# The Establishment of the Fields Medal in Mathematics

# The Establishment of the Fields Medal in Mathematics

*Overview*

In the period between the two World Wars, despite frequently divisive international bitterness and rancor, Canadian mathematician John Charles Fields (1863-1932) worked to establish an award for mathematical achievement designed to underscore the international character of mathematics and promote promising academic talent. For mathematicians, the International Medal for Outstanding Discoveries in Mathematics, more generally known as the Fields Medal in mathematics, eventually became an equivalent of a Nobel Prize.

*Background*

Fields undertook his studies in mathematics in Canada, France, and Germany before spending the majority of his academic career at the University of Toronto. This international experience made him sensitive to political intrusion into academic life and aware of the need to unify the mathematics community. In 1924 Fields became president of the International Congress of Mathematics in Toronto. Far from being a truly international conference, however, Fields found that the scars of World War I had not yet sufficiently healed to permit the inclusion of German mathematicians. Disturbed at this, Fields proposed establishing an award that would renew international ties within the mathematics community by recognizing exceptional work and promise in mathematics. (Traditionally, Fields medals have gone only to younger mathematicians under the age of 40.) Following Fields's death in 1932, his will endowed such an award. The 1932 International Congress of Mathematicians in Zurich, Switzerland, adopted Fields's plan for an international award in mathematics, and the first Fields medals were awarded at the 1936 congress in Oslo, Norway.

It was Fields's intent, as expressed in a letter that preceded his will, that at least two gold medals be awarded at each meeting of the International Congress of Mathematics (which are held every four years). Fields stressed that the awards should be open to mathematicians from all countries and that the final decision on the awarding of the medals must be left to an international committee. In 1966 the ICM increased the possible number of medals awarded to four.

Although Fields intended the international committee awarding the medals to have wide discretion in their choices, he also made it clear that he wished the medals awarded not simply for academic achievement (i.e., work and research already performed) but also as an "encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others." In addition to bestowing academic honor, the monetary award accompanying the Fields Medal (established from funds left over from the fractionated 1924 Toronto Congress) offers medal recipients a sum to facilitate their further work and research in mathematics.

Because Fields so adamantly wanted the medals to be a unifying incentive and influence for the mathematics community, he specified that no country, institution, or individual should be linked to the medals. Ironically, of course, the International Congress award eventually became known—contrary to Fields's own wishes—as the Fields Medal. Officially, the medal is titled the International Medal for Outstanding Discoveries in Mathematics. Even more bitterly ironic for an award designed to promote peaceful discourse and recognition, the Fields Medals awarded in 1936 were the last given until 1950, when international tensions following World War II subsided enough to allow the resumption of awards.

With his characteristic attention to detail, Fields also specified the nature of the medals, asserting that they should be of significant value (i.e., worth more than $200 in gold—a considerable sum during the Depression). Although Fields did not specify any special inscription, the medals bear the likeness of Archimedes (c. 287-212 b.c.) and carried a Greek inscription translated as a charge to "transcend one's spirit and to take hold of the world."

*Impact*

The International Congress of Mathematics committee that selects Fields Medal recipients eventually wielded a profound power to shape mathematics. This result was not unforeseen by Fields, who intended from the outset that the medals be used to stimulate new research and promote contributions to mathematics in areas selected by the committee. In this regard the international congress, working through the Fields Medal selection committee, was able to significantly influence the evolution of and emphasis on certain areas of mathematics. Subsequent to the establishment of the Fields Medal, various branches and subdisciplines of mathematics, for example topological analysis, received a substantial boost from the awarding of Fields medals.

Beyond directly influencing the course of work in mathematics though monetary awards, the Fields Medal also exerts a more subtle—yet substantial—influence. Serving essentially as a Nobel Prize for mathematics, the Fields Medal also garners public attention and stimulates a broader dissemination of information. Many academic mathematicians receive their only brush with public recognition and exposure through the receipt of a coveted Fields Medal. In addition, the committee's shifting recognition of work in certain fields attracts talented young mathematicians and helps focus attention on rapidly developing areas. The spotlight of attention cast by the Fields Medal has also illuminated work done at a few leading institutions, especially Princeton University's Institute for Advanced Study, where a substantial number of medal winners have held appointments.

The awarding of Fields medals is not without controversy. The selection committees are, of course, comprised primarily of mathematicians able to evaluate the difficult and often abstract mathematical concepts involved in nominated work. Although the committee mitigates individual academic biases and prejudices, some academicians charge that areas of work such as logic are too often ignored. In the later half of the twentieth century, especially during the height of the Cold War tensions, other mathematicians expressed a growing concern regarding bias for, or prejudice against, certain mathematicians or schools of mathematics based on nationalism or parochial scholarly interest.

As a result of attention to various fields and subdisciplines, the Fields Medal both reflects and stimulates trends in the highest levels of mathematics research. Because the nominating and selecting committees have wide latitude, they are often able to reward and stimulate work in areas that are far removed from practical application. As a direct result of the awarding the Fields Medal—especially as its influence and prestige grew throughout the course of twentieth century—diverse and abstract fields such as algebraic topology received a considerable boost. Especially for theorists, the challenge of mathematics that involved solutions to German mathematician David Hilbert's (1862-1943) famous list of 23 problems (posted at the International Congress held in Paris in 1900) found renewed appreciation.

As the twentieth century progressed, advances in science, especially in physics and cosmology, became increasingly dependent upon advances and application of mathematics. Accordingly, stimulation of some areas of mathematics work, particularly in topological geometry, was needed to keep up with growing demands of science. In 1936 the medals were awarded for work dealing with Riemann surfaces and the plateau problem. In the 1950s work in functional analysis, number theory, algebraic geometry, and topology drew favor from the committee. In the 1960s research in partial differential equations, topology, set theory, and algebraic geometry found favor. In the 1970s, in addition to continued stimulation of topology, new and important work in number theory, group theory, Lie groups, and algebraic K-theory found recognition. In the 1980s research in topology continued to garner recognition as did work with the Mordell and Poincaré conjectures. Links to modern physics were strengthened with the 1990s recognition of work with superstring theory and mathematical physics.

By the end of the twentieth century the Fields Medal would recognize work in chaos theory, a fusion of scientific and mathematical efforts to seek order in complex and seemingly unpredictable systems and phenomena including population growth, the spread of disease (epidemiology), explosion dynamics, meteorology, and highly complex and intricate complex chemical reactions.

**K. LEE LERNER**

*Further Reading*

Monastyrsky, M. *Modern Mathematics in the Light of the**Fields Medal.* Wellesley, MA: A.K. Peters, 1997.

Tropp, Henry S. "The Origins and History of the Fields Medal." *Historia Mathematica* 3 (1976): 167-181.

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# The Establishment of the Fields Medal in Mathematics

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