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|Marin Mersenne. H Loeffel, Blaise Pascal,
Basel: Birkhäuser 1987, CC BY-NC
|Cartoon from the weblog Occupy the Math|
- A Tribute to Maryam Mirzakhani, American Mathematical Society
- Salerno, Adriana. “Remembering Maryam Mirzakhani,” AMS Blogs, posted July 24, 2017.
- Wilkinson, Amie. “With Snowflakes and Unicorns, Marina Ratner and Maryam Mirzakhani Explored a Universe in Motion,” New York Times, August 7, 2017.
- Lamb, Evelyn. “Mathematics World Mourns Maryam Mirzakhani, Only Woman to Win Fields Medal,”, Scientific American blog, posted July 17, 2017
- Myers, Andrew and Bjorn Carey. “Maryam Mirzakhani, Stanford mathematician and Fields Medal winner, dies”, Stanford University News, July 15, 2017.
- Klarreich, Erica. A Tenacious Explorer of Abstract Surfaces,, Quanta Magazine (a video produced by the Simons Foundation in which Mirzakhani discusses the mathematics problems she studies).
- Carey, Bjorn. “Stanford’s Maryam Mirzakhani wins Fields Medal”, Stanford Report, August 12, 2014.
- “The Work of Maryam Mirzakhani”, Notices of the American Mathematical Society, Vol. 61, No. 9 (October 2014), 1079-1081. (Reprint of the IMU article above)
- McMullen, Curtis. “The work of Maryam Mirzakhani,” Laudatio delivered during the 2014 International Congress of Mathematicians.
- Svoboda, Elizabeth. “Maryam Mirzakhani“, PopSci’s Fourth Annual Brilliant 10, PopSci.com.
- MathSciNet [subscription required]
- Mathematics Genealogy Project
This article has been chosen from: Mathematically Gifted and Black
Dr. Talitha M. Washington is an Associate Professor of Mathematics at Howard University in Washington, DC, and is currently on detail as a Program Officer at the National Science Foundation. Prior to coming to Howard, she was an Assistant Professor of Mathematics at the University of Evansville (2005-2011), an Assistant Professor of Mathematics at The College of New Rochelle (2003-2005), and a VIGRE Research Associate at Duke University (2001-2003). Dr. Washington holds a 2001 Ph.D. in mathematics from the University of Connecticut with specialization in applied mathematics.
Dr. Washington grew up in Evansville, Indiana and appreciates the familial atmosphere of this community. While growing up, she enjoyed learning about all subjects. In high school, her favorite classes were AP Chemistry and an English course on classical plays. She graduated from Bosse High School a semester early and studied abroad in Costa Rica for six months on an exchange program with American Field Service. She then entered Spelman College as an engineering major simply because engineering is a popular field in Indiana. However, this major appeared to require too much work. Thus, she chose to major in math because of its connection to engineering, and it appeared to be the easier major because there was no laboratory component. Little did she know, she would fall in love with the subject and lead a long, distinguished career in mathematics.
As an undergraduate, she studied abroad at the Universidad Autónoma de Guadalajara in Mexico where she took a proof-based linear algebra course and earned a 0 on her first assignment. Determined to succeed, she rallied her new classmates to study in the library. By the end of the semester, she earned 100’s on her assignments. This experience taught her that obstacles can be overcome by cultivating and building a community of learners. Upon returning from Mexico to Spelman, she researched the Black-Scholes option pricing model under the direction of Dr. Jeffrey Ehme. After a summer internship at CIGNA in Hartford, Connecticut, she became interested in pursuing a career in actuarial science. Dr. Ehme encouraged her to apply to graduate school in addition to seeking positions in industry. From the support and assistance from Dr. Ehme and other members in the Spelman math department, she gained acceptance at the University of Connecticut and funding from the Packard Foundation. Thus, she declined an actuarial science position from CIGNA and began her graduate studies. Since she had secured no employment the summer before entering graduate school, she decided to backpack through southern Mexico, Belize, and Guatemala for two months – sola.
Dr. Washington faced many obstacles along the way but she met them with vigor, resilience, and determination. To her surprise, she entered a graduate program with students who already had earned advanced degrees from other universities. She met this challenge by actively seeking help and guidance from students and professors. Diligently, she worked night and day to learn the background material necessary for success. While she was in graduate school, she gave birth to her first child. She gained assistance with this transitional time in her life by securing a doula, a professional labor support person, and by participating in a network of mothers through La Leche League. She now has three vivacious STEM teenagers who are in pursuit of computer science, engineering, and biology.
