Aleksandr Gelfond made significant contributions to the theory of transcendental numbers and the theory of interpolation and approximation of the functions of a complex variable. He established the transcendental character of any number of the form a:ssb:ks, where ais an algebraic number different from 0 or 1 and b is any irrational algebraic number, which is now known as Gelfond's theorem.
Gelfond was born in St. Petersburg (later Leningrad); his father was a physician who also dabbled in philosophy. Gelfond entered Moscow University in 1924 and completed his undergraduate degree in mathematics in 1927. He pursued postgraduate studies from 1927 to 1930 under the direction of A.J. Khintchine and V.V. Stepanov.
Gelfond's first teaching assignment was at the Moscow Technological College. He quickly won a more prestigious appointment at Moscow University, where he began teaching mathematics in 1931. He became a professor of mathematics in 1931, a position he held until his death. For several years, Gelfond served as the chairman of the mathematics department specializing in the theory of numbers. His enthusiasm for the history of mathematics was reflected not only by his own works on Leonhard Euler, but by his incorporating a "history of mathematics" division into the theory department he chaired.
In 1933, Gelfond was also appointed to a post in the Soviet Academy of Sciences Mathematical Institute. He completed a doctorate in mathematics and physics in 1935 and was elected a corresponding member of the Academy of Sciences of the U.S.S.R. in 1939.
Gelfond found his greatest inspiration in the past. In 1748, Euler proposed that logarithms of rational number with rational bases are either rational or transcendental; in 1900, David Hilbert developed a 23-problem series on the rationality or transcendence of the logarithms of such numbers. For three decades, mathematicians were unable to trace a solution to the puzzle posed by Hilbert's seventh problem--the assumption that a:ssb:ks is transcendental if a is any algebraic number other than 0 or 1 and b is any irrational algebraic number.
In 1929, Gelfond established connections between the properties of an analytic function and the arithmetic nature of its values, publishing his first paper on the topic, "Sur les nombres transcendant," in 1929. He built on this discovery to unravel Hilbert's seventh riddle by using linear forms of exponential functions. Gelfond published the results of his work, "Sur le septieme probleme de Hilbert," in 1934. He continued his explorations, using his knowledge of functions to develop theorems related to rational integers, transcendental numbers (he was able to construct new classes in this area), mutual algebraic independence, and analytic theory.
Gelfond's interest in function theory was probably shaped by the Luzitania--an informal academic and social organization clustered around Nikolai Nikolaevich Luzin, a noted mathematician in the 1920s. Gelfond was a contemporary and colleague of Nina Karlovna Bari, a Luzin protegee; although Gelfond's name does not appear on the list of those who declared themselves Luzitanians. Luzin's prominence and the intellectual vigor of the students he attracted influenced the philosophy and direction of mathematics at the university. By 1930, the Luzitania movement sputtered and died, and Luzin left Moscow State for the Academy of Science's Steklov Institute.
In 1936, during the dictatorship of Josef Stalin, Luzin was charged with ideological sabotage. Luzin's trial was abruptly and surprisingly canceled, but he was officially reprimanded and withdrew from academia. Luzin's fall demonstrated--in a way that could not be ignored--the inextricable interweaving of politics and academic achievement. Gelfond was permitted to pursue his studies in peace in part because of his political connections.
"He was a member of the Communist Party," wrote Ilya Piatetski-Shapiro, for whom Gelfond was an instructor, mentor and advisor. "His father was personally acquainted with Lenin . . . he said that his father and Lenin had disagreements in public life, but in private life they were friends. Being a member of the Communist Party, Gelfond felt that he had some influence. . . ." Such influence could not overcome the deep wave of anti-Semitism that swept over Russia after World War II. Despite Gelfond's recommendation, Piatetski-Shapiro, who was Jewish, was denied admission to Moscow University's graduate school by the party committee of the mathematics department.
But Gelfond "was a very warm person, very humane and sensitive to me and to the other students," Piatetski-Shapiro wrote, and Gelfond was reluctant to let a promising student--winner of the Moscow Mathematical Society award for young mathematicians--languish. Although his sponsorship could have had dire implications for his own career, Gelfond persisted, and finally secured admission to the graduate program for Piatetski-Shapiro at the Moscow Pedagogical Institute.
Gelfond's most comprehensive publications were released in 1952. Transtsendentnye i albegraicheskie chisla provided an overview of his work in transcendental numbers, and his work on the theory of the functions of a complex variable is compiled in Ischislenie knoechnyko raznostey.
In 1968, Gelfond was named a corresponding member of the International Academy of the History of Science. He also served as chair of the scientific council of the Soviet Academy of Sciences Institute of the History of Science and technology, which refereed works on the history of physics and mathematics.
Gelfond's drive to expand the understanding of mathematics theory persisted to the day of his death. "When he died . . . I was present in the hospital," wrote Piatetski-Shapiro. "I remember he was trying to write some formula and tell me something which was clearly related to the zeta function. He could not because he was already paralyzed."
Gelfond died in Moscow; most sources list the year of his death as 1968, but Piatetski-Shapiro records it as 1966.
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