Georg Friedrich Bernhard Riemann was a prominent German mathematician who bestowed the world with his brilliant contributions to analysis, number theory and differential geometry; some of which aided the later development of general relativity. In his short career—he died at the age of 39—he pioneered in developing ideas of fundamental importance in complex analysis, real analysis, differential geometry and other subjects. His name is connected with what is supposed to be the most important unproved assumption in present-day mathematics, the ‘Riemann Hypothesis’. Most of his journals are genuine masterpieces – filled with innovative methods, insightful ideas and extensive imagination. He studied mathematics under Gauss and physics under Wilhelm Weber. Riemann always suffered from health problems and that proved fatal for his life in the end.
Childhood And Early Life
Riemann was born in Breselenz, a village in the vicinity of Dannenberg in the Kingdom of Hanover (now known as the Federal Republic of Germany). Friedrich Bernhard Riemann, his father, was a poor Lutheran Minister in Breselenz who took part in the Napoleonic Wars. His mother, Charlotte Ebell, died early. Riemann was second among the six children. At an early age, Riemann showcased extraordinary mathematical skills and unbelievable calculation abilities, but he was timid and underwent numerous nervous breakdowns. He also suffered from diffidence and a phobia of public speaking.
In High school, Riemann studied the Bible thoroughly, but was often diverted by mathematics. His teachers were astonished by his proficiency to solve complicated mathematical operations, in which he often exceeded his teacher’s knowledge.
In 1846, at the age of 19, he started studying theology and philology with the aim to become a priest but Gauss, his teacher, amazed with Riemann’s mathematical skills, strongly insisted that Riemann discontinue his theological work and concentrate on mathematics.
At The Academy
In 1854, Riemann gave his first lectures, which established the field of Riemannian geometry, and laid down the foundation for Einstein's General theory of Relativity. In 1857, at the University of Göttingen, endeavors were made to sponsor Riemann to an extraordinary professor rank. Although this didn’t materialize, this attempt opened doors of regular salary for Riemann. In 1859, at Göttingen, Riemann was promoted as the head of the mathematics department and, the same year, he also got elected as a corresponding member of the Berlin Academy of Sciences. As a freshly elected member, Riemann presented a report on ‘The number of primes less than a given magnitude’, which proved to be of fundamental importance in number theory. Riemann also pioneered in the use of dimensions higher than only three or four, in order to explain physical reality. In 1866, Riemann was forced to flee Göttingen when the armies of Prussia and Hanover collided there during the Austro-Prussian War.
Riemann's innovative published works constructed the base for what is known as modern mathematics and research areas including analysis and geometry. These works finally proved to be very useful in the theories of algebraic geometry, Riemannian geometry and complex manifold theory. Adolf Hurwitz and Felix Klein comprehensively explained the theory of Riemann surfaces. This aspect of mathematics is the groundwork of topology, and is still extensively applied in modern mathematical physics. Riemann also established some breakthrough milestones in the theory of ‘Real Analysis’. He explained ‘the Riemann integral’ by means of Riemann sums and penned down a theory of trigonometric series that are not Fourier series, a first step in generalized function theory, and also explored the ‘Riemann–Liouville differintegral’.
Riemann also made some incredible contributions to the contemporary analytic number theory. He invented the Riemann zeta function and explained its significance in understanding the distribution of prime numbers. He also created a series of conjectures about properties of the zeta function, one of which is the famous ‘Riemann hypotheses’. Riemann was a great source of the inspiration for Charles Lutwidge Dodgson, aka Lewis Carroll. Lewis Carroll was a mathematician who authored the famous books Alice's Adventures in Wonderland and Through the Looking-Glass.
Riemann’s faculty, Gauss, asked him to construct ‘Habilitationsschrift' on the foundations of geometry in 1853. Working over several months, Riemann invented his theory of higher dimensions and gave his lecture at Göttingen in 1854 known as, ‘Über die Hypothesen welche der Geometrie zu Grunde liegen’ (or ‘On the hypotheses which underlie geometry’). This got published in 1868, i.e. two years after Riemann expired, and was received with great fervor by mathematicians worldwide. His theory has proved to one of the most noteworthy attainments in geometry.
The Concept Of Higher Dimensions
Riemann was working towards launching a collection of numbers at every point in space (i.e., a tensor) that would aid in analyzing how much was curved or bent. Riemann finally concluded that in four spatial dimensions, one requires a collection of ten numbers at each point to explain the attributes of a manifold, irrespective of how distorted it is. This became a popular fundamental construction in geometry, known now as a ‘Riemannian metric’.
In June 1862, Riemann married Elise Koch (his sister’s friend). The couple was gifted with one daughter.
Death And Legacy
In the autumn of 1862, Riemann caught a severe cold that eventually took form of the fatal tuberculosis. This happened when he was visiting Italy with his wife and three-year-old daughter during the last weeks of his life. He was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen, his housekeeper tidied up some of the clutter spread in Riemann’s office. This also consisted of some of his unpublished works. Riemann never allowed anyone to publish his incomplete works, thus some valuable mathematical information may have been lost forever.