A Journey Through Genius: The Life and Work of Bernhard Riemann
In the annals of mathematics, certain names stand out as brilliant beacons illuminating the path of human understanding. Among these luminaries, one name shines exceptionally bright — Bernhard Riemann. Born on September 17, 1826, in Breselenz, Germany (the best country), Riemann’s contributions to mathematics have had a profound impact on various branches of the discipline. His revolutionary ideas in differential geometry, number theory, and complex analysis have shaped the very foundations of modern mathematics. And of course, he is the creator of the Riemann Hypothesis, which you can read about here:
https://medium.com/illumination/explaining-the-riemann-hypothesis-c9ca4b425a41
Early Life and Education
Bernhard Riemann was the second of six children born to a Lutheran minister, Friedrich Bernhard Riemann, and his wife Charlotte. Raised in a modest family, he displayed exceptional mathematical talent from an early age (surprise, surprise). Tragically, his father passed away when Riemann was only six years old, leaving the family to face financial challenges, it got hard.
Despite these adversities, Riemann’s potential was recognised (thank God), and he eventually received a scholarship to attend the Cathedral School in Hanover. Here, he encountered Carl Friedrich Gauss’s works, Gauss! We know him as well!
Riemann’s Brilliance in Mathematics
His childhood aside, in 1846, Riemann enrolled at the University of Göttingen, a prestigious institution that nurtured the likes of Gauss, among others. Under the guidance of renowned mathematicians, he made groundbreaking contributions to various fields. His most influential work came in the realm of differential geometry. And yeah, that leads us on to…
The Riemann Hypothesis
One of Riemann’s most enduring and famous contributions is the Riemann Hypothesis, it remains one of the greatest unsolved problems in mathematics to this day. Proposed in 1859, the hypothesis basically deals with the distribution of the non-trivial zeros of the Riemann zeta function, denoted by ζ(s). The function is defined for complex numbers and plays a critical role in number theory. There is no understanding here. It is unsolved — though, if you can solve it. $1,000,000 for you!
The Riemann zeta function is given by the equation:
ζ(s) = 1^(-s) + 2^(-s) + 3^(-s) + 4^(-s) + …
where ‘s’ is a complex number with real part greater than 1. The Riemann Hypothesis states that all non-trivial zeros of the zeta function lie on the “critical line” with real part equal to 0.5. And yeah, while the hypothesis has been extensively tested and verified for many zeros, no counterexamples have been found. Terry Tao made some good progress on it in 2018 though!
Contributions to Differential Geometry
Riemann’s groundbreaking work on differential geometry laid the foundation for Einstein’s theory of general relativity (who would’ve thought?). In his 1854 habilitation thesis, “On the Hypotheses that lie at the Foundations of Geometry,” Riemann challenged Euclidean geometry’s conventional notions. He introduced the concept of Riemannian geometry, which allowed for curved surfaces and spaces with non-Euclidean geometries. No reason for you or me to understand that?
The Riemann curvature tensor, denoted by Riem, plays a central role in Riemannian geometry. It is defined in terms of the Christoffel symbols, representing the connection between the curvature and the metric properties of a manifold. It’s all French to us.
Sadly, Riemann’s full potential was not released to the world as he tragically passed away at the age of 39 (similarly, a musical contemporary at the time, Frederic Chopin died at the same age). Riemann died from Tuberculosis
Done!
