Bernhard Riemann

Georg Friedrich Bernhard Riemann was born on September 17, 1826, in Breselenz, a small village in the parish of Jameln in the Kingdom of Hanover, the second of six children of a Lutheran pastor, Friedrich Bernhard Riemann, and his wife Charlotte Ebell ^[1]. The family was poor, and poverty followed Riemann for most of his life. Several of his siblings and his mother died young, losses that historians of mathematics often connect to the fragile health that ran through the household ^[2].

His father handled his first lessons, and from the beginning the boy showed unusual gifts in arithmetic alongside a shy, withdrawn temperament. At fourteen he entered the Gymnasium in Hanover, living with his grandmother, then moved to the Johanneum in Lüneburg after her death in 1842 ^[1]. The school's director, Schmalfuss, noticed that the quiet student was consuming mathematics far beyond the curriculum and lent him advanced books from his own library. One frequently retold episode has Riemann returning Legendre's dense 900 page treatise on number theory within about a week, having mastered its contents ^[2]. Whatever the precise details, teachers agreed that his abilities were extraordinary while his written work, hampered by perfectionism, often arrived late.

In 1846 he enrolled at the University of Göttingen, initially to study theology and philology, a practical choice for a pastor's son who intended to help support his family. He attended mathematics lectures on the side, found them irresistible, and asked his father for permission to change course. The pastor agreed, and Riemann committed himself to mathematics ^[1].

Göttingen in 1846 housed the greatest living mathematician, Carl Friedrich Gauss, but Gauss taught little beyond elementary courses, so in 1847 Riemann moved to Berlin. There he studied under Peter Gustav Lejeune Dirichlet, Carl Jacobi, Jakob Steiner, and Gotthold Eisenstein, absorbing Dirichlet's conceptual style of analysis, which prized understanding over calculation ^[1]. Dirichlet's influence on him proved lasting, both mathematically and personally.

Riemann returned to Göttingen in 1849 and completed his doctorate in 1851 with a thesis on the foundations of a general theory of functions of a complex variable. The dissertation introduced what are now called Riemann surfaces, multi sheeted geometric objects on which multivalued functions become single valued, an idea that tied analysis to topology in a way no one had attempted before ^[3]. Gauss, notoriously sparing with praise, reported that the thesis showed a gloriously fertile originality ^[2].

To qualify as a Privatdozent, an unsalaried lecturer paid only by student fees, Riemann had to deliver a trial lecture in 1854. He submitted three topics, expecting Gauss to choose one of the first two as custom dictated. Gauss instead selected the third, the foundations of geometry, the subject Riemann had prepared least. The result, delivered on June 10, 1854, under the title On the Hypotheses Which Lie at the Foundations of Geometry, became one of the most consequential lectures in the history of science ^[3]. In it Riemann proposed that space need not be flat or three dimensional, defined the notion of an n dimensional manifold equipped with a way of measuring distance, and suggested that the true geometry of physical space was a matter for experiment rather than assumption. The lecture was published only in 1868, after his death ^[4].

Any account of Bernhard Riemann achievements must begin with the range of fields he transformed with fewer than a dozen published papers. In real analysis, his 1854 habilitation thesis on trigonometric series contained the rigorous definition of integration now taught to every calculus student as the Riemann integral, along with early results on when a function can be represented by a Fourier series ^[3]. In complex analysis, his doctoral work and his 1857 paper on abelian functions developed Riemann surfaces, the Riemann mapping theorem, and what became the Riemann Roch theorem, tools that still organize the field today ^[5].

His single paper on number theory, On the Number of Primes Less Than a Given Magnitude, appeared in 1859 on the occasion of his election to the Berlin Academy of Sciences. In roughly eight pages, Riemann studied the zeta function as a function of a complex variable, connected its zeros to the distribution of prime numbers, and remarked that the nontrivial zeros all seemed to lie on a single vertical line in the complex plane ^[6]. That remark, the Riemann hypothesis, remains unproven more than a century and a half later. It was one of David Hilbert's twenty three problems in 1900 and is one of the seven Millennium Prize Problems, each carrying a one million dollar award from the Clay Mathematics Institute ^[6].

