Friedrich Bernhard Riemann




When I was a boy, Germany was thought of as the leading country in science, and it inspired many American universities. I recalled those days on reading John Gehl's bio of  the 19th-century German mathematician George Friedrich Bernhard Riemann (1826-1866), who made important contributions in many areas of mathematics, including differential geometry, complex variables, and analytic number theory. He laid the foundations for modern topology, and his 1854 paper "On the Hypotheses Which Lie at the Foundation of Geometry" has become a classic treatise, the results of which Albert Einstein incorporated into his relativistic theory of gravitation. When Riemann first expanded geometry to use n-dimensional space to deal with concepts like distance and curvature, he produced a geometry that might have been considered a fanciful exercise in pure mathematics, with little relation to practical reality. As it turned out, however, Riemannian non-Euclidian geometry would have useful applications in physics and his view of geometry as the general study of "curved spaces" provided Einstein with some of the mathematical tools he needed to construct his theory of relativity.

Riemann was born the second of six children of a Lutheran pastor, who gave him an excellent early education. Later, while attending the local Gymnasium he displayed a special talent for mathematics and to the amazement of his teachers mastered on his own calculus and Legendre's Theory of Numbers, topics that far exceeded the school's curriculum. Originally, Riemann intended to study theology and follow his father into the ministry, but his evident interest in mathematics persuaded him, with his father's kind approval, to abandon theology for math and science.

Beginning in 1846 he studied at the universities of Göttingen and Berlin, where he was interested in problems concerning the theory of prime numbers, elliptic functions and geometry. Following studies in experimental physics and natural philosophy, which sought to derive universal principles from all natural phenomena, he concluded that mathematical theory could secure a connection between magnetism, light, gravitation, and electricity, and he suggested field theories, in which the space surrounding electrical charges may be mathematically described. This was the beginning of his mature studies that would become so useful for the later development of modern mathematical physics.

After an interruption to serve in the Prussian army during the campaign to suppress the 1848 Revolution, Riemann earned a doctorate at Göttingen in 1851, based on a thesis that was warmly praised by the then elderly, renowned mathematician Carl Friedrich Gauss. In 1853 Riemann joined the faculty at Göttingen, becoming professor of mathematics there in 1859. Despite his delicate health, Riemann enjoyed a productive university career. In 1857 he became professor extraordinarius (associate professor) and in 1859 professor, succeeding the mathematician Peter Gustav Lejeune Dirichlet, who had succeeded Gauss four years earlier. Despite persistent health problems, Riemann produced a small but steady stream of papers, which contained his highly original ideas. He was only 39 years old when he died of tuberculosis in 1866. At the time of his death his friend, the famous mathematician Julius Dedekind, said of him: "The gentle mind which had been implanted in him in his father's house remained with him all his life, and he served his God faithfully, as his father had, but in a different way."

[See <http://www.amazon.com/exec/obidos/tg/detail/-/0677001401/newsscancom/ref=nosim> for S. Chowla's "The Riemann Hypothesis and Hilbert's Tenth Problem,"

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Ronald Hilton 2004

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last updated: October 8, 2004