Bernhard Riemann
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 Contributions
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.
Riemannian Geometry
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’.
Personal Life
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.
