Bernhard Riemann

Georg Friedrich Bernhard Riemann (1826 - 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. Though he published relatively little during his short life, his work introduced concepts that became foundational to much of modern mathematics and theoretical physics, including Riemannian geometry, the Riemann integral, and the still-unproven Riemann hypothesis.^[1]

Early life and education

Riemann was born on 17 September 1826 in Jameln, a village in the Kingdom of Hanover, the second of six children of a Lutheran pastor.^[1] A shy and often sickly child, he displayed exceptional mathematical ability from an early age. He attended the Gymnasium in Lüneburg and in 1846 enrolled at the University of Göttingen, initially to study theology at his father's wish before turning fully to mathematics. He continued his studies at the University of Berlin, where he learned from leading figures such as Dirichlet and Jacobi, before returning to Göttingen to complete his doctorate under Carl Friedrich Gauss in 1851.^[2]

Academic career

In 1854 Riemann delivered his habilitation lecture, "Über die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses which lie at the Foundations of Geometry"), which laid the groundwork for what is now called Riemannian geometry.^[2] He was appointed a professor at Göttingen in 1857 and, in 1859, succeeded Dirichlet as the chair of mathematics, the same year he was elected a corresponding member of the Berlin Academy of Sciences.

Major work

Riemann's output, though small, was extraordinarily influential. In complex analysis he introduced Riemann surfaces, a geometric way of understanding multi-valued functions. In real analysis his rigorous definition of the integral, now known as the Riemann integral, remains standard in introductory calculus. His 1859 paper on the distribution of prime numbers studied the Riemann zeta function and stated the Riemann hypothesis, a conjecture about the location of the function's non-trivial zeros that endures as one of the most important unsolved problems in mathematics.^[3]

Legacy

Riemann's health declined in the 1860s, and he died of tuberculosis on 20 July 1866 at Selasca, near Verbania on Lake Maggiore in Italy, at the age of 39.^[1] Despite his early death, his ideas transformed mathematics. Riemannian geometry provided the mathematical framework that Albert Einstein used to formulate the general theory of relativity in 1915.^[2] Elected a Foreign Member of the Royal Society, Riemann is commemorated across the discipline in terms such as the Riemann hypothesis, Riemann surfaces, the Riemann–Roch theorem, and the Cauchy–Riemann equations.