While in graduate school, she enjoyed the vibrant atmosphere of working through mathematical problems and in particular, the many applications of math. With her advisor, Dr. Yung-Sze Choi, she researched a theoretical protein-protein interaction model. They also became involved with the National Resource for Cell Analysis and Modeling at University of Connecticut Health Center. Dr. Choi pushed Dr. Washington to further her work in mathematics by securing a postdoctoral position. Thus, after graduating, she headed down to Duke University where she researched a hormone secretion model with Dr. Michael Reed and Dr. Joseph Blum, in the mathematics and cell biology departments, respectively.
Recently, she has researched a fellow Evansville native, Dr. Elbert F. Cox, who is the first Black in the world to earn a Ph.D. in mathematics. She has shared his story on radio and television stations, as well as in the Notices of the American Mathematical Society, easily the most visible journal to all mathematicians. With her applied background, she led various undergraduate and graduate research projects from modeling the Tacoma Narrows Bridge to modeling calcium homeostasis to the construction of nonstandard finite difference schemes. With her passion for education, she led a youth conference, Stepping Up, that encouraged youth to pursue viable careers through higher education. She also led a one-week research-based summer camp for middle schoolers to explore current trends in mathematics and the sciences.
Dr. Washington enjoys her work at Howard University, in the same mathematics department Elbert Cox once led. As a program officer at the National Science Foundation, she is responsible for shaping the nation’s science and engineering enterprise and ensuring that underrepresented groups and diverse institutions across all geographic regions are included in the scientific enterprise of the nation. In her spare time, she serves on a number of boards and shares motivational speeches on diversity and mathematics to a wide range of audiences. She balances stress by maintaining a rigorous exercise regimen with fitness guru Cathe Friedrich, that includes kickboxing, step aerobics, strength training, and yoga. During the warm seasons, you may even find her running down the Rock Creek Park trails in Washington, DC.
Each time I read something about UFOs or some aliens, I ask myself who will get the benefit from these kind of rumors? Here is one the answers:
I enjoy reading this article so much. Dan Ashlock is about to show why Math is more similar to humanities rather experimental sciences. I strongly recommend you to read it:
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In the movie Contact based on the novel of the same name by Carl Sagan, Dr. Ellie Arroway searches for intelligent extraterrestrial life by scanning the sky with radio telescopes. When Arroway, played by Jodie Foster, recognizes prime numbers in an interplanetary signal, she believes it’s proof that an alien intelligence has sent the human race a message.
A number is considered prime if it is only divisible by one and itself. For example, two, three, five and seven are prime. The number 15, which is three times five, is not prime. It’s no coincidence that Arroway believes the aliens in Contact use prime numbers as a cosmic “hello” — they are building blocks of other numbers. Every number is a product of primes.
In December 2017, the largest known prime number was discovered using a computer search. The prime was discovered by Jonathan Pace, an electrical engineer who currently works at FedEx. Why is this important? Because without prime numbers your banking information, Paypal transactions or Amazon purchases could be compromised.
Large primes, like the one just discovered, play a critical role in cyber-security. Cryptography is the science of encoding and decoding information, and many of its algorithms, such as RSA, rely heavily on prime numbers.
While there are infinitely many primes, there is no known formula to generate them all. A race is ongoing to find larger primes using a mixture of math techniques and computation.
One way to get large primes uses a mathematical concept discovered by the 17th-century French monk and scholar, Marin Mersenne.
A Mersenne prime is one of the form 2ⁿ – 1, where n is a positive integer. The first four of these are three, seven, 31 and 127.
Not every number of the form 2ⁿ – 1 is prime, however; for example, 2⁴ – 1 = 15. If 2ⁿ – 1 is prime, then it can be shown that n itself must be prime. But even if n is prime, there is no guarantee the number 2ⁿ – 1 is prime: 2¹¹ – 1 = 2,047, which is not prime as it equals 23 times 89.
There are only 50 known Mersenne primes. An unresolved conjecture is that there is an infinite number of them.
The search for new primes
The Great Internet Mersenne Prime Search (or GIMPS) is a collaborative effort of many individuals and teams from around the globe to find new Mersenne primes. George Woltman began GIMPS in 1996, and in 2018 it includes more than 183,000 volunteer users contributing the collective power of over 1.6 million CPUs.