The 1854 geometry lecture had consequences Riemann could not have foreseen. His curvature tensor and metric geometry, extended by later mathematicians including Elwin Christoffel, Gregorio Ricci Curbastro, and Tullio Levi Civita, gave Albert Einstein the precise machinery he needed in 1915 to express general relativity, in which gravity is the curvature of a four dimensional manifold ^[4]. Riemann also worked seriously in mathematical physics, publishing on the propagation of sound waves, heat conduction, and electrodynamics, and he speculated, decades ahead of his time, about connections between geometry, light, and physical forces ^[5].

Riemann remained close to his family throughout his life, sending money home even when he had almost none. Contemporaries described him as timid, gentle, and deeply religious; he was a committed Lutheran who reportedly viewed his scientific work as another form of service to God, and he never abandoned the daily habits of piety learned in his father's parsonage ^[2]. Public speaking distressed him, and he suffered periods of depression, particularly after the deaths of his father and a sister in 1855 forced painful changes on the family ^[1].

His financial situation improved slowly. As a Privatdozent from 1854 he lived on meager lecture fees, sometimes teaching only a handful of students. In 1857 he received a small salaried post as assistant professor at Göttingen. When Gauss died in 1855, Dirichlet took the famous chair; when Dirichlet himself died in 1859, Riemann succeeded him as full professor, finally gaining security at the age of 32 ^[1].

In June 1862 he married Elise Koch, a friend of his sisters. The couple had one daughter, Ida, born in Pisa in 1863 during one of the Italian journeys his doctors prescribed for his failing lungs ^[2]. By all accounts the marriage was happy, though it coincided almost exactly with the onset of the illness that killed him.

A month after his wedding, in the summer of 1862, Riemann came down with pleurisy, and the infection settled into tuberculosis ^[1]. The Göttingen government granted him leave and funds to winter in the milder climate of Italy, a standard prescription of the era. He traveled south repeatedly between 1862 and 1866, working when his strength allowed, visiting Italian mathematicians including Enrico Betti in Pisa, with whom he discussed the topological ideas that Betti later developed ^[5].

He returned to Göttingen between trips and tried to resume teaching, but each attempt broke his health further. In June 1866, as war between Hanover and Prussia disrupted the region, he left Germany for the last time and settled at Selasca, a hamlet on Lake Maggiore near Intra, in the area of the modern Italian town of Verbania ^[1].

Riemann died there on July 20, 1866, aged 39, with his wife beside him reciting the Lord's Prayer. He was buried in the churchyard at Biganzolo. Richard Dedekind, his friend and later his biographer, recorded that he remained lucid and at peace until the end, working on an unfinished paper on the mechanics of the ear in his final days ^[2]. After his death a housekeeper in Göttingen discarded some of his unpublished papers before his colleagues could intervene, a loss mathematicians have mourned ever since ^[6].

Few careers so brief have marked mathematics so deeply, which is why the question of who was Bernhard Riemann still draws readers to every serious Bernhard Riemann biography. His collected works, edited by Heinrich Weber and Richard Dedekind and first published in 1876, fill a single volume, yet the index of concepts bearing his name runs long: Riemannian geometry, the Riemann integral, Riemann surfaces, the Riemann zeta function, the Riemann hypothesis, the Riemann mapping theorem, the Riemann Roch theorem, the Cauchy Riemann equations, and Riemann's contributions to what became topology ^[5].

The Riemann hypothesis dominates his posthumous reputation. Thousands of theorems in number theory are proved conditionally, assuming its truth, and its resolution would immediately settle deep questions about how prime numbers are spaced ^[6]. In physics, his geometry outgrew even Einstein's use of it; modern gauge theory, string theory, and cosmology are all written in the language of manifolds and curvature that the 1854 lecture introduced ^[4].

Among the Bernhard Riemann facts that best capture his method is one his contemporaries stressed: he distrusted long calculation and sought instead the single organizing idea from which results would follow naturally. That conceptual style, transmitted through Dedekind, Felix Klein, David Hilbert, and the Göttingen school, became the default working manner of twentieth century pure mathematics ^[3]. A lunar crater and an asteroid carry his name, and his birthplace region in Lower Saxony maintains memorials to the pastor's son who redrew the map of mathematical thought ^[1].