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This Mathematical Month - December: A Brief Look at Past Events and Episodes in the Mathematical Community

**Monthly postings of vignettes on people, publications, and mathematics to inform and entertain.**

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**Featured Item for December **
**December 1952:** On the 31st of that month, **Vaughan F. R. Jones** was born in Gisborne, New Zealand. He is known for his discovery of the eponymous Jones polynomial, which supplied a new and richer invariant for knots than had previously been known. This work came out of a surprising connection between knot theory and von Neumann algebras. For this work he was awarded the Fields Medal in 1990 at the International Congress of Mathematicians in Kyoto. Jones received his PhD from the University of Geneva in 1979, under the direction of André Haefliger. He held a couple of positions in the United States before moving to the University of California, Berkeley, in 1985, where he has remained. He also serves as Distinguished Alumni Professor of the University of Auckland.

**December 1997:** The establishment of the **Beal Prize** for the solution of a conjecture in number theory was announced in the December 1997 issue of the *Notices of the AMS*. The conjecture was made by Andrew Beal, a prominent banker who is also a mathematics enthusiast. Beal's original prize for the solution of the conjecture, US$50,000, has been raised to US$100,000. The AMS is custodian of the funds and uses the income to support the annual Paul Erdos Memorial Lecture, presented at AMS meetings, and other activities. Beal's conjecture asserts that if *A*^{x} +B^{y} = C^{z} , where *A, B, C, x, y*, and *z* are positive integers and *x, y*, and *z* are all greater than 2, then *A, B*, and *C* must have a common prime factor. Is it true, or is it false? No one knows for sure, as neither a proof nor a counterexample has been found to date (December 2005). Read more about the Beal prize and conjecture in the Notices article and at the Beal Conjecture web site.

**December 1994: John Nash** received the Nobel Prize in Economics Sciences from the King of Sweden. Nash, John C. Harsanyi, and Reinhard Selten were honored for "their pioneering analysis of equilibria in the theory of non-cooperative games". The story of Nash's struggle with mental illness is movingly told in the best-selling biography *A Beautiful Mind* by Sylvia Nasar, which was made into a widely acclaimed movie. The book recounts the cliff-hanger deliberations of the Nobel Prize committee as it considered whether to give Nash the prize. Some were concerned that his mental illness meant that he was no longer the person who actually did the work to be honored; others feared an embarassing scene during the prize ceremony with the King of Sweden. By a narrow margin, the Nobel committee voted to give Nash the prize. "All the Swedes' fears. .. about how Nash would cope with the pomp in Stockholm proved groundless," Nasar wrote. "Everything went swimmingly." Read a brief autobiography of Nash on the Nobel Prize web site.

**December 1988:** **Everett Pitcher** retired as Secretary of the AMS. He was elected as Secretary in 1967 and served in that capacity for 22 years. "Professor Pitcher ends a 22-year tenure marked by a sincere belief in the value of the Society and a profound affection for the field of mathematics," says a December 1988 article in the *Notices of the AMS*, published on the occasion of his retirement as Secretary. "By all accounts, he consistently conducted the business of the Society with diplomacy, efficiency, and fairness." Pitcher was born in 1912 and received his PhD in 1935 from Harvard University, under the direction of Marston Morse. He joined the faculty at Lehigh University in 1938, retiring as Distinguished Professor Emeritus of Mathematics in 1978. Each spring Lehigh University sponsors the Pitcher Lectures, presented by an outstanding mathematician; a special lecture was held in July 2002 to mark his 90th birthday. Pitcher died on December 4, 2006, at the age of 94.[Read an obituary by Steven Weintraub in the November 2007 *Notices* and information on the Pitcher Lectures. ]

**December 4, 1984: Douglas Fuenteseca**, a mathematics instructor at the University of Antofagasta in Chile, was banished by government security forces. In November 1984, the government of General Agosto Pinochet had declared a state of seige, which invested the military with powers to arrest and detain people without judicial intervention. Among the many arrested were some scientists, engineers, and medical professionals. The arrests drew the attention of the Committee on Human Rights of the National Academy of Sciences, which sent a delegation to Chile in March 1985. The AMS expressed support for this visit and asked for information about cases of particular concern to society members. One of these was Fuenteseca, whose arrest was brought to the attention of the AMS Council by C. Herbert Clemens, a member of the AMS Committee on Human Rights. According to an NAS report, Fuenteseca collected money and established a fund to pay for breakfasts for students when the university dining hall was closed following student strikes. A few days after his arrest, he was banished to a village 650 miles from his home and, while blindfolded, made to sign and fingerprint a "confession". The Chilean Mathematical Society held a conference to show solidarity with Fuenteseca and collected funds to support him while he was in exile. Various mathematical organizations---including the AMS, the French Mathematical Society, the International Union, and the Latin American Federation of Mathematics---wrote letters of support. Fuenteseca was released in March 1985, a week before the visit of the NAS committee.

