Bernhard Riemann

Georg Friedrich Bernhard Riemann (Breselenz, Germany, September 17, 1826 - Verbania, Italy, July 20, 1866) was a German mathematician who made very important to differential analysis and geometry, some of which paved the way for the further development of general relativity. His name is connected with the zeta function, the Riemann hypothesis, the Riemann integral, the Riemann lemma, Riemann manifolds, Riemann surfaces, and Riemann geometry. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.

Biography

In 1840 Bernhard went to Hanover to live with his grandmother and to visit the Lyceum. After his grandmother's death in 1842 he entered the Johanneum Lüneburg. Since he was little, he showed a fabulous capacity for calculation together with an almost sickly shyness. During his high school studies he learned so fast that he quickly outpaced all of his teachers. It is also said that Riemann spoke with his professors so that the didactics in the teaching of mathematics would adapt to him. That is why many historians suggest that he suffered from Asperger's Syndrome.

In 1846 at the age of 19, he began to study philology and theology at the University of Göttingen, his idea was to please his father and be able to help his family by becoming a pastor. He attended Gauss's lectures on the Method of Least Squares. In 1847 his father raised enough money for him to start studying mathematics.

In 1847 he moved to Berlin, where Jacobi, Dirichlet and Steiner taught. In 1848 demonstrations and labor movements broke out throughout Germany, Riemann was recruited by the student militias, even helped protect the king in his palace in Berlin. He stayed there for two years and returned to Göttingen in 1849.

In 1859, after receiving a doctorate in mathematics from Gauss in 1851, he first formulated the Riemann hypothesis, which is one of the most famous and important unsolved problems in mathematics.

Riemann gave his first lectures in 1854, in which he founded the field of Riemannian geometry. He was promoted to extraordinary professor at the University of Göttingen in 1857 and he became ordinary professor in 1859. In 1862 he married Elise Koch. He died of tuberculosis on his third trip to Italy in Selasca.

Main works

• Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Grösse (Basic concepts for a general theory of complex variable functions 1851). Published in Werke: Dissertation on the general theory of complex variable functions, based on today's so-called Cauchy-Riemann equations. In it, he invented the Riemann surface instrument.

• Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe (On the representation of a function by a trigonometric series, 1854) Published in Werke: Made to access his post as Assistant Professor and in which he analyzed the conditions of Dirichlet for the problem of representation of serial functions of Fourier. With this work, he defined the concept of Riemann integral and created a new branch of mathematics: The theory of functions of a real variable.

• Ueber die Hypothesen, Welche der Geometrie zu Grunde liegen (On the hypothesis in which the geometry is based, 1854) Published in Werke: Transcription of a master class given by Riemann at the request of Gauss which deals with the fundamentals of geometry. It develops as a generalization of the principles of Euclidean and non-Euclidean geometry. The unification of all geometries is now known as Riemann's geometry and is basic for the formulation of Einstein's relativity theory.

• Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (About the number of minor cousins than a given amount, 1859) Published in Werke: Riemann's most famous work. It's his only essay on the theory of numbers. Most of the article is dedicated to prime numbers. It introduces the Riemann zeta function.

In our language, there is an edition of Riemann's mathematical, physical and philosophical writings: Riemanniana Selecta, edited by J. Ferreirós (Madrid, CSIC, 2000; Classics of Thought collection). The last three mentioned works are included, as well as other material, preceded by an introductory study of about 150 pages.

Eponymy

In addition to the different mathematical concepts that bear his name, one must:

• Lunar crater Riemann bears this name in his memory.
• The asteroid (4167) Riemann also commemorates its name.
• The geometry of Riemann