Bernhard Riemann

Bernhard Riemann
Public domain · source
Name	Bernhard Riemann
Birth date	1826-09-17
Birth place	Breselenz, Kingdom of Hanover
Death date	1866-07-20
Death place	Selasca, Kingdom of Italy
Nationality	German
Fields	Mathematics
Alma mater	Göttingen University; University of Berlin
Doctoral advisor	Carl Friedrich Gauss
Influenced	Henri Poincaré, Felix Klein, Bernhard Bolzano, David Hilbert, Emmy Noether

Bernhard Riemann Bernhard Riemann was a German mathematician whose work established foundations for complex analysis, differential geometry, and analytic number theory, notably the Riemann hypothesis. His ideas reshaped subsequent developments in mathematical physics, topology, and the study of manifolds. Riemann's brief career at Göttingen University produced concepts later central to the work of figures such as Carl Friedrich Gauss, Hermann Weyl, and Albert Einstein.

Early life and education

Riemann was born in the village of Breselenz within the Kingdom of Hanover and raised in a household influenced by Georg Friedrich Bernhard Riemann (father) and Charlotte Ebell (mother), before moving to Hannover. He attended school in Hannover and later the Lyceum in Göttingen where encounters with lecturers introduced him to ideas from Johann Friedrich Pfaff and Leopold Kronecker. Riemann enrolled at Göttingen University and then pursued further studies at the University of Berlin, where he heard lectures by Carl Gustav Jacobi, Peter Gustav Lejeune Dirichlet, Karl Weierstrass, and Jacobi and encountered the influential work of Augustin-Louis Cauchy and Niels Henrik Abel.

Academic career and positions

After returning to Göttingen University, Riemann completed a habilitation defended before Carl Friedrich Gauss, who nominated him to a lectureship; he later succeeded to a professorship at Göttingen. He delivered notable lectures at institutions including Göttingen, attracting attendees from across Europe such as Felix Klein and Leopold Kronecker, and corresponded with scholars at the Prussian Academy of Sciences and the Royal Society. During his tenure he held positions that connected him with mathematicians like Hermann Hankel and applied scientists linked to Joseph Fourier and Lord Kelvin.

Contributions to mathematics

Riemann introduced the concept of Riemann surfaces to rigorously treat multi-valued functions in complex analysis, generalizing ideas from Gauss and Augustin-Louis Cauchy and influencing Hermann Weyl and André Weil. His formulation of what became known as the Riemann integral refined integration theory following work by Bernard Bolzano and Niels Henrik Abel, impacting studies by Karl Weierstrass. In differential geometry he proposed Riemannian metrics and curvature tensors that extended notions from Carl Friedrich Gauss and anticipated formulations later central to Albert Einstein's general relativity, influencing Élie Cartan and Hermann Weyl. In analytic number theory his introduction of the zeta function and the statement now called the Riemann hypothesis connected zero distributions to prime counting, building on methods from Leonhard Euler and informing later efforts by G.H. Hardy, John Edensor Littlewood, Atle Selberg, and André Weil. Riemann's work on potential theory and conformal mappings drew on predecessors like Simeon Denis Poisson and Pierre-Simon Laplace and informed progress by Bernhard Riemann's intellectual descendants including Ludwig Bieberbach and Oswald Teichmüller.

Major works and publications

Riemann's habilitation lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" laid out the foundations of what became Riemannian geometry, later influencing publications by Élie Cartan and Hermann Weyl. His memoir on abelian functions synthesized ideas related to Abel and Jacobi and shaped subsequent expositions by Felix Klein and Henri Poincaré. The 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Größe" introduced the Riemann zeta function and the Riemann hypothesis, which catalyzed research by G.H. Hardy, Bernhard Riemann commentators, and 20th-century analysts such as Atle Selberg and Alan Turing. His collected lectures in complex analysis, conformal mapping, and potential theory circulated as manuscripts and were edited posthumously influencing treatises by Karl Weierstrass and Felix Klein.

Influence and legacy

Riemann's concepts of manifold, metric, and curvature undergirded modern differential geometry and shaped the mathematical language used by Albert Einstein in formulating general relativity, prompting developments by Élie Cartan, Hermann Weyl, and Marcel Berger. His zeta function problem motivated an extensive body of analytic number theory pursued by G.H. Hardy, John Edensor Littlewood, Atle Selberg, André Weil, Enrico Bombieri, and Andrew Wiles, and inspired computational projects by Alan Turing and J. van de Lune. The theory of Riemann surfaces spawned by his work propelled advances by Henri Poincaré, Felix Klein, Oswald Teichmüller, and André Weil, while his influence appears in modern fields explored by Michael Atiyah and Isadore Singer. Riemann's name endures in many eponyms studied at institutions such as Göttingen University, the British Royal Society, and the Prussian Academy of Sciences.

Personal life and death

Riemann maintained correspondence with peers including Carl Friedrich Gauss, Leopold Kronecker, and Felix Klein and led a private life with limited public engagement, often residing in Göttingen and traveling to cities such as Berlin and Milan for work and health. He suffered from persistent illness, often identified as tuberculosis, and sought treatment in locales including Nice and the Italian Alps, ultimately dying on the shores of Lake Maggiore near Selasca. Posthumous publications and manuscript collections were curated by contemporaries at Göttingen University and editors like Richard Dedekind and Hermann Hankel who preserved his legacy.

Category:1826 births Category:1866 deaths Category:German mathematicians Category:Göttingen University faculty