🇩🇪Bernhard
Riemann

He published fewer than a dozen papers in his entire career, died of tuberculosis at thirty-nine before his most important work had been fully understood, and still managed to reshape the foundations of mathematics more profoundly than almost anyone who came before or after. Georg Friedrich Bernhard Riemann was born on September 17, 1826, in the small village of Breselenz in the Kingdom of Hanover, Germany, the second of six children of a Lutheran pastor. The family was poor and Riemann was chronically ill for much of his life, but his mathematical gifts were evident from childhood. His first teacher, a local schoolmaster, reportedly told his parents that he had nothing left to teach the boy. At fourteen, Riemann was reading Legendre's 900-page text on number theory — a graduate-level work — and returning it in six days, having apparently committed it to memory. He went to Göttingen to study theology, in deference to his father's wishes, and then switched to mathematics after a semester, with his father's blessing. He never looked back.

At Göttingen, Riemann came under the influence of Carl Friedrich Gauss — the greatest mathematician of the preceding generation — and Karl Wilhelm Jacobi in Berlin. His doctoral thesis, submitted in 1851 when he was twenty-five, introduced the concept of what are now called Riemann surfaces: a way of making multi-valued functions in the complex plane single-valued by imagining them as living on specially constructed geometric surfaces with multiple sheets. Gauss, who read thousands of doctoral theses in his role as examiner, praised Riemann's work with extraordinary enthusiasm — the old man, famously sparing with compliments, called it proof of a "gloriously fertile originality." The thesis created the foundation of modern complex analysis and topology in a single document. It remains one of the most important doctoral dissertations in the history of mathematics.

In 1854, Riemann delivered his Habilitation lecture — required for the title of Privatdozent, which would allow him to teach at Göttingen — on "the hypotheses which lie at the foundations of geometry." The lecture, titled Über die Hypothesen, welche der Geometrie zu Grunde liegen, was one of the most consequential lectures ever delivered in the history of science. In it, Riemann generalized Gauss's work on the curvature of surfaces to n-dimensional curved spaces, introducing the framework now called Riemannian geometry: a way of doing geometry on any curved manifold of any dimension, in which distances are measured by a metric tensor that can vary from point to point. It was not merely a technical advance. It was a complete reconceptualization of what geometry is. Gauss, who was in the audience and already old and frail, was visibly moved. The geometry of the universe would no longer need to be Euclidean. Space itself could curve.

Five years later, in 1859, Riemann published his single most influential paper: a short, eight-page work "On the Number of Primes Less Than a Given Magnitude." In it, he extended Euler's zeta function to the complex plane, analyzed its properties, and stated — almost casually, as a "likely" fact he had not fully proved — the conjecture now known as the Riemann hypothesis: that all non-trivial zeros of the zeta function have real part equal to 1/2. This conjecture, which encodes the precise distribution of prime numbers along the number line, has resisted all attempts at proof for over 165 years. It is one of the Clay Mathematics Institute's seven Millennium Prize Problems, with a prize of one million dollars attached. When Andrew Wiles proved Fermat's Last Theorem in 1995, he described the Riemann hypothesis as a problem of incomparably greater depth. Every attempt to prove it has led to profound new mathematics, and the conjecture is now known to connect to quantum mechanics, random matrix theory, and the spacings of energy levels in heavy atomic nuclei. It seems to describe something fundamental about how order and randomness coexist in the universe.

Riemann's health deteriorated through the early 1860s. He suffered from pleurisy and then tuberculosis, and spent his final years in Italy seeking a warmer climate. He died on July 20, 1866, in Selasca on Lake Maggiore, while his wife read the Lord's Prayer to him. He was thirty-nine years old. His housekeeper, not understanding the significance of what she was destroying, burned most of his unpublished manuscripts after his death. What remained was enough to redirect the course of mathematics for the next century. Einstein, searching in the 1910s for a geometric language to describe gravity as the curvature of spacetime, found everything he needed in Riemann's 1854 lecture — a framework that had been waiting sixty years for the physics to catch up with the geometry. When Einstein published general relativity in 1915, he acknowledged that the mathematical backbone of his theory had been provided, in its entirety, by a German pastor's son who had died of tuberculosis at thirty-nine.