The most recently discovered Mersenne prime is succinctly written as 2⁷⁷²³²⁹¹⁷ – 1; that’s two multiplied by itself 77,232,917 times, minus one. Jonathan Pace’s discovery took six days of computation on a quad-core Intel i5-6600 CPU, and was independently verified by four other groups.
The newly discovered prime has a whopping 23,249,425 digits. To get a sense of how large that is, suppose we filled up a book with digits, each digit counted as a word and each book having a hundred thousand words. Then the digits of 2⁷⁷²³²⁹¹⁷ – 1 would fill up about 232 books!
How does GIMPS find primes?
GIMPS uses the Lucas-Lehmer test for primes. For this, form a sequence of integers starting with four, and whose terms are the previous term squared and minus two. The test says that the number 2ⁿ – 1 is prime if it divides the (n-2)th term in the sequence.
While the Lucas-Lehmer test looks easy enough to check, the computational bottleneck in applying it comes from squaring numbers. Multiplication of integers is something every school-aged kid can do, but for large numbers, it poses problems, even for computers. One way around this is to use Fast Fourier Transforms (FFT), algorithms that speed up computations.
Anyone can get involved with GIMPS — as long as you have a decent computer with an internet connection. Free software to search for Mersenne primes can be found on the GIMPS website.
While the largest known prime is stunningly massive, there are infinitely many more primes beyond it waiting to be discovered. Like Ellie Arroway did in Contact, we only have to look for them.
Une association de professionnels du zinc veut faire entrer les bistrots et terrasses de Paris au patrimoine culturel immatériel de l’Unesco.
Déjà, en 2010, c’était le repas gastronomique des Français qui avait intégré la liste du patrimoine culturel immatériel. Il ne s’agissait pas en l’occurrence de la haute gastronomie mais du rituel populaire du repas, quotidien ou exceptionnel, celui des fêtes ou bien celui où l’on reçoit, le plat du jour au bistrot ou la cuisine bourgeoise. Soit à la fois un art de vivre et une forme éprouvée du lien social. Demain, ce sont donc les cafés et terrasses de Paris qui pourraient figurer dans la liste de l’Unesco. Continue reading “La vie au café”
Like Olympic medals and World Cup trophies, the best-known prizes in mathematics come around only every four years. Already, maths departments around the world are buzzing with speculation: 2018 is a Fields Medal year.
While looking forward to this year’s announcement, I’ve been looking backwards with an even keener interest. In long-overlooked archives, I’ve found details of turning points in the medal’s past that, in my view, hold lessons for those deliberating whom to recognize in August at the 2018 International Congress of Mathematicians in Rio de Janeiro in Brazil, and beyond.
Since the late 1960s, the Fields Medal has been popularly compared to the Nobel prize, which has no category for mathematics1. In fact, the two are very different in their procedures, criteria, remuneration and much else. Notably, the Nobel is typically given to senior figures, often decades after the contribution being honoured. By contrast, Fields medallists are at an age at which, in most sciences, a promising career would just be taking off.
When it began in the 1930s, the Fields Medal had very different goals. It was rooted more in smoothing over international conflict than in celebrating outstanding scholars. In fact, early committees deliberately avoided trying to identify the best young mathematicians and sought to promote relatively unrecognized individuals. As I demonstrate here, they used the medal to shape their discipline’s future, not just to judge its past and present.
As the mathematics profession grew and spread, the number of mathematicians and the variety of their settings made it harder to agree on who met the vague standard of being promising, but not a star. In 1966, the Fields Medal committee opted for the current compromise of considering all mathematicians under the age of 40. Instead of celebrity being a disqualification, it became almost a prerequisite.
I think that the Fields Medal should return to its roots. Advanced mathematics shapes our world in more ways than ever, the discipline is larger and more diverse, and its demographic issues and institutional challenges are more urgent. The Fields Medal plays a big part in defining what and who matters in mathematics.
The committee should leverage this role by awarding medals on the basis of what mathematics can and should be, not just what happens to rise fastest and shine brightest under entrenched norms and structures. By challenging themselves to ask every four years which unrecognized mathematics and mathematicians deserve a spotlight, the prizegivers could assume a more active responsibility for their discipline’s future.
Born of conflict
The Fields Medal emerged from a time of deep conflict in international mathematics that shaped the conceptions of its purpose. Its chief proponent was John Charles Fields, a Canadian mathematician who spent his early career in a fin de siècle European mathematical community that was just beginning to conceive of the field as an international endeavour5.
