Bernhard Riemann

German mathematician

• Born: September 17, 1826; Breselenz, German Confederation
• Died: July 20, 1866; Selasca, Italy

Nineteenth-century German mathematician Bernhard Riemann built on the work of German mathematician Gottfried Leibniz and Swiss mathematician Leonhard Euler to define position mathematically, as a corollary to defining magnitude algebraically. Riemann’s algebraic functions and Riemannian geometry laid the foundations of the field of topology, central to physics, quantum theory, and general relativity.

Primary field: Mathematics

Specialty: Geometry

Early Life

Georg Friedrich Bernhard Riemann was born on September 17, 1826 in the Hanoverian village of Breselenz (now Jameln, Germany). His father, a Lutheran pastor, had two sons and four daughters with Riemann’s mother, Charlotte Ebell. When Riemann was young, his mother and several other members of his family contracted tuberculosis and died. Bernhard started studying arithmetic around the age of six and quickly became a prodigy. He was homeschooled and began developing his own mathematical problems. At age ten, Riemann’s father arranged for a tutor to instruct his son in more advanced math and geometry.

At fourteen, Riemann left home for the first time to live with his grandmother in Hanover, where he attended school for two years. At sixteen, his grandmother died, and Riemann moved to Lüneburg to continue his secondary education.

Riemann’s father wanted him to become a preacher, and Bernhard studied Hebrew in Lüneburg. An outstanding student of mathematics, he was excused from regular math classes and invited to read from the instructor’s library. He read mathematician Adrien-Marie Legendre’s 859-page Theory of Numbers in six days. He mastered Legendre’s work and became interested in prime numbers. The so-called Riemann’s hypothesis, a problem he developed later and that mathematicians have yet to solve, stems from Legendre’s work concerning how many prime numbers exist. Bernhard also studied Euler’s calculus. At nineteen, he went to study theology and philology at the University of Göttingen. His passion for numbers drew him to mathematical lectures in equation theory, definite integrals, terrestrial magnetism, and Carl Friedrich Gauss’s least squares. Bernhard eventually gained his father’s permission to forego becoming a preacher to study mathematics in Berlin.

Life’s Work

For two years in Berlin, Riemann studied modern mathematics. After attending lectures by mathematician Gustav Lejeune Dirichlet, he became interested in analysis and number theory. Riemann also studied modern geometry, including elliptic functions. He developed the idea that elliptic functions could be derived using complex variables. This spawned one of his most outstanding contributions to mathematical history, an analytic function of a complex variable.

In his early twenties, Riemann returned to Göttingen to study philosophy. At Göttingen, Riemann worked with physicist Wilhelm Weber, a student of mathematician Carl Friedrich. Riemann became Weber’s lab assistant in experimental physics. In 1851, Riemann, at the age of twenty-five, submitted his dissertation to Gauss with the title “Foundations for a general theory of functions of one complex variable.” The dissertation revolutionized elliptic functions and algebraic geometry and laid the foundations for modern holomorphic functions, the system of topology, Abelian integrals, and Riemann manifolds. Biographers have since theorized that Riemann was seeking some kind of unified field theory. Riemann’s work was not officially proved accurate until mathematician David Hilbert examined his work in 1900.

In 1853, Riemann’s interest in physics, first stimulated by Weber, once again flowered. In December 1853, he prepared papers on gravity, light, magnetism, and electricity and applied for his Habilitationsschrift (a required lecture and paper for academic advancement in Germany). Success in one’s Habilitationsschrift would gain a scholar permission to lecture at the university for no pay. Although Riemann hoped his work in physics would be accepted, Gauss encouraged Riemann to write on the foundations of geometry instead. Riemann had to start afresh, and his progress was slowed by health problems. It was spring before his work was ready. Riemann’s probationary work included a lecture given June 10, 1854. This paper, “On the hypotheses which lie at the foundations of geometry,” matched algebra with geometric proofs and made possible the application of geometry to physics. Gauss approved of this Riemann earned a lectureship. In later decades, physicist Albert Einstein would recognize in Riemann’s work the material he was looking for to explain a universe of curvature.

