Bernhard Riemann

Bernhard Riemann was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. He is considered by many to be one of the greatest mathematicians of all time. Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover. His father was a Lutheran pastor who taught him mathematics and philosophy. Riemann was a shy and introverted child who suffered from nervous breakdowns. He showed exceptional talent in mathematics and languages from an early age. Riemann attended high school in Lüneburg and then studied at the University of Göttingen and the University of Berlin. He was influenced by the works of Gauss, Euler, Dirichlet, Jacobi, and others. He obtained his doctorate in 1851 with a dissertation on complex analysis. He became a lecturer at Göttingen in 1854 and a professor in 1859. Riemann's most famous work is his 1859 paper on the distribution of prime numbers, where he introduced the Riemann zeta function and stated the Riemann hypothesis, one of the most important unsolved problems in mathematics. He also developed the theory of Riemann surfaces, which are complex manifolds that can represent multi-valued functions. He made fundamental contributions to the theory of functions, complex analysis, harmonic analysis, Fourier series, and potential theory. Riemann also revolutionized the field of differential geometry with his 1854 habilitation lecture on the foundations of geometry. He introduced the concept of a Riemannian manifold, which is a space that can have curvature and dimension different from that of Euclidean space. He defined the notion of a metric tensor, which measures distances and angles on a manifold. He also derived the Riemann curvature tensor, which characterizes the intrinsic curvature of a manifold. His ideas laid the groundwork for the development of general relativity by Einstein. Riemann married Elise Koch in 1862 and they had no children. He suffered from poor health throughout his life and died of tuberculosis on 20 July 1866 in Selasca, Italy. He was buried in the Biganzolo cemetery in Verbania. His collected works were published posthumously by Dedekind and Weber in 1876.