26 Facts About Riemann

Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry.

Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover.

Riemann'sfather, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars.

Riemann was the second of six children, shy and suffering from numerous nervous breakdowns.

Riemann exhibited exceptional mathematical talent, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.

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In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics.

Riemann'steachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge.

Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his father's approval, Riemann transferred to the University of Berlin in 1847.

Riemann fled Gottingen when the armies of Hanover and Prussia clashed there in 1866.

Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God.

Riemann's published works opened up research areas combining analysis with geometry.

The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz.

Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Gottingen in 1854 entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.

Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium.

For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on a manifold, no matter how distorted it is.

The topological "genus" of the Riemann surfaces is given by, where the surface has leaves coming together at branch points.

The generalization of the theorem to Riemann surfaces is the famous uniformization theorem, which was proved in the 19th century by Henri Poincare and Felix Klein.

Karl Weierstrass found a gap in the proof: Riemann had not noticed that his working assumption might not work; the function space might not be complete, and therefore the existence of a minimum was not guaranteed.

When Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish it.

Riemann had been in a competition with Weierstrass since 1857 to solve the Jacobian inverse problems for abelian integrals, a generalization of elliptic integrals.

Riemann used theta functions in several variables and reduced the problem to the determination of the zeros of these theta functions.

Riemann however used such functions for conformal maps in his 1859 lecture on hypergeometric functions or in his treatise on minimal surfaces.

Riemann gave an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet.

Riemann's essay was the starting point for Georg Cantor's work with Fourier series, which was the impetus for set theory.

The Riemann hypothesis was one of a series of conjectures he made about the function's properties.

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Riemann knew of Pafnuty Chebyshev's work on the Prime Number Theorem.