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ike Olympic medals and World Cup trophies, the best-known prizes in come around only every Lfour years. Already, maths departments around the world are buzzing with specula- tion: 2018 is a year. While looking forward to this year’s announcement, I’ve been looking backwards with an even keener interest. In long-over- looked archives, I’ve found details of turning points in the medal’s past that, in my view,

KARL NICKEL/OBERWOLFACH PHOTO COLLECTION PHOTO KARL NICKEL/OBERWOLFACH hold lessons for those deliberating whom to recognize in August at the 2018 Inter­ national Congress of in in , and beyond. Since the late 1960s, the Fields Medal has been popularly compared to the , which has no category for mathematics1. In fact, the two are very different in their proce- dures, criteria, remuneration and much else. Notably, the Nobel is typically given to senior figures, often decades after the contribution being honoured. By contrast, Fields medal- lists are at an age at which, in most sciences, a promising career would just be taking off. This idea of giving a top prize to rising stars who — by brilliance, luck and circum- stance — happen to have made a major mark when relatively young is an accident of history. It is not a reflection of any special connection between maths and youth — a myth unsupported by the data2,3. As some mathematicians have long recognized4, this accident has been to mathematics’ detriment. It reinforces biases within the discipline and in the public’s attitudes about mathemati- was on the Fields Medal shortlist in 1958. cians’ work, career pathways and intellectual and social values. All 56 winners so far have been phenomenal mathematicians, but such biases have contributed to 55 of them being male, most being from the and The Fields Medal Europe and most working on a collection of research topics that are arguably unrepre- sentative of the discipline as a whole. When it began in the 1930s, the Fields should return to Medal had very different goals. It was rooted more in smoothing over international conflict than in celebrating outstanding scholars. In fact, early committees delib- its roots erately avoided trying to identify the best young mathematicians and sought to pro- Forgotten records of mathematics’ best-known mote relatively unrecognized individuals. As I demonstrate here, they used the medal prize hold lessons for the future of the discipline, to shape their discipline’s future, not just to argues historian Michael Barany. judge its past and present.

