surface area: a so-called minimal surface. In 1931, Douglas discovered and proved a general solution, for which he came to win the Fields medal (although this was also done independently by Tibor Radó: Radó also created the busy beaver family of Turing machines, see Will this article ever end). Before this contribution, only some special cases had been proven by such mathematicians as Schwarz, Weierstrass, and Riemann.
Around the same time, Douglas was working at the Massachusetts Institute of Technology as an assistant professor, later to be promoted to an associate professor. He continued to work on Plateau’s problem even after solving it, focusing on further generalisations of the problem. He published 11 papers between 1939 and 1940 on these generalisations.
After working on Plateau’s problem for many years, Douglas diverted his attention to group theory, making significant contributions to the field. He spent the last 10 years of his life as a professor, back at the City College of New York.
Ahlfors started his university studies at the University of Helsinki and was taught by two of the best Finnish mathematicians at the time: Ernst Lindelöf and Rolf Nevanlinna. After this, he followed Nevanlinna to Zürich and discovered mathematics can be taught in a different way. Lars came to understand “that [he] was supposed to do mathematics, not just learn it”. During his time in Zürich, Ahlfors encountered Denjoy’s conjecture (now known as the Denjoy–Carleman–Ahlfors theorem) which he proved in 1929. Loosely, the theorem determines the number of values an entire function (a function that is differentiable everywhere in the complex plane) can take at infinity. This is what gave Ahlfors international recognition, though he himself always credits the significant help of Nevanlinna and Pólya as the main influences that lead to his proof. Though impressive, this was not what earned Ahlfors his Fields medal. That award was specifically credited to a single paper. This paper shed some light on Nevanlinna’s theory of meromorphic functions and quasiconformal mappings (more stuff to do with complex functions). Currently this paper is recognised as one of the starting points for essentially a new branch of analysis called metric topology.
Having already gone through the First World War as a child, Ahlfors was just about to go through the next challenge—the Second World War. But, surprisingly, his research benefited. He was able to devote himself to his work completely, even though libraries had closed due to the lack of students. In 1944, Lars was offered a position in Zürich, opportune timing since the Soviet Union was attacking Finland and Ahlfors’s own health was poor. So he, his wife, and two young children planned to flee to Switzerland, via Stockholm, the UK and then Paris.
Times were tough, and Ahlfors was only able to take 10 crowns with him. So what did he do on arrival in Stockholm? He smuggled his Fields medal across the border and sold it in a pawn shop! He later reflected, “I’m sure it is the only Fields medal that has been in a pawn shop. As soon as I got a little money some people in Sweden helped me retrieve it.” What a relief! Imagine if someone had actually bought it!
Luckily, the family made it to Switzerland safely. Slowly, though, Ahlfors began to feel that his invitation had been not an honour, but simply an attempt to fill a position they couldn’t find anyone else for. So when an offer came in from Harvard, Ahlfors gladly accepted it and remained there for more than 30 years until his retirement.
Since Douglas and Ahlfors first won in 1936, 58 other mathematicians have been awarded the medal, including the only person to decline the award—Grigori Perelman—and the first and so far only woman to receive the award—Maryam Mirzakhani. The next ICM is due to be held in 2022 in St Petersburg, Russia, and the story of the Fields medal and its winners will continue.
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