From 1855 to 1859, Riemann and Dirichlet collaborated on Abelian functions, hypergeometric series, and differential equations. During this time, Riemann had a nervous breakdown and took a vacation in the mountains. Richard Dedekind, a fellow mathematician and good friend of his, joined him. Riemann recovered and his theory on Abelian functions was published in 1857. That same year, he gained the title of assistant professor. Although the position resulted in more earnings for Riemann, he continued to struggle financially.

Dirichlet died May 5, 1859. The government made Riemann his successor, partly due to Dirichlet’s constant support and partly to Riemann’s own growing reputation. He was granted an apartment, and other European mathematicians started to recognize his contributions. Riemann traveled to Paris in 1860 and began his memoir. In this, he developed quadratic differential forms, which are basic to relativity theory. Later, Einstein’s 1916 publication on general relativity scrutinized Riemann’s ideas. Now well paid, Riemann married one of his sisters’ friends, Elise Koch, in July 1862. He was thirty-six. Two months later, he began suffering from pleurisy, a symptom of tuberculosis and pneumonia.

Riemann convalesced in Italy, and for the rest of his life travelled between home and southern Italy, seeking a full recovery that never came. His daughter, Ida, was born in Pisa, 1863. Early in 1866, Riemann was named a foreign member of the Berlin Academy. In March 1866, he became a member of the French Academy of Sciences, and in June, he was elected to the Royal Society of London. He worked when his strength allowed him to until his last day, studying the mechanics of the inner ear’s workings. Riemann died July 20, 1866 in Selasca, Lake Maggiore. He was buried in the village of Boganzola.

Impact

Non-Euclidian geometry, relatively new in the nineteenth century, was not immediately accepted in academic communities during Riemann’s era. Riemann observed that the Cartesian plane, comparable to a piece of paper, was not truly flat. When observed in minute detail, the Cartesian plane is a jumble of complex planes, and Riemann was fascinated with the mathematics of these tiny spaces. This was a revolutionary contrast to Euclid, and many academics became angry when Riemann challenged time-treasured theories. Riemann’s theories were called curious and absurd. However, they changed existing knowledge of geometry and space from flat and limited to infinite and point-crammed, making more knowledge of these spaces and geometries accessible to geometers. Riemann’s definition of curvature presaged Einstein’s theory of relativity.

Much more productive research has been done in the wake of almost everything Riemann developed. His flair was for making connections among things that seemed disparate at the time, but which now are taken for granted. For example, before Riemann, math and physics were distinctly separate fields, with empirical research heavy on the physics side, and the possibility of pure research tempting on the other. Today, the two fields are deeply interlinked. Riemann’s scope of genius was fostered no doubt by his wide-ranging studies in mathematics, philosophy, physics, electricity, gravity, the human ear, theology, and language. His so-called Riemann surfaces are widely considered the most productive and influential of his scientific contributions.

Bibliography

Jost, Jürgen. Riemann Geometry and Geometric Analysis. 5th ed. Berlin: Springer, 2008. Print. Presents Riemannian geometry from the vantage point of Einstein’s theory of general relativity. Includes a discussion of quantum field theory.

Knoebel, Arthur, et al. Mathematical Masterpieces: Further Chronicles by the Explorers. New York: Springer, 2007. Print. Reviews mathematical histories of discrete/continuous algorithms, curvature, and number history. Each section contains reproductions of primary documents and concludes with problems to solve.

Odifreddi, Piergiorgio. The Mathematical Century: The 30 Greatest Problems of the Last 100 Years. Trans. Arturo Sangalli. Princeton: Princeton UP, 2004. Print. Presents a review of the work of mathematician David Hilbert, whose work utilizes Riemannian geometry.