#
This Mathematical Month - June: 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.**

< **May** o **July** >
**Featured Item for June**
**June 1993:** At a conference at the Isaac Newton Institute, **Andrew Wiles** gave his first ever lecture on his proof of Fermat's Last Theorem. The institute buzzed with rumors that his lecture would contain a big surprise, but few realized just how big it would be. Using a regular chalkboard, Wiles sketched his proof for the audience of experts, who burst into explosive applause when he came to the end and concluded that he had proved Fermat's Last Theorem. It was a few months later that a gap appeared in the proof, and it wasn't until sometime after that that the proof was finally complete. But many remember Wiles's historic lecture as a high point for mathematics.

**June 1993:** From the 16th to the 19th of that month, the then-newly-formed Palestinian Society of Mathematical Sciences held its first general meeting 16-19 June 1993 in Jerusalem and Ramallah. International participation was invited. The AMS Council sent the following greetings: "The Council of the American Mathematical Society welcomes the formation of the Palestine Society of Mathematical Sciences and extends congratulations to its inaugural meeting. It hopes for many years of fruitful cooperation between the societies in the furtherance of mathematical research, education, and scientific exchange." The Palestinian Society of Mathematical Sciences is a reciprocity member of the AMS. Professor Marwan Awartani of Birzeit University, was the founding president of the Palestinian Society for Mathematical Sciences and the founding chairman of the Palestinian Mathematical Olympiad.

**June 1992:** The Fields Institute in Canada opens its doors. Originally established on the campus of the University of Waterloo, the Fields Institute is now located at the University of Toronto. It has become one of the world's major mathematics institutes, with an active and diverse program across all areas of mathematics. In 2004, **Barbara Lee Keyfitz** was named director of the Fields Institute, the first woman to serve as director of a major mathematics institute (the only possible exception is **Cathleen Morawetz**, who in the 1980s was director of the Courant Institute of Mathematical Sciences at New York University; however, Courant differs from a national mathematics institute in that it functions more like a school of mathematics in offering courses and awarding degrees). "The idea of refreshing the stream of research in Canada by interacting with other countries is an important theme at this institute," Keyfitz remarked. Read about the founding of the Fields Institute in "Canada's Fields Institute Opens Its Doors," *Notices*, July/August 1992, and about Keyfitz's appointment in "Keyfitz Named Director of Fields Institute," *Notices*, September 2004.

**June 1992:** The Isaac Newton Institute celebrated its inauguration. **Sir Michael Atiyah**, founding director of the Newton Institute, said at the time: "I think the view is at the present time that a lot of the future development of mathematics will probably be to use more advanced mathematics in related fields and to bring problems from other fields into mathematics. So our aim is really to bridge the gap by bringing people together." Since then, it has become one of the world's main institutes for research in pure and applied mathematics. Visit the Newton Institute's web site.

**June 1937:** On the 12th of that month, **Vladimir Arnold**, one of the most brilliant and influential mathematicians of the 20th century, was born in Odessa. He came from a long line of scientists, and his interest in and talent for mathematics showed early. In 1954 he began studying in the legendary Faculty of Mathematics and Mechanics (known as Mekh-Mat) at Moscow State University, joining a group of brilliant students---among them **Yuri Manin, Yakov Sinai,** and **Sergei Novikov**---who were taught by extraordinary teachers. Arnold was awarded the equivalent of a PhD in 1961 with a thesis that was written under the direction of **Andrei Kolmogorov** and that contained a solution of Hilbert's 13th Problem. In 1965 he became a professor at Mekh-Mat and then in 1986 took up the position of Principal Researcher at the Steklov Institute of Mathematics in Moscow. In addition to his Russian positions, in 1993 he was appointed professor at the University Paris-Dauphine, a position he held until 2005. Arnold became well-known early on for his work in Hamiltonian dynamics---he was one of the founders of KAM (Kolmogorov-Arnold-Moser) Theory and was the discoverer of Arnold diffusion. But his work stretches across an almost unbelievable variety of fields, including differential equations, symplectic geometry, real algebraic geometry, the calculus of variations, hydrodynamics, and magnetohydrodynamics. Arnold's strong and thought-provoking views have been the subject of many articles and books; see for example the article in the April 1997 issue of the *Notices of the AMS*, in which Arnold is interviewed by S. H. Lui. "Our brain has two halves: one is responsible for the multiplication of polynomials and languages, and the other half is responsible for orientation of figures in spaces and all the things important in real life," Arnold says at the end of the interview. "Mathematics is geometry when you have to use both halves." See also the entry about Arnold on the MacTutor History of Math web site. Arnold died on June 3, 2010; an obituary appeared in the *New York Times*.