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As the mathematics profession grew organizers’ discussions, years later, over what mostly through letters sent to Bohr, who and spread, the number of mathemati- to do with these leftover funds. summarized the key points in letters he sent cians and the variety of their settings made Fields forced the issue from his death- back. The committee conducted most of it harder to agree on who met the vague bed in 1932, endowing two medals to be these exchanges in the second half of 1949, standard of being promising, but not a star. awarded at each ICM. The 1932 ICM in agreeing on the two winners that December. In 1966, the Fields Medal committee opted appointed a committee to select the The letters suggest that Bohr entered the for the current compromise of consider- 1936 medallists, but left no instructions as to process with a strong opinion about who ing all mathematicians under the age of 40. how the should proceed. Instead, early should win one of the medals: the French Instead of celebrity being a disqualification, committees were guided by a memorandum , who it became almost a prerequisite. that Fields wrote shortly before his death, had blown Bohr away with an exciting new I think that the Fields Medal should return titled ‘International Medals for Outstand- theory at a 1947 conference6. The Second to its roots. Advanced mathematics shapes ing Discoveries in Mathematics’. World War meant that Schwartz’s career our world in more ways than ever, the dis- Most of the memorandum is procedural: had got off to an especially rocky start: he cipline is larger and more diverse, and its how to handle the funds, appoint a commit- was Jewish and a Trotskyist, and spent part demographic issues and institutional chal- tee, communicate its decision, design the of the French Vichy regime in hiding using lenges are more urgent. The Fields Medal medal and so on. In a false name. His long-awaited textbook had plays a big part in defining what and who fact, Fields wrote, the “Fields still not appeared by the end of 1949, and matters in mathematics. committee “should be stipulated there were few major new results to show. The committee should leverage this role left as free as possible” that the medal Bohr saw in Schwartz a charismatic leader by awarding medals on the basis of what to decide winners. To should not be of mathematics who could offer new con- mathematics can and should be, not just minimize national named after nections between pure and applied fields. what happens to rise fastest and shine bright- rivalry, Fields stipu- any person or Schwartz’s theory did not have quite the est under entrenched norms and structures. lated that the medal place.” revolutionary effects Bohr predicted, but, by By challenging themselves to ask every four should not be named promoting it with a Fields Medal, Bohr made years which unrecognized mathematics after any person or a decisive intervention oriented towards his and mathematicians deserve a spotlight, place, and never intended for it to be named discipline’s future. the prizegivers could assume a more active after himself. His most famous instruction, The best way to ensure that Schwartz won, responsibility for their discipline’s future. later used to justify an age limit, was that the Bohr determined, was to ally with Marston awards should be both “in recognition of Morse of the Institute for Advanced Study in BORN OF CONFLICT work already done” and “an encouragement Princeton, who in turn was promoting his The Fields Medal emerged from a time of for further achievement”. But in context, this Norwegian colleague, . The path deep conflict in international mathematics instruction had a different purpose: “to avoid to convincing the rest of the committee was that shaped the conceptions of its purpose. invidious comparisons” among factious not straightforward, and their debates reveal Its chief proponent was , national groups over who deserved to win. a great deal about how the members thought a Canadian mathematician who spent his The first medals were awarded in 1936, to about the Fields Medal. early career in a fin de siècle European math- mathematicians from Committee members started talking ematical community that was just beginning and from the United States. about criteria such as age and fields of study, to conceive of the field as an international The Second World War delayed the next even before suggesting nominees. Most endeavour5. medals until 1950. They have been given thought that focusing on specific branches The first International Congress of every four years since. of mathematics was inadvisable. They enter- Mathematicians (ICM) took place in 1897 tained a range of potential age considera- in Zurich, , followed by ICMs BLOOD AND TEARS tions, from an upper limit of 30 to a general in in 1900, Heidelberg in in The Fields Medal selection process is sup- principle that nominees should have made 1904, Rome in 1908 and Cambridge, UK, in posed to be secret, but mathematicians are their mark in mathematics some time since 1912. The First World War derailed plans human. They gossip and, luckily for histori- the previous ICM in 1936. Bohr cryptically for a 1916 ICM in , and threw ans, occasionally neglect to guard confiden- suggested that a cut-off of 42 “would be a mathematicians into turmoil. tial documents. Especially for the early years rather natural limit of age”. When the dust settled, aggrieved research- of the Fields Medal, before the International By the time the first set of nominees was ers from and took the reins Mathematical Union became more formally in, Bohr’s cut-off seemed a lot less arbitrary. and insisted that Germans and their war- involved in the process, such ephemera may It became clear that the leading threat to time allies had no part in new international well be the only extant records. Bohr’s designs for Schwartz was another endeavours, congresses or otherwise. They One of the 1936 medallists, Ahlfors, French mathematician, André Weil, who planned the first postwar meeting for 1920 served on the committee to select the 1950 turned 43 in May 1949. Everyone, Bohr in Strasbourg, a city just repatriated to winners. His copy of the committee’s corre- and Morse included, agreed that Weil was France after half a century of German rule. spondence made its way into a mass of docu- the more accomplished mathematician. In Strasbourg, the US delegation won ments connected with the 1950 ICM, largely But Bohr used the question of age to try to the right to host the next ICM, but when hosted by Ahlfors’s department at Harvard ensure that he didn’t win. its members returned home to start fund- University in Cambridge, Massachusetts; As chair, Bohr had some control over the raising, they found that the rule of German these are now in the university’s archives. narrative, frequently alluding to members’ exclusion dissuaded many potential sup- The 1950 Fields Medal committee had views that “young” mathematicians should porters. Fields took the chance to bring the broad international membership. Its chair, be favoured while framing Schwartz as the ICM to Canada instead. In terms of inter- Harald Bohr (younger brother of the prime example of youth. He asserted that national participation, the 1924 Toronto physicist Niels), was based in Denmark. Weil was already “too generally recognized” congress was disastrous, but it finished Other members hailed from Cambridge, and drew attention to Ahlfors’s contention with a modest financial surplus. The idea UK, Princeton in , Paris, War- that to give a medal to Weil would be “maybe for an international medal emerged in the saw and Bombay. They communicated even disastrous” because “it would make the

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impression that the Committee has tried to to defend the move were repeated across designate the greatest mathematical genius.” major media outlets, and the ‘Nobel prize of Their primary objective was to avoid mathematics’ moniker was born. international conflict and invidious com- This coincidence — comparing the Fields parisons. If they could deny having tried to Medal to a higher-profile prize at the same select the best, they couldn’t be accused of time that a rule change allowed the med- having snubbed someone better. allists to be much more advanced — had a But Weil wouldn’t go away. Commit- lasting impact in mathematics and on the tee member Damodar Kosambi thought it award’s public image. It radically rewrote the

