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This Mathematical Month - October: 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 October **
**October 1811:** On the 25th of that month, **Evariste Galois** was born. The brief and torrid life of this extraordinary man has become the stuff of legend---and indeed many accounts of his life contain a liberal dose of fiction. He was born near Paris during the period of political upheaval that followed the French Revolution. He showed exceptional promise in mathematics while he was a student at the Ecole Normale Supérieure, and he published his first mathematics paper at the age of 17. Galois continued to do highly original research in mathematics despite being imprisoned for political activities and despite absorbing the devastating blow of the death of his father by suicide. Galois gave birth to the notion of a *group*, which has become fundamental in mathematics. Legend has it that Galois worked out the entirety of *group theory*, now a major branch of the field, the night before his death in a duel. This is an exaggeration; he had been developing the theory in the years before his death, and, the night before the duel, he wrote an outline of the theory in a letter to his friend Auguste Chevalier. There have also been incorrect accounts of the reasons for the duel; the circumstances were complex and are not entirely understood but they were related to a romantic attachment. After Galois's death on 31 May 1832, Chevalier, together with Galois's brother, collected and recopied Galois's mathematical papers. These eventually came into the hands of Joseph Liouville, who became convinced of their correctness and published them. Galois solved a problem that had perplexed mathematicians, namely, the question of why there are formulas for solving polynomial equations of degree two, three, and four, but no such formulas for equations of degree five and higher. His work on this question led to Galois Theory, which elucidates connections between group theory and field theory. In his article that won the Lester R. Ford Award from the Mathematical Association of America, Tony Rothman attempts to sort out the facts and legends in accounts of the life of Galois. He writes: "The fact that he could work through such a turbulent life is testimony to the extraordinary fertility of his imagination. There is no question that Galois was a great mathematician who developed one of the most original idea in the history of mathematics. The invention of legends does not make him any greater." Rothman's article, which originally appeared in 1982 in the *American Mathematical Monthly*, is available on his web site.

**October 2005:** The AMS hosts the first *Einstein Public Lecture in Mathematics*, given on October 21, 2005, at the AMS Sectional Meeting in Lincoln, Nebraska. The speaker is **Sir Michael Atiyah**, 1966 Fields Medalist and a renowned expositor known for his breadth of vision and clarity of style. The lecture, called "The Nature of Space", examines the efforts of mathematicians, philsophers, and physicists working over more than 2000 years to come to grips with the nature of space. This lecture contributes to a worlwide celebration of the 100th anniversary of Albert Einstein's *annus mirabilis*, the year 1905 when he wrote three fundamental papers that changed the course of modern physics. James G. Arthur of the University of Toronto, AMS President for 2005-2006, initiated the idea of public lectures hosted by the AMS, as a way of raising public awareness of mathematics. Read the biography of Atiyah at the MacTutor web site, and an interview with James Arthur in the *Notices of the AMS*.

**October 1994:** Emails began to circulate among mathematicians about a stunning new breakthrough in topology spurred by an observation of theoretical physicists **Edward Witten** and **Nathan Seiberg**. More than a decade before, **Simon Donaldson** spurred great interest in gauge theory when he discovered deep connections between four-dimensional topology and Yang-Mills theory from mathematical physics. The so-called Seiberg-Witten equations simplified many of the massive technical difficulties posed by Donaldson theory and led to important new insights. The equations had actually been around for a while, but it took the insight of Seiberg and Witten to understand the equations' potential impact in gauge theory. Many mathematicians kicked themselves for not having seen the connection before. [See "Gauge Theory is Dead!---Long Live Gauge Theory!" *Notices of the AMS,* March 1995].

**October 1990:** The European Mathematical Society (EMS) was founded. Sir Michael Atiyah was one of the main initiators, and Friedrich Hirzebruch served as the first EMS president. The purpose of the EMS is the development of all aspects of mathematics in the countries of Europe, particularly the promotion of research in mathematics and its applications. The EMS membership base consists of European mathematical societies as well as individual members. The society has helped to foster mathematical activity within Europe, primarily through meetings and publications. The main EMS meeting, the European Congress of Mathematics, was first held in Paris in 1992, and ECMs have been held every four years since then. The ECM is the occasion for awarding the prestigious EMS Prizes, which honor outstanding work by young European mathematicians. The EMS Article Competition is intended to raise public understanding and awareness of mathematics by honoring articles written by mathematicians and intended for a wide audience. The EMS Publishing House is the publications arm of the society, producing books and journals as well as the membership publication the *EMS Newsletter*. More information may be found on the EMS web site.