**June 1912: Alan M. Turing** was born in London. He died in 1954, not long before his 42nd birthday. Turing is widely considered to be one of the greatest intellects of the twentieth century. He established the basis for modern computers and originated what is now called the "Turing machine," a mathematical model of an all-purpose computer. Much of his thinking about computing anticipated problems that became important later on; one example is the well known "Turing test" that bears his name. He played a decisive role in helping the British to break German codes during World War II and for this work he received an O.B.E. After the war, Turing continued to work in computing and codes, and his interests expanded to include pattern formation and morphogenesis. He had all his life been openly homosexual, and in 1952 he was arrested and found guilty of homosexuality. Because of his work on code-breaking, he had high-level clearance, but this was withdrawn after his conviction. In June 1954, he was found dead of cyanide poisoning. The poison was in an apple he had been eating. The definitive work about Alan Turing is the compelling and highly readable *Alan Turing: The Enigma* by Andrew Hodges (1983). Hodges has created an extensive web site about Turing's life and work. During 2012, many events took place around the world to mark the centenary of Turing's birth; see for example 2012 The Alan Turing Year and A.M. Turing Centenary Celebration of the Association for Computing Machinery. The *AMS Notices* carried two articles to mark the occasion, "Why Are There No 3-Headed Monsters? Mathematical Modeling in Biology", by J. D. Murray, and "Incomputability after Alan Turing", by S. Barry Cooper

**June 1911:** A review of Luther Eisenhart's book *A Treatise in Differential Geometry* appeared in the *Bulletin of the AMS*. The reviewer was Gilbert Ames Bliss, who, like Eisenhart, served as President of the AMS. This book had an impact in the United States by introducing American students to modern methods in an important area of research. In his review, Bliss wrote: "The writer of this review has found the book an exceedingly useful one to have in the hands of students in a course in differential geometry... In these days of the popularity and elegant methods of the theory of functions of a real variable, it is interesting to note that much of the theory of surfaces can be developed without the use of imaginaries, and to see also that the existence and uniqueness theorems for real differential equations can be applied with economy in many places." Eisenhart's book remains in print as a Dover publication. The AMS makes all issues of the Bulletin freely available to all mathematicians through the generosity of its members. The full review is available on the *Bulletin* web site.

**June 1903**: **Sir W. V. D. Hodge** was born in Edinburgh on the 17th of that month. He was one of the outstanding geometers of the 20th century and had a profound influence on the field. Spending most of his career at Cambridge University, he did a great deal to revitalize geometry at that institution and indeed throughout the United Kingdom. Hodge was a pioneer of the application of analytic methods in geometry. His theory of harmonic forms, now commonly called Hodge theory, brought the analysis of the Laplace equation to bear on questions of the topology and geometry of manifolds. These ideas have since been used with great success in many different branches of geometry but have had their most profound impact in algebraic geometry, where they have spawned new concepts surrounding for example the so-called Hodge Conjecture. This conjecture is one of the six Millennium Prize Problems for which the Clay Mathematics Institute has offered a prize of US$ 1 million apiece for their solutions. Many of the most celebrated advances in the mathematics of the second half of the 20th century, such as the index theory of elliptic operators and the applications of gauge theory in topology, have roots in Hodge theory. Some of these advances were carried out by Hodge's mathematical descendents, in particular **Michael Atiyah** and Atiyah's student **Simon Donaldson**, both of whom received Fields Medals. Hodge died in 1975 at the age of 72. An unsigned {\it Times of London} obituary called him "jovial, informal and down-to-earth". Among Hodge's many honors were election to the Royal Society of London and of Edinburgh, election to the U.S. National Academy of Sciences, the Royal Medal, the Copley Medal, and several honorary doctorates. He was knighted in 1959. Read more about Hodge in the biography on the MacTutor History of Mathematics website.

**June 24, 1880: Oswald Veblen** was born in Decorah, Iowa. A gifted mathematician who produced influential works, Veblen also had a large impact on the development and organization of mathematical and scientific research in the United States. He received his Ph.D. in mathematics in 1903 from the University of Chicago, under the direction of E. H. Moore. Veblen's doctoral thesis, *A System of Axioms for Geometry*, clarified aspects of David Hilbert's then-recent work on the foundations of geometry. After two postdoctoral years at Chicago, Veblen took a position at Princeton University, where his mathematical reputation steadily grew and he was made a professor in 1910. Among his most important contributions are the first rigorous proof of the Jordan Curve Theorem and his efforts to make the work of Henri Poincaré accessible to the pioneers of algebraic topology. After the start of the first World War, Veblen was commissioned as an army reserve captain. As the head of experimental ballistics at the Aberdeen Proving Ground in Maryland, he made important contributions to that area. After the war, Veblen worked hard at fund-raising for the support of mathematical research, in particular bringing in funds to rescue the AMS from a budget shortfall at a crucial time in the Society's history. Veblen served as AMS President during 1923-1924. Perhaps his most notable work in nurturing scientific research was his service as first director of the Institute for Advanced Study, which was founded in 1933 with a faculty consisting of, in addition to Veblen as director, James Alexander, Albert Einstein, and John von Neumann; shortly thereafter Hermann Weyl joined the faculty as well. Veblen's early stewardship of the Institute was crucial in making it the world-renowned scholarly center it is today. Much of this summary of Veblen's life comes from "The Vision, Insight, and Influence of Oswald Veblen" by Steve Batterson, which appears in the May 2007 issue of the *AMS Notices*. Veblen was, as Batterson puts it, a "statesman of mathematics"---an individual who had an important impact through his own work as a mathematician, through his mentorship of young people, and through his advocacy for the field.

< **May** o **July** >