MFO/OBERWOLFACH PHOTO COLLECTION PHOTO MFO/OBERWOLFACH would be “ridiculous” to deny him a medal medal’s purpose, divorcing it from the origi- — a comment Bohr gossiped about to a nal goal of international reconciliation and Danish colleague but did not share with embracing precisely the kinds of judgement the committee. Member William Hodge Fields thought would only reinforce rivalry. worried “whether we might be shirking Any method of singling out a handful of our duty” if Weil did not win. Even Ahlfors honorees from a vast discipline will have argued that they should expand the award shortcomings and controversies. Social and to four recipients so that they could include structural circumstances affect who has the Weil. Bohr wrote again to his Danish confi- opportunity to advance in the discipline at all dant that “it will require blood and tears” to stages, from primary school to the professori- seal the deal for Schwartz and Selberg. ate. Selection committees themselves need to Bohr prevailed by cutting the debate short. be diverse and attuned to the complex values He argued that Weil would open a floodgate and roles of mathematics in society. to considering prominent older mathemati- But, however flawed the processes were cians, and asked for an up or down vote on before 1966, they forced a committee of elite the pair of Schwartz and Selberg. Finally, at mathematicians to think hard about their the awards ceremony at the 1950 ICM, Bohr discipline’s future. The committees used the praised Schwartz for being recognized and medal as a redistributive tool, to give a boost eagerly followed by a younger generation of to those who they felt did not already have mathematicians — the very attributes he had every advantage but were doing important used to exclude Weil. work nonetheless. Our current understanding of the social FURTHER ENCOURAGEMENT impact of mathematics and of barriers to Another file from the Harvard archives diversity within it is decidedly different to shows that the 1950 deliberations reflected that of mathematicians in the mid-twentieth broader attitudes towards the medal, not just century. If committees today were given the one zealous chair’s tactics. Harvard math- same licence to define the award that early ematician kept a selection of committees enjoyed, they could focus on letters from his service on the 1958 commit- The nomination of French mathematician André mathematicians who have backgrounds and tee in his private collection. Weil divided the 1950 Fields Medal committee. identities that are under-represented in the Zariski’s committee was chaired by math- discipline’s elite. They could promote areas ematician of the Swiss Federal descent (). Ultimately, the of study on the basis of the good they do in Institute of Technology in Zurich. Its first 1958 awards went to and René the world, beyond just the difficult theorems round of nominations produced 38 names. Thom, both of whom the committee consid- they produce. was the clear favourite, ered promising but not too accomplished — In my view, the medal’s history is an invi- proposed by five of the committee members. unlikely to provoke invidious comparisons. tation for mathematicians today to think Hopf began by crossing off the list the creatively about the future, and about what two oldest nominees, Lars Gårding and A SWEEPING EXPEDIENT they could say collectively with their most . His next move proved that it By 1966, the adjudication of which young famous award. ■ was not age per se that was the real disquali- mathematicians were good but not too good fying factor, but prior recognition: he ruled had become testing. That year, committee Michael Barany is a postdoctoral fellow in out Hirzebruch and one other who, having chair Georges de Rham adopted a firm age the Society of Fellows and Department of recently taken up professorships at pres- limit of 40, the smallest round number that History, Dartmouth College, Hanover, New tigious institutions, “did not need further covered the ages of all the previous Fields Hampshire, USA. encouragement”. Nobody on the committee recipients. e-mail: [email protected] seems to have batted an eyelid. Suddenly, mathematicians who would 1. Barany, M. J. Not. Am. Math. Soc. 62, 15–20 Of those remaining, the committee previously have been considered too accom- (2015). agreed that was plished were eligible. Grothendieck, presum- 2. Stern, N. Soc. Stud. Sci. 8, 127–140 (1978). 3. Hersh, R. & John-Steiner, V. Loving + Hating the most talented, but few of his results were ably ruled out as too well-known in 1962, Mathematics: Challenging the Myths of published and they considered him a shoo- was offered the medal in 1966, but boycotted Mathematical Life 251–272 (Princeton Univ. in for 1962. John Nash, born in the same its presentation for political reasons. Press, 2011). 4. Henrion, C. AWM Newsletter 25 (6), 12–16 year as Grothendieck (1928), came third in The 1966 cohort contained another politi- (1995). the final ballot. Although the 1958 shortlist cally active mathematician, . 5. Riehm, E. M. & Hoffman, F. Turbulent Times in also included Olga Ladyzhenskaya and Har- He went to accept his medal in rather Mathematics: The Life of J.C. Fields and the History of the Fields Medal (American Mathematical ish-Chandra, it would take until 2014 for the than testify before the US House Un-Ameri- Society & Fields Institute, 2011). Fields Medals to go to a woman (Maryam can Activities Committee about his activism 6. Barany, M. J, Paumier, A.-S. & Lützen, J. Hist. Mirzakhani) or a mathematician of Indian against the War. Colleagues’ efforts Math. 44, 367–394 (2017).

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