Bernhard Riemann

Bernhard Riemann
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Name	Bernhard Riemann
Caption	Portrait of Bernhard Riemann
Birth date	17 September 1826
Birth place	Breselenz, Kingdom of Hanover
Death date	20 July 1866
Death place	Selasca, Kingdom of Italy
Fields	Mathematics, Physics
Alma mater	University of Göttingen, University of Berlin
Doctoral advisor	Carl Friedrich Gauss
Known for	Riemannian geometry, Riemann hypothesis, Riemann integral
Spouse	Elise Koch

Bernhard Riemann was a pioneering German mathematician whose profound and wide-ranging work fundamentally reshaped modern mathematics and theoretical physics. His contributions span analysis, number theory, and differential geometry, with concepts like Riemannian geometry providing the mathematical foundation for Albert Einstein's general relativity. Despite a life cut short by illness, his innovative ideas, particularly the unsolved Riemann hypothesis, continue to drive central research in pure mathematics.

Early life and education

Born in the village of Breselenz in the Kingdom of Hanover, he was the second of six children to Friedrich Bernhard Riemann, a Lutheran pastor. He showed exceptional mathematical talent early, often surpassing his teachers, and in 1846 began studying philology and theology at the University of Göttingen to please his father. His passion for mathematics soon prevailed, leading him to transfer to the University of Berlin in 1847, where he studied under renowned figures like Peter Gustav Lejeune Dirichlet, Carl Gustav Jacob Jacobi, and Jakob Steiner. Returning to Göttingen in 1849, he completed his doctoral dissertation in 1851 under the supervision of the legendary Carl Friedrich Gauss, who praised its groundbreaking insights into the theory of complex analysis.

Mathematical contributions

His doctoral thesis, founded on the concept of Riemann surfaces, revolutionized the understanding of complex functions by introducing a topological perspective. In his seminal 1854 habilitation lecture, "On the Hypotheses Which Lie at the Foundations of Geometry," delivered before Gauss, he laid the groundwork for Riemannian geometry, defining the Riemann curvature tensor and the notion of an *n*-dimensional manifold. This work provided the essential mathematical language for non-Euclidean geometry and later for general relativity. In real analysis, he refined the definition of the integral, leading to the Riemann integral, and made significant discoveries in Fourier series. His 1859 paper "On the Number of Primes Less Than a Given Magnitude" introduced the Riemann zeta function and formulated the famous Riemann hypothesis, a central, unresolved conjecture in analytic number theory concerning the distribution of prime numbers.

Later years and death

Appointed as a professor at the University of Göttingen in 1859, succeeding his mentor Dirichlet, he continued his research despite increasingly poor health. He married Elise Koch, a friend of his sister, in 1862, and they had one daughter. Suffering from tuberculosis, he frequently traveled to warmer climates in Italy on the advice of his doctor, spending time in places like Pisa and Genoa in an attempt to recuperate. His final journey was to the village of Selasca on the shores of Lake Maggiore in the Kingdom of Italy. He died there in 1866 at the age of 39, with his wife by his side, and was buried in the cemetery in Biganzolo.

Legacy and influence

His work has had an immeasurable impact across mathematics and physics. Riemannian geometry became the cornerstone of Albert Einstein's theory of general relativity and modern differential geometry. The Riemann hypothesis remains one of the most important open problems in mathematics, featured as one of the Clay Mathematics Institute's Millennium Prize Problems. His ideas in complex analysis and topology influenced generations of mathematicians, including Felix Klein, David Hilbert, and Henri Poincaré. Concepts bearing his name, such as the Riemann sphere, Riemann-Roch theorem, and Riemannian manifold, are fundamental to contemporary research. His collected works, published posthumously, continue to be a vital source of inspiration.

Selected publications

* "Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse" (1851 doctoral dissertation) * "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe" (On the representability of a function by a trigonometric series, 1854) * "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (On the Hypotheses Which Lie at the Foundations of Geometry, 1854 habilitation lecture) * "Theorie der Abel'schen Functionen" (1857) * "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" (On the Number of Primes Less Than a Given Magnitude, 1859)

Category:German mathematicians Category:1826 births Category:1866 deaths