Israel Moiseevich Gelfand, Part I Vladimir Retakh, Coordinating Editor Israel Moiseevich Gelfand, a age twelve Gelfand understood that some problems mathematician compared by in geometry cannot be solved algebraically and Henri Cartan to Poincaré and drew a table of ratios of the length of the chord to Hilbert, was born on Septem- the length of the arc. Much later it became clear ber 2, 1913, in the small town for him that in fact he was drawing trigonometric of Okny (later Red Okny) near tables. Odessa in the Ukraine and died From this period came his Mozartean style in New Brunswick, New Jersey, and his belief in the unity and harmony of USA, on October 5, 2009. mathematics (including applied mathematics)— Nobody guided Gelfand in the unity determined not by rigid and loudly his studies. He attended the proclaimed programs, but rather by invisible only school in town, and his and sometimes hidden ties connecting seemingly mathematics teacher could different areas. Gelfand described his school years offer him nothing except and mathematical studies in an interview published I. M. Gelfand encouragement—and this was in Quantum, a science magazine for high school very important. In Gelfand’s students [1]. In Gelfand’s own words: “It is my deep own words: “Offering encouragement is a teacher’s conviction that mathematical ability in most future most important job.” In 1923 the family moved professional mathematicians appears…when they to another place and Gelfand entered a vocational are 13 to 16 years old…. This period formed my school for chemistry lab technicians. However, style of doing mathematics. I studied different he was expelled in the ninth grade as a son of subjects, but the artistic form of mathematics that a “bourgeois element” (“netrudovoi element” in took root at this time became the basis of my taste Soviet parlance)—his father was a mill manager. in choosing problems that continue to attract me After that Gelfand (he was sixteen and a half at to this day. Without understanding this motivation, that time) decided to go to Moscow, where he had I think it is impossible to make heads or tails of some distant relatives. the seeming illogicality of my ways in working Until his move to Moscow in 1930, Gelfand lived and the choice of themes in my work. Because of in total mathematical isolation. The only books this motivating force, however, they actually come available to him were secondary school texts and together sequentially and logically.” several community college textbooks. The most ad- The interview also shows how a small provincial vanced of these books claimed that there are three boy was jumping over centuries in his mathematical kinds of functions: analytical, defined by formulas; discoveries. At the age of fifteen Gelfand learned empirical, defined by tables; and correlational. Like of a series for calculating the sine. He described Ramanujan, he was experimenting a lot. Around this moment in the “Quantum” interview: “Before this I thought there were two types of mathematics, Vladimir Retakh is professor of mathematics at Rutgers Uni- algebraic and geometric…When I discovered that versity. His email address is [email protected]. the sine can be expressed algebraically as a series, He thanks Mark Saul for his help in the preparation of this the barriers came tumbling down, and mathematics collection. became one. To this day I see various branches of The AMS wants to thank M. E. Bronstein, Elena Ermakova, mathematics, together with mathematical physics, Tatiana I. Gelfand, Tatiana V. Gelfand, and Carol Tate for as parts of a united whole.” their help with photographs for this article. After arriving in Moscow, Gelfand did not DOI: http://dx.doi.org/10.1090/noti937 have steady work and lived on earnings from 24 Notices of the AMS Volume 60, Number 1 occasional odd jobs. At some point he had the stabilized, anti-Semitism be- good fortune to work at the checkout counter at came part of the Soviet the State (Lenin’s) Library. This gave Gelfand a rare system, and Gelfand be- opportunity to talk with mathematics students came a full member of from Moscow University. He also started to attend the Soviet Academy only in university seminars, where he found himself under 1984 after being elected to intense psychological stress: new breezes were leading foreign academies. blowing in mathematics with the new demands In 1989 Gelfand moved for rigorous proofs. It was so different from his to the United States. Af- “homemade” experiments and his romantic views ter spending some time of mathematics. He also learned that none of at Harvard and MIT, he his discoveries were new. Neither this nor other became a professor at Rut- circumstances of his life deterred him, and his gers University, where he interest in mathematics continued to grow. worked until his death. Gelfand’s