The first International Congress of Mathematicians (ICM) took place in 1897 in Zurich, Switzerland, followed by ICMs in Paris in 1900, Heidelberg in Germany in 1904, Rome in 1908 and Cambridge, UK, in 1912. The First World War derailed plans for a 1916 ICM in Stockholm, and threw mathematicians into turmoil.
When the dust settled, aggrieved researchers from France and Belgium took the reins and insisted that Germans and their wartime allies had no part in new international endeavours, congresses or otherwise. They planned the first postwar meeting for 1920 in Strasbourg, a city just repatriated to France after half a century of German rule.
In Strasbourg, the US delegation won the right to host the next ICM, but when its members returned home to start fundraising, they found that the rule of German exclusion dissuaded many potential supporters. Fields took the chance to bring the ICM to Canada instead. In terms of international participation, the 1924 Toronto congress was disastrous, but it finished with a modest financial surplus. The idea for an international medal emerged in the organizers’ discussions, years later, over what to do with these leftover funds.
Fields forced the issue from his deathbed in 1932, endowing two medals to be awarded at each ICM. The 1932 ICM in Zurich appointed a committee to select the 1936 medallists, but left no instructions as to how the group should proceed. Instead, early committees were guided by a memorandum that Fields wrote shortly before his death, titled ‘International Medals for Outstanding Discoveries in Mathematics’.
Most of the memorandum is procedural: how to handle the funds, appoint a committee, communicate its decision, design the medal and so on. In fact, Fields wrote, the committee “should be left as free as possible” to decide winners. To minimize national rivalry, Fields stipulated that the medal should not be named after any person or place, and never intended for it to be named after himself. His most famous instruction, later used to justify an age limit, was that the awards should be both “in recognition of work already done” and “an encouragement for further achievement”. But in context, this instruction had a different purpose: “to avoid invidious comparisons” among factious national groups over who deserved to win.
The first medals were awarded in 1936, to mathematicians Lars Ahlfors from Finland and Jesse Douglas from the United States. The Second World War delayed the next medals until 1950. They have been given every four years since.
Blood and tears
The Fields Medal selection process is supposed to be secret, but mathematicians are human. They gossip and, luckily for historians, occasionally neglect to guard confidential documents. Especially for the early years of the Fields Medal, before the International Mathematical Union became more formally involved in the process, such ephemera may well be the only extant records.
One of the 1936 medallists, Ahlfors, served on the committee to select the 1950 winners. His copy of the committee’s correspondence made its way into a mass of documents connected with the 1950 ICM, largely hosted by Ahlfors’s department at Harvard University in Cambridge, Massachusetts; these are now in the university’s archives.
The 1950 Fields Medal committee had broad international membership. Its chair, Harald Bohr (younger brother of the physicist Niels), was based in Denmark. Other members hailed from Cambridge, UK, Princeton in New Jersey, Paris, Warsaw and Bombay. They communicated mostly through letters sent to Bohr, who summarized the key points in letters he sent back. The committee conducted most of these exchanges in the second half of 1949, agreeing on the two winners that December.
The letters suggest that Bohr entered the process with a strong opinion about who should win one of the medals: the French mathematician Laurent Schwartz, who had blown Bohr away with an exciting new theory at a 1947 conference6. The Second World War meant that Schwartz’s career had got off to an especially rocky start: he was Jewish and a Trotskyist, and spent part of the French Vichy regime in hiding using a false name. His long-awaited textbook had still not appeared by the end of 1949, and there were few major new results to show.
Bohr saw in Schwartz a charismatic leader of mathematics who could offer new connections between pure and applied fields. Schwartz’s theory did not have quite the revolutionary effects Bohr predicted, but, by promoting it with a Fields Medal, Bohr made a decisive intervention oriented towards his discipline’s future.
The best way to ensure that Schwartz won, Bohr determined, was to ally with Marston Morse of the Institute for Advanced Study in Princeton, who in turn was promoting his Norwegian colleague, Atle Selberg. The path to convincing the rest of the committee was not straightforward, and their debates reveal a great deal about how the members thought about the Fields Medal.
Committee members started talking about criteria such as age and fields of study, even before suggesting nominees. Most thought that focusing on specific branches of mathematics was inadvisable. They entertained a range of potential age considerations, from an upper limit of 30 to a general principle that nominees should have made their mark in mathematics some time since the previous ICM in 1936. Bohr cryptically suggested that a cut-off of 42 “would be a rather natural limit of age”.