**December 1953: Friedrich Hirzebruch** completed his proof of a far-reaching generalization of the Riemann-Roch Theorem. The proof was announced in the *Proceedings of the National Academy of Sciences* in a paper communicated by Solomon Lefschetz on December 21, 1953. Originally proved in the 19th century, the Riemann-Roch Theorem uncovers a deep connection between the complex analysis of a Riemann surface or algebraic curve and its topological invariants. Hirzebruch extended the theorem from its original setting of algebraic curves to algebraic varieties of arbitrary dimensions. Known today as the Hirzebruch-Riemann-Roch Theorem, this landmark result and the methods Hirzebruch employed have had a profound impact on mathematics and theoretical physics over the past sixty years. Hirzebruch proved the theorem during a two-year stay at the Institute for Advanced Study (IAS) in Princeton, a place that provided part of the inspiration for his founding of the Max Planck Institute for Mathematics in Bonn, Germany, in 1980. The article "1953" by D. Kotschick, which appeared in the Spring 2013 issue of the *IAS Newsletter*, describes Hirzebruch's work in the context of other significant world events of 1953.

**December 1934:** The first meeting of the Bourbaki group was held. Nicolas Bourbaki is the pseudonym for a group of mathematicians (most of them French) who collaborated on writing mathematical books to provide modern tools for the working mathematician. The founders of Bourbaki were Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonne, and Andre Weil. Although the members of Bourbaki were outstanding mathematicians, their identities were kept secret and the individual members did not claim credit for the works the group produced. Contributing time and effort to a publication that does not bear one's own name is highly unusual in mathematics, and in science in general; indeed the Bourbaki group may be the only such example. Bourbaki had a profound impact on mathematics, especially in the 1950s and 1960s; later on the Bourbaki books were often criticized as being too abstract and formal. The group's impact was also felt through the Bourbaki Seminar, which has taken place in Paris since 1948. [See "Twenty-Five Years with Nicolas Bourbaki, 1949-1973," by Armand Borel, *Notices of the AMS*, March 1998.]

**December 1900:** On the 17th of that month, **Mary Lucy Cartwright** was born in Aynho, Northamptonshire, England. As a young woman she was especially interested in the study of history but decided to specialize in mathematics, as it required less memorization of facts. She entered Oxford University just as World War I was ending, when the university was overcrowded with young men returning from the war and starting or resuming their studies. She was one of only five women studying mathematics. Her love of the subject grew, and she earned a first class degree from Oxford and graduated in 1923. After four years teaching in schools, she returned to Oxford to write a PhD dissertation. G. H. Hardy started out as her adviser, but E. C. Titchmarsh took over when Hardy visited Princeton for a year. One of Cartwright's examiners was J. E. Littlewood, with whom she would collaborate later on. After earning her D.Phil. with a thesis on the zeros of integral functions, she held a research fellowship at Oxford. With the recommendation of Hardy and Littlewood, she obtained a lectureship at Cambridge University, where she remained for the rest of her academic career. Her research explored the sometimes strange behavior of complex functions, which is often depicted nowadays in images of fractal phenomena. Cartwright was ahead of her time in many ways. In an article to appear in the February 2009 issue of the *AMS Notices*, Freeman Dyson recalls a 1943 lecture by Cartwright in which she described the concept of chaos. She had perceived this crucial phenomenon decades before it was recognized by Edward Lorenz, who launched the study of chaos theory by identifying chaotic behavior in weather patterns. Cartwright received many honors in her lifetime, including election to the Royal Society, London, and receipt of its Sylvester Medal. She is the only woman to have served as president of the London Mathematical Society. In 1969 she became Dame Mary Cartwright, Commander of the Order of the British Empire. She died in 1998, at the age of 97. Read more about Cartwright in the MacTutor biography.

**December 1889**: On the first of this month, **Henri Poincaré** wrote a letter acknowledging a serious problem in his work on the *n*-body problem, for which he had received a prize from King Oscar II of Sweden. A suggestion by the Swedish mathematician **Gösta Mittag-Leffler** had led the king to announce that, for his sixtieth birthday, he would award a prize for a paper dealing with any of four mathematical topics. One was the *n*-body problem of celestial mechanics, which asks for a complete description of how *n* celestial bodies will move under the influence of their mutual gravitational attraction. In 1888, Poincaré submitted for the prize a 160-page paper that, while it did not solve the problem completely, constituted a major advance. The prize was duly awarded to Poincaré and plans were laid to publish his paper in *Acta Mathematica*, a journal founded by Mittag-Leffler. During the printing of the paper, **Lars Edvard Phragmén**, a young mathematical collaborator of Mittag-Leffler, found a mistake. Poincaré initially tried to correct it, but after giving the matter further thought realized his error was fundamental and his conclusion that the solar system is stable had been wrong. Mittag-Leffler recalled all the issues of *Acta* containing the erroneous paper, and in 1890 Poincaré paid for a corrected version to be printed. This corrected paper was the first to note the extreme sensitivity to initial conditions that is the hallmark of chaos. The story related here is summarized from "Henri Poincaré. A Life in the Service of Science", by Jean Mawhin, which appeared in the October 2005 issue of the *Notices of the AMS*.

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