**October 1927: Friedrich Hirzebruch** was born on 17 October 1927, in Hamm, Germany. An outstanding researcher with an international reputation, Hirzebruch was a key figure in revitalizing mathematics in postwar Germany. He received his PhD in 1949 from the University of Münster under the direction of Heinrich Behnke. Another major influence was Heinz Hopf, whom Hirzebruch encountered while visiting the Eidgenössisches Technische Hochschule in Zurich. During 1952-54, Hirzebruch visited the Institute for Advanced Study in Princeton (1952-54), where he came into contact with other leaders in algebraic geometry and topology, the subjects in which he has made his mark. His major results include the signature theorem and the Riemann--Roch theorem, which were an important influence on and starting point for Michael Atiyah and Isadore Singer in their development of the general index theorem for elliptic operators on manifolds. Hirzebruch's groundbreaking monograph on topological methods in algebraic geometry is still widely read, decades after its publication. Together with Atiyah and Alexander Grothendieck, Hirzebruch is one of the architects of K-Theory, which has grown into an important subject in its own right, straddling the fields of algebra, geometry, and topology. In 1957, having settled at the University of Bonn, Hirzebruch started an annual meeting called the *Arbeitstagung*, which has a unique structure and which quickly established itself as an important venue for presenting new research. In 1980, he founded the Max Planck Institute for Mathematics in Bonn, which has since developed into one of the world's main mathematical research centers. Among Hirzebruch's many honors are the Wolf Prize (1988) and the Cantor Medal of the German Mathematical Society (2004). Hirzebruch died on May 27, 2012; an obituary appeared in the *New York Times* as well as in newspapers in Germany. A video interview with Hirzebruch is available on the Science Lives web site of the Simons Foundation. The entry about Hirzebruch provides further details about his life and work.

**October 1911:** On the 26th of that month, **Shiing-Shen Chern** was born in Jiaxing, China. When he was a youngster, China was just starting to establish western-style universities. Chern entered Nankai University at the age of 15 and attended graduate school at Tsinghua University. In 1934, he obtained a three-year scholarship to study in the west, and he decided to go to Hamburg. He completed his PhD in two years and spent the third year of his scholarship in Paris, working with the great geometer Elie Cartan. After his return to China, Chern was quite isolated, but he studied reprints that Cartan had sent him, and he continued to publish. His work drew international attention, and in 1943 he was invited to the Institute for Advanced Study in Princeton. It was there that he completed his proof of the generalized Gauss-Bonnet theorem, which marries local geometry to global topological invariants. In an interview with the *AMS Notices*, Chern spoke of the differential geometry of fiber bundles, which led to his proof of Gauss-Bonnet, as being what he considered to be his most important work. Among his other outstanding achievements is the discovery of what are now known as Chern characteristic classes in fiber spaces, which have proved to be of great importance not only in mathematics but also in mathematical physics. In 1949 Chern became a professor at the University of Chicago and later moved to the University of California, Berkeley, where he was founding director of the Mathematical Sciences Research Institute. After his retirement he returned to China and was a major figure in building mathematical research there, founding the Nankai Institute for Mathematics in Tianjin in 1985. Chern died at his home in Tianjin on December 3, 2004, at the age of 93. A memorial article for Chern appears in the October 2011 issue of the *AMS Notices*.

**October 1900: Charlotte Angas Scott** wrote a report about the 1900 International Congress of Mathematicians that appeared in the *Bulletin of the AMS* later that year. In the report, Scott described many aspects of the Congress, including David Hilbert's seminal lecture in which he presented 10 problems from his list of 23 problems that would go on to have a major impact on the direction of research in mathematics. The Congress had sections on history and teaching in which there was discussion, by impassioned proponents and detractors, of the idea of promoting Esperanto as the language in which scientists should communicate. Scott describes a couple of the mathematical talks, including one by Gosta Mittag-Leffler, which reported on work presented in papers he published in two volumes of *Acta Mathematica* in 1899 and 1900. Mittag-Leffler was also one of the speakers at the closing session; the other was Henri Poincaré, who delivered his now-famous talk on the role of intuition and logic in mathematics. At the end of the report Scott presented some of her own opinions about the Congress. For example, she wrote: "One thing very forcibly impressed on the listener is that the presentation of papers is usually shockingly bad." Some lecturers read from printed materials in a bored monotone; some enunciated their native languages so poorly that they were very difficult to understand (the languages of the Congress were French, English, German, and Italian). Scott pointed to Mittag-Leffler's section lecture as an example of how good a lecture can be. "It is not given to everyone to do it with this charm," she admitted, "but there is no excuse for any normally constituted human being, sufficiently versed in mathematics, failing to interest a suitable audience for a reasonable time in that which interests himself, always provided that it be of sufficient novelty either in matter or in mode of treatment to justify him in presenting it at all." Click here to read Scott's report.

**October 1815: Karl Theodor Wilhelm Weierstrauss** was born on October 31 in Ostenfelde, Westphaslia (now Germany). Although his father arranged for him to study finance and planned on his working in the Prussian administration, Weierstrauss was always torn between the wishes of his father and pursuing the subject he loved--mathematics. He studied the subject on his own, while qualifying to become a teacher and holding various jobs. Upon publication of his paper "Zur Theorie der Abelischen Functionen" in *Crelle's Journal* in 1854, mathematicians took notice. The University of Königsberg awarded him an honorary doctor's degree in 1854, after which he became a lecturer, published a full version of his theory of inversion of hyperelliptic integrals, and received offers of positions at several universities. He eventually accepted the position of professor then chair at the University of Berlin. When Sofia Kovalevskaya came to Berlin he taught her privately as she was not allowed admission to the university. (She went on to receive an honorary doctorate from Göttingen and the two corresponded for 20 years.) He edited the complete works of Jakob Steiner and Carl Gustav Jacob Jacobi, and is widely regarded as the father of modern analysis. For more details on his life and work, see his biography on the MacTutor website.

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