By the time the first set of nominees was in, Bohr’s cut-off seemed a lot less arbitrary. It became clear that the leading threat to Bohr’s designs for Schwartz was another French mathematician, André Weil, who turned 43 in May 1949. Everyone, Bohr and Morse included, agreed that Weil was the more accomplished mathematician. But Bohr used the question of age to try to ensure that he didn’t win.
As chair, Bohr had some control over the narrative, frequently alluding to members’ views that “young” mathematicians should be favoured while framing Schwartz as the prime example of youth. He asserted that Weil was already “too generally recognized” and drew attention to Ahlfors’s contention that to give a medal to Weil would be “maybe even disastrous” because “it would make the impression that the Committee has tried to designate the greatest mathematical genius.”
Their primary objective was to avoid international conflict and invidious comparisons. If they could deny having tried to select the best, they couldn’t be accused of having snubbed someone better.
But Weil wouldn’t go away. Committee member Damodar Kosambi thought it would be “ridiculous” to deny him a medal — a comment Bohr gossiped about to a Danish colleague but did not share with the committee. Member William Hodge worried “whether we might be shirking our duty” if Weil did not win. Even Ahlfors argued that they should expand the award to four recipients so that they could include Weil. Bohr wrote again to his Danish confidant that “it will require blood and tears” to seal the deal for Schwartz and Selberg.
Bohr prevailed by cutting the debate short. He argued that Weil would open a floodgate to considering prominent older mathematicians, and asked for an up or down vote on the pair of Schwartz and Selberg. Finally, at the awards ceremony at the 1950 ICM, Bohr praised Schwartz for being recognized and eagerly followed by a younger generation of mathematicians — the very attributes he had used to exclude Weil.
Another file from the Harvard archives shows that the 1950 deliberations reflected broader attitudes towards the medal, not just one zealous chair’s tactics. Harvard mathematician Oscar Zariski kept a selection of letters from his service on the 1958 committee in his private collection.
Zariski’s committee was chaired by mathematician Heinz Hopf of the Swiss Federal Institute of Technology in Zurich. Its first round of nominations produced 38 names. Friedrich Hirzebruch was the clear favourite, proposed by five of the committee members.
Hopf began by crossing off the list the two oldest nominees, Lars Gårding and Lipman Bers. His next move proved that it was not age per se that was the real disqualifying factor, but prior recognition: he ruled out Hirzebruch and one other who, having recently taken up professorships at prestigious institutions, “did not need further encouragement”. Nobody on the committee seems to have batted an eyelid.
A sweeping expedient
By 1966, the adjudication of which young mathematicians were good but not too good had become testing. That year, committee chair Georges de Rham adopted a firm age limit of 40, the smallest round number that covered the ages of all the previous Fields recipients.
Suddenly, mathematicians who would previously have been considered too accomplished were eligible. Grothendieck, presumably ruled out as too well-known in 1962, was offered the medal in 1966, but boycotted its presentation for political reasons.
The 1966 cohort contained another politically active mathematician, Stephen Smale. He went to accept his medal in Moscow rather than testify before the US House Un-American Activities Committee about his activism against the Vietnam War. Colleagues’ efforts to defend the move were repeated across major media outlets, and the ‘Nobel prize of mathematics’ moniker was born.
This coincidence — comparing the Fields Medal to a higher-profile prize at the same time that a rule change allowed the medallists to be much more advanced — had a lasting impact in mathematics and on the award’s public image. It radically rewrote the medal’s purpose, divorcing it from the original goal of international reconciliation and embracing precisely the kinds of judgement Fields thought would only reinforce rivalry.
Any method of singling out a handful of honorees from a vast discipline will have shortcomings and controversies. Social and structural circumstances affect who has the opportunity to advance in the discipline at all stages, from primary school to the professoriate. Selection committees themselves need to be diverse and attuned to the complex values and roles of mathematics in society.
But, however flawed the processes were before 1966, they forced a committee of elite mathematicians to think hard about their discipline’s future. The committees used the medal as a redistributive tool, to give a boost to those who they felt did not already have every advantage but were doing important work nonetheless.
Our current understanding of the social impact of mathematics and of barriers to diversity within it is decidedly different to that of mathematicians in the mid-twentieth century. If committees today were given the same licence to define the award that early committees enjoyed, they could focus on mathematicians who have backgrounds and identities that are under-represented in the discipline’s elite. They could promote areas of study on the basis of the good they do in the world, beyond just the difficult theorems they produce.
In my view, the medal’s history is an invitation for mathematicians today to think creatively about the future, and about what they could say collectively with their most famous award.