Bernhard Riemann was a German mathematician whose revolutionary ideas of multi-dimensional space challenged Euclidean geometry perspectives, and ultimately led mathematics to radically new approaches, and found applications in many fields, such as General Relativity.

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German

Mathematics

Analysis, number theory, differential geometry

Georg-August University of Göttingen, 1846, where he studied under Gauss, and 1849-. Head of Mathematics, 1859-

Berlin University, 1847-48

Paper on prime-counting function, in which the Riemann Hypothesis appears for the first time. This paper was a seminal work in analyticial number theory.

With his untimely death while travelling in Italy, many papers were disposed of from his Göttingen office. It is suspected that many insights were consequently lost.

*Über die Hypothesen welche der Geometrie zu Grunde liegen*, 1854 (On the hypotheses which lie at the foundation of geometry), a milestone in geometry.

Lecture, 1859, on hypergeometric functions, conformal mapping and minimal surfaces.

Riemann integral: he demonstrated that every function which is continous at all points, can be integrated. The set of these theorems is call the *Riemann-Stieltjes integral*.

Fourier analysis: leading to the Riemann-Lebesgue lemma and set theory (Georg Cantor).Complex analysis: Riemann surfaces, Riemann mapping theorem

Riemann hypothesis

Differential geometry, studies in non-Euclidean geometries which were abstract for his time, but which Einstein more than half a century later found to be of fundamental utility in his General Relativity.

Riemann curvature tensor and Riemannian metric, which prescribes the ten numbers to describe any point in a manifold.

Analytical Number Theory: Riemann zeta function, which describes the distribution of prime numbers, and forms the Riemann Hypothesis. He improved Gauss's prime-counting function.

Expression for the line-element in multi-dimensional space: $1/{1 + 1/{4}α∑x^2}&√{∑dx^2}$

$g = w/2 - n +1$ : a description of the topology of surfaces, where n leaves at w branch points, and g is termed the topological genus.

Riemannian geometry, 1854 (pub. 1868). Riemann laid the foundations for topology, with work on algebraic geometry, complex manifold theory, and Riemann surfaces.

Multi-dimensional space (greater than 3)

Riemann's name is used for many theorems and geometrical terms. Riemann Space is fundamental to the New Physics of Einstein, while his Riemann's Hypothesis concerning prime numbers remains one of the greatest unsolved mysteries which has fascinated generations of mathematicians ever since.

After a brief and stunning career, pushing the fundamentals of geometry and number theory onto a new plane, figuratively and in practice, Riemann died suddenly at the age of 40, while travelling in Italy. The tragedy was compounded by his housekeeper in Göttingen destroying his papers, not realising that among them were many unpublished works in progress of one of mathematics' greatest geniuses.

(Biographies of famous scientists no. 74)

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