Riemann is the man of brilliant intuition.

— Felix Klein, Lectures on the Development of Mathematics in the 19th Century (1926)

Riemann’s collected works, edited by Dedekind and Weber in 1876, fit into a volume of a little over five hundred pages, including the posthumous works and the notes. Few mathematicians left so little text and so many consequences.

What bears his name

  • The Riemann integral and Riemann sums, the definition of the integral in calculus courses.
  • Riemann surfaces and the Riemann sphere, the complex plane with a point at infinity, closed up like a sphere.
  • The Cauchy–Riemann equations and the Riemann mapping theorem.
  • Riemannian geometry, Riemannian manifolds, the metric and the Riemann curvature tensor, the language of general relativity.
  • The Riemann–Roch theorem, which counts the functions that can be defined on a Riemann surface, completed by his student Gustav Roch in 1865.
  • The Riemann zeta function and the Riemann hypothesis.
  • The Riemann rearrangement theorem, on conditionally convergent series.

A new style

More than the results, Riemann changed the style of mathematics. Before him, a function was above all a formula, and mathematicians worked by manipulating formulas. Riemann thought in concepts: a complex function defined by its properties, not by its expression; a surface defined without being drawn anywhere; a space defined by its rule of measurement. Twentieth-century mathematicians, from Hilbert to Weyl, saw in him the founder of the “conceptual” mathematics that became dominant.

This style had a price: Riemann often stated results without complete proof, trusting his intuition. Weierstrass’s criticism of Dirichlet’s principle showed the risk, and several generations of mathematicians worked to put Riemann’s claims on rigorous foundations. Almost all of them were right.

Bernhard Riemann in 1863, three years before his death.
Bernhard Riemann in 1863, three years before his death.Unknown author · Wikimedia Commons · public domain

In physics

Riemann’s geometry remained pure mathematics for more than half a century, until Einstein, helped by his mathematician friend Marcel Grossmann, realized between 1912 and 1915 that it was the tool for general relativity. Today the same language describes black holes, gravitational waves and the expansion of the universe. The zeta function, in turn, appears in statistical physics, in quantum field theory and in strange coincidences: the spacings between the zeros of zeta follow the same statistics as the energy levels of heavy atomic nuclei, an observation made in 1972 by Hugh Montgomery and Freeman Dyson.

The places

There is a crater Riemann on the Moon and an asteroid, 4167 Riemann. In Breselenz, his home village, a stone with a plaque remembers the mathematician, and in Göttingen a plaque marks one of the houses where he lived.

Exercise: the Riemann sphere

The Riemann sphere identifies the complex plane with a sphere, by projection from the north pole. Where do the points $z$ with very large $|z|$ go?

Stereographic projection links each point of the sphere, except the north pole, to a point of the plane by the line through the north pole. Points of the plane very far from the origin correspond to points of the sphere very close to the north pole. The north pole itself is associated with a single “point at infinity”, $\infty$. In this way, functions such as $\frac{1}{z}$ become defined on the whole sphere: $\frac{1}{0} = \infty$ and $\frac{1}{\infty} = 0$.

Exercise: a productive life

Riemann published his first great work, the thesis, in 1851, and died in 1866. If the collected works have about 550 pages, how many pages did he produce, on average, per year of his career?

From 1851 to 1866 is 15 years. $\frac{550}{15} \approx 37$ pages per year, less than a page a week. Few cases show so well that, in mathematics, what counts is the depth of the ideas, not the volume.

Sources: Richard Dedekind and Heinrich Weber (eds.), Bernhard Riemann’s Gesammelte Mathematische Werke (1876) · Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (1926–1927) · Detlef Laugwitz, Bernhard Riemann 1826–1866 (1999) · Hugh L. Montgomery, “The pair correlation of zeros of the zeta function”, Proceedings of Symposia in Pure Mathematics 24 (1973).

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