parents. Just as his abrupt expulsion from school, the The articles in the Notices next twist in Gelfand’s fate was also one of many present various descriptions paradoxes of life in the Soviet Union. On one hand, of Gelfand’s multifaceted re- as a son of a “bourgeois element” he could not be a search, his way of doing university student. On the other hand, at eighteen mathematics, and interaction he was able to obtain a teaching position at one with people. A leading Amer- of many newly created technical colleges and at ican expert once told me that nineteen to enter the Ph.D. program at Moscow after reading all definitions University. The reasons were simple: the Soviet in Gelfand’s papers, he could state needed knowledgeable instructors to educate easily prove all his theorems. its future engineers and scientists of the proper Well, this easiness was based “proletarian origin”. But at that time the system on long computations and a was not rigid enough to purge or even strictly thorough consideration of a regulate graduate schools. As a result, a talented variety of carefully selected boy was able to enter a Ph.D. program without a examples. Gelfand himself college or even a high school diploma. liked to repeat a statement At the beginning of his career Gelfand was by a Moscow mathematician: influenced by several Moscow mathematicians, “Gelfand cannot prove hard especially his thesis adviser, A. N. Kolmogorov. theorems. He just turns any Gelfand at age 3, 1916. In the Quantum interview Gelfand said that from theorem into an easy one.” Kolmogorov he learned “that a true mathematician Gelfand always was surrounded by numerous must be a philosopher of nature.” Another influence collaborators attracted by his legendary intuition was the brilliant L. G. Shnirelman. In 1935 Gelfand and the permanent flow of new ideas: in every defended his “candidate” (Ph.D.) thesis and in 1940 decade he was establishing a new area of research. obtained the higher degree of Doctor of Science. He had completely different approaches to different In 1933 he began teaching at Moscow University, people, as described by A. Vershik, A. Zelevinsky, where he became a full professor in 1943 and and me in our recollections. His students and started his influential seminar. He lost this position collaborators represent an unusual variety of styles temporarily in 1952 during the infamous “anti- and interests. The only similarity for members of cosmopolitans” (in fact, anti-Semitic) campaign but the Gelfand school was their inherited passion for was allowed to continue the seminar. Gelfand also mathematics. worked at the Steklov Institute and for many years One cannot write about Gelfand without men- at the Institute for Applied Mathematics. There tioning his legendary seminar, which started in he took part in the secret program related to the 1943 and continued until his death. Gelfand con- Soviet version of the Manhattan Project and its sidered the seminar one of his most important extensions. Andrei Sakharov mentioned his work creations. It is hard to describe the seminar in a few with Gelfand in [1]. words: it was a “mathematical stock exchange”, a In 1953 Gelfand was elected a Corresponding breeding ground for young scientists, a demonstra- Member of the Academy of Science (an important tion of how to think about mathematics, a one-man title in the Soviet hierarchy). This happened right show, and much more. It was not about functional after Stalin’s death and at the end of the anti- analysis or geometry, it was about mathematics. cosmopolitans campaign. According to Gelfand, Some would come to the seminar just to hear the political uncertainty of the times made his Gelfand’s jokes and paradoxes. For example, after election possible. Later the situation in the USSR his visit to the U.S. he stated, “[The] Mathematical January 2013 Notices of the AMS 25 world is not a metric space: recollections by Sergei Tabachnikov in the second The distance from Harvard to part of this article). Gelfand founded, ran, and MIT is greater than the sum wrote several textbooks for the school. His uni- of distances from Harvard to versity textbooks Linear Algebra and Calculus of Moscow and from Moscow to MIT.” Variations (written with S. Fomin) also bear the One should add that sometimes imprint of his style and personality. Gelfand’s jokes were rather sharp, It is hard to describe all of Gelfand’s achieve- but for a young person to become ments in mathematics. He left his unique and a subject of Gelfand’s joke meant powerful imprint everywhere (excluding, probably, to be noticed, to be knighted. The mathematical logic). Some (but far from all) of seminar was also the right place to his breakthroughs are described here by Simon find out about fresh preprints com- Gindikin, David Kazhdan, Bertram Kostant, Peter ing from the inaccessible West.