World of Scientific Discovery on Georg Friedrich Bernhard Riemann
Riemann was born on September 17, 1826 at Breselenz, Hanover, Germany, the son of a Lutheran minister. He demonstrated a remarkable talent for mathematics at an early age. While still a child, he is said to have tried to demonstrate the validity of the stories contained in the Book of Genesis by mathematical means. When Riemann enrolled at Göttingen in 1846, he planned to study theology and philology. However, he also attended course in and eventually asked for his father's permission to concentrate on the field he loved most, mathematics. He stayed at Göttingen for only one year before transferring to the University of Berlin, which had a stronger program in mathematics than did Göttingen. In 1849 he returned to Göttingen, however, and eventually received his Ph.D. degree there two years later. His thesis dealt with the theory of complex functions.
For two years, Riemann served as an unpaid lecturer at Göttingen. Then, in 1854 he prepared to give a demonstration lecture to senior faculty members that would earn him a paid position. The most senior member of Riemann's committee was Carl Gauss, who asked Riemann to speak on the last of the three topics proposed by the candidate. That topic, non-Euclidean geometry, was one about which Gauss himself had been thinking for many years. One biographer has described the "unusual emotion" with which Gauss praised Riemann's presentation of this topic. Riemann's lecture did not appear in print until 1867, a year after his death. When it did, it slowly became one of the most important papers on the subject ever written. Riemann's approach was to develop a global system of geometry that did not necessarily deal with points, lines or space in the ordinary sense. As one example, he briefly discussed a "spherical" geometry in which every line through a point not on a given line meets that given line. This condition violates Euclid's fifth postulate, namely that through a point external to a line, one and only one parallel can be drawn to that line. In Riemann's example, no parallel lines are possible. Riemann went beyond just describing a geometry on the sphere, however, and his discussion of a general study of curved space made possible Einstein's theory of general relativity.
Riemann made contributions in a number of other fields of mathematics during his short lifetime. For example, he followed up on the work of Niels Abel, Jacob, and Augustin-Louis Cauchy in the area of complex analysis. His method for the solution of some of these problems are now known as the Cauchy-Riemann equations. He also clarified the concept of the integral by defining what is now called the Riemann integral. The Riemann hypothesis about the zeros of the zeta function is still one of mathematic's famous unsolved problems.
Riemann became full professor at Göttingen in 1859. He soon became seriously ill with consumption (tuberculosis), however, and spent the last years of his life travelling between Göttingen and Italy, where he could take advantage of the better climate. His condition continued to deteriorate, however, and he died in Selasca, on the shores of Lake Maggiore in Italy on June 16, 1866. Although he lived less than forty years, he made a great impact on mathematical thought and has been described by a biographer as "one of the most profound and imaginative mathematicians of all time."
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