Bernhard Riemann

Georg Friedrich Bernhard Riemann was a German mathematician. He lived from 1826 to 1866. Riemann made many important discoveries in math. His ideas still help us understand the world today.

One of his biggest ideas was a new way to describe area under curves. This is called the Riemann integral. He also worked with special shapes called Riemann surfaces. These shapes helped mathematicians study complex numbers in new ways.

In 1859, Riemann wrote a famous paper about numbers called primes. Prime numbers can only be divided by 1 and themselves. In this paper, he made a guess called the Riemann Hypothesis. This guess is still one of the biggest unsolved problems in math today. His work also helped ideas about space and gravity. These ideas later led to general relativity.

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

Early years

Bernhard Riemann was born on 17 September 1826 in Breselenz. This village is near Dannenberg in the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, was a pastor. He fought in the Napoleonic Wars. Riemann was very good at math when he was young. However, he was shy and had health problems.

Education

Bernhard Riemann went to school in Hanover and later in Lüneburg. He was very good at mathematics. In 1846, he started at the University of Göttingen to study to become a pastor. But he changed to mathematics after Carl Friedrich Gauss encouraged him. He studied in Berlin and then returned to Göttingen to keep studying mathematics.

Academia

Bernhard Riemann began giving lectures in 1854. These lectures helped start the field of Riemannian geometry. His ideas later supported Albert Einstein's general theory of relativity. In 1859, Riemann became the head of the mathematics department at the University of Göttingen. He was also the first to suggest using dimensions higher than merely three or four to help understand our world. 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 papers were lost after his death.

Riemannian geometry

Bernhard Riemann's work connected analysis and geometry. It opened new areas of study that grew into Riemannian geometry, algebraic geometry, and complex manifold theory. These ideas form the basis of topology and 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. He introduced 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 using shapes called Riemann surfaces. This made hard math problems easier to solve. For example, some functions that used to have many answers now had just one answer on these special surfaces.

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

Real analysis

In real analysis, Bernhard Riemann discovered the Riemann integral. He showed that many useful functions can be added up in a special way. He also linked this work to Fourier series, a method that uses sine and cosine waves to show functions. His ideas helped another famous mathematician, Georg Cantor, with his work in set theory.

Number theory

Bernhard Riemann made important contributions to number theory in a short paper. 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. Here are some of his key writings:

• 1851 – Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, his first major work.
• 1857 – Theorie der Abelschen Functionen, published in a math 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, entered in 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.