Bernhard Riemann

Georg Friedrich Bernhard Riemann was a German mathematician who lived from 1826 to 1866. He made many important discoveries in math that still affect how we understand the world today. One of his biggest ideas was a new way to describe area under curves, called the Riemann integral. He also worked with special shapes called Riemann surfaces, which helped mathematicians study complex numbers in a new way.

Riemann wrote a famous paper in 1859 about numbers called primes, which are numbers only divisible by 1 and themselves. In this paper, he made a guess called the Riemann Hypothesis, which is still one of the biggest unsolved problems in math. His work also helped lay the groundwork for modern physics, especially ideas about space and gravity that led to general relativity.

Because of his clever ideas and careful thinking, Riemann is remembered as one of the best mathematicians who ever lived. His work continues to inspire new discoveries in many areas of science and math.

Early years

Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, was a pastor who fought in the Napoleonic Wars. Riemann showed great talent in math from a young age, though he was shy and had health problems.

Education

Bernhard Riemann attended school in Hanover and later Lüneburg, where he showed great talent for mathematics. In 1846, he began studying at the University of Göttingen to become a pastor, but soon switched to mathematics after encouragement from Carl Friedrich Gauss. He then studied in Berlin before returning to Göttingen to continue his mathematical studies.

Academia

Bernhard Riemann began giving lectures in 1854, which helped start the field of Riemannian geometry. These ideas later supported Albert Einstein's general theory of relativity. In 1859, after the passing of another mathematician, Dirichlet, Riemann became the head of the mathematics department at the University of Göttingen. He was also the first to propose using dimensions higher than merely three or four to help understand the world around us. In 1862, he married Elise Koch, and they had a daughter.

Protestant family and death in Italy

Bernhard Riemann left Göttingen in 1866 when soldiers from Hanover and Prussia were fighting there. He traveled to Italy and passed away from tuberculosis in a place called Selasca, near Lake Maggiore in Verbania. He was buried in a cemetery in Biganzolo.

Riemann grew up in a strong Christian family; his father was a Protestant minister. He believed that his work as a mathematician was a way to honor God. Sadly, some of his unpublished papers were lost after his death because his housekeeper threw them away.

Riemannian geometry

Bernhard Riemann's work connected analysis and geometry, opening new areas of study that grew into Riemannian geometry, algebraic geometry, and complex manifold theory. These ideas form the basis of topology and continue to help scientists understand mathematical physics.

In 1853, his teacher Gauss asked Riemann to explore the foundations of geometry. After many months of work, Riemann gave an important lecture in 1854. Though it was published after his death, this work is now seen as one of the greatest advances in geometry. He showed how to describe curved spaces in any number of dimensions, introducing ideas like the Riemannian metric and Riemann curvature tensor. These tools help us measure distances and curves in spaces that are not flat.

Complex analysis

Bernhard Riemann created a new way to study complex numbers by using shapes called Riemann surfaces. This helped make tricky math problems easier to solve. For example, functions that used to have many answers now had just one answer on these special surfaces.

Riemann also discovered important ideas, like the Riemann mapping theorem, which shows how certain shapes in complex numbers can match perfectly. His work inspired many other mathematicians to explore these ideas further.

Real analysis

In real analysis, Bernhard Riemann discovered the Riemann integral during his advanced studies. He showed that many useful functions can be integrated and connected this work to Fourier series, a way to represent functions using sums of sine and cosine waves. His ideas helped lead to important work by another famous mathematician, Georg Cantor, in the field of set theory.

Number theory

Bernhard Riemann made important contributions to number theory, especially in a short paper he wrote. He studied the zeta function, which helps mathematicians understand how prime numbers are spread out. One of his big ideas was the Riemann hypothesis, a guess about how this function works.

Riemann also showed clever ways to connect the zeta function to other math ideas. He worked with a famous mathematician named Leonhard Euler and learned from Pafnuty Chebyshev’s research on the Prime Number Theorem. His work opened many new doors in solving puzzles about numbers.

Writings

Bernhard Riemann wrote many important mathematical works throughout his life. Some of his key writings include:

• 1851 – Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, his first major work.
• 1857 – Theorie der Abelschen Functionen, published in a mathematics journal.
• 1859 – Über die Anzahl der Primzahlen unter einer gegebenen Größe, where he made a famous guess about prime numbers.
• 1861 – Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae, submitted for a contest.
• 1867 – Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe, about representing functions.
• 1868 – Über die Hypothesen, welche der Geometrie zugrunde liegen, his ideas about the foundations of geometry.
• 1876 – Bernhard Riemann's Gesammelte Mathematische Werke und wissenschaftlicher Nachlass, a collection of his works.
• 1882 – Vorlesungen über Partielle Differentialgleichungen.
• 1901 – Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen.
• 2004 – Collected papers, a modern collection of his papers.