Bernhard Riemann was a German mathematician who had a profound impact on mathematics and physics. His ideas in non-Euclidean geometry helped set the stage for Einstein’s work on relativity. He also came up with what came known as the Riemann zeta function and the Riemann hypothesis, a hypothesis that has not been proven despite a million dollar prize for the proof.
Riemann died young and although he did not publish a lot of papers, those papers that he did publish had a profound impact on the field. Riemann believed in a conceptual approach to mathematics and did not use a lot of equations to try and explain his work. This made his work much more accessible than other mathematicians who used algorithms to explain their work.
Riemann was born on September 17, 1826, in Breselenz, Hanover (modern day Germany). His father was a Lutheran minister (Friedrich Bernhard Riemann) and his mother was Charlotte Ebell. Riemann’s father married in middle age and the couple had six children. Riemann was the second child. His mother died while Riemann was still quite young.
Riemann was initially educated at home by his father. Riemann was taught by his father until he was ten years old and then a local school teacher took over Riemann’s education.
Riemann was enrolled into the third grade of Hannover’s Lyceum in 1840. Riemann stayed with his grandmother while he attended the Lyceum, but she died two years later in 1842. As a result, Bernhard had to change schools. He enrolled in the Johanneum Gymnasium in Lüneburg. He was a good student and worked hard but he did not come across as an outstanding student.
While in the Gymnasium, he began to show a lot of interest in mathematics. The director of the Gymnasium began to lend his books on mathematics to Riemann. It is claimed that when the director lent Riemann a book on the theory of numbers written by Adrien-Marie Legendre, Riemann read the nine hundred pages in six days.
In 1846, Riemann entered the University of Gottingen. Riemann’s father wanted him to study theology so that’s what he did although he did attend some classes in mathematics. He enjoyed the mathematics classes so he asked his father if it was okay to switch to philosophy. This would allow him to study mathematics. Riemann was close to his family and needed his father’s permission to switch his studies. His father agreed to the switch and Riemann was able to take classes in mathematics from two prominent mathematicians—Moritz Stern and Carl Gauss.
Even though both Stern and Gauss taught at the University of Gottingen, it was not a great school for studying mathematics. Gauss only gave courses in elementary math to Riemann and did not seem to recognize Riemann’s mathematical talent, although, Stern claims that he did recognize Riemann’s talent.
In 1847, Riemann enrolled in the Berlin University which was popular at the time for mathematical studies, mainly because of the professors. While at the Berlin University, Riemann was able to study under Jakob Steiner, a Swiss mathematician who focused on geometry; Carl Jacobi, a German mathematician who studied differential equations and number theory (among other things); Gotthold Eisenstein, a German mathematician who focused on number theory and analysis; and Gustav Dirichlet, a German mathematician who also focused on number theory and analysis.
Riemann learned a lot from Eisenstein but the teacher who had the most influence on him was Dirichlet. Riemann and Dirichlet had similar ideas around explaining their ideas. Dirichlet would avoid long computations if at all possible when explaining his work and would try to make himself clear based on intuitive reasoning. Riemann liked this method and adopted it in his own work even though the approach was not strong enough to make the conclusions airtight.
While in Berlin, Riemann came up with his general theory of complex variables. This theory was the basis for Riemann’s later work.
In 1849, Riemann returned to the University of Gottingen. The German physicist W. E. Weberhad returned to the university to teach physics and Riemann worked as his assistant for eighteen months. Johann Listing was also a new professor of physics. Through these two men, Riemann learned a lot about theoretical physics. He also learned about topology from Listing.
Riemann spent three years at the university learning about physics, philosophy, and education while working on his PhD thesis under the guidance of his old teacher, Carl Gauss. Riemann submitted his thesis in November 1851. The thesis was on complex function theory and Riemann surfaces (although they weren’t known as this yet). Riemann’s thesis built on the work of Cauchy and Puiseux but is still regarded as an amazing piece of original work. Riemann defended his thesis and was awarded his PhD in December of 1851.
After gaining his PhD, Riemann continued his studies to gain his Habilitation. This is a degree that would allow him to become a lecturer at a university. It spent approximately two and a half years working on his dissertation for the Habilitation. His dissertation was on Fourier series and what one can say about the behaviour of a function represented by a trigonometric series. In this dissertation, he also discussed what was to become known as Riemann integrability.
The Dissertation Lecture
Once he completed his dissertation, Riemann needed to give a lecture to complete the requirements for his Habilitation. Riemann came up with three different lectures and his advisor, Carl Gauss, got to choose which one Riemann was to give. Riemann prepared two lectures on electricity and one on geometry. Riemann expected Gauss to choose the lectures on electricity but to his surprise, Gauss chose the geometry lecture.
Riemann delivered his lecture on June 10, 1954 and during the lecture, Riemann looked at two main issues. The first issue was how to define n-dimensional space. It was during this lecture that he defined what came to be known as Riemannian space. The second issue of the lecture was on how geometry relates to the world. He looked at real space and what the dimension of real space were. Riemann argued that the truth about space would come from actual experience, not from studying old books on geometry. Riemann stated that up close, space could be irregular but appear smother as we move away. If we move far enough away, Riemann thought that space might curve and could possible bend completely around into a closed system as if it were a ball.
Gauss was moved by the lecture and was amazed at the insight shown by Riemann. The papers from both his dissertation and his lecture were published a number of years later and the paper on the lecture became a mathematical classic.
Riemann was now a Privatdozent. A Privatdozent is a university lecturer who is not paid by the university and is instead paid by the students. Riemann gave his first course on partial differential equations and how they applied to physics. During the courses Riemann gave between the years 1855 and 1856, Riemann explained his theory of Abelian functions. Riemann’s theory of Abelian functions was published in 1857 and is also viewed as a mathematical masterpiece.
Riemann became a full professor at the University of Gottingen in 1859 after the death of both Gauss and Dirichlet. A few days after this, Riemann was elected a member of the Berlin Academy of Sciences. As a new member of the Academy, Riemann was required to present his most recent research.
In his presentation, Riemann looked at a zeta function and again, his work had a significant impact on the field of mathematics and physics.
Riemann made a number of important discoveries that have made him famous. He did not publish a lot of material but his ideas and the material that he did publish were all of the highest standard.
Riemann came up with elliptic geometry. This geometry is a non-Euclidean geometry and does not have parallel lines nor does the sum of a triangle’s angles add up to one hundred and eighty degrees. Riemann’s geometry unified and generalized Euclidean geometry, hyperbolic geometry and elliptic geometry. It also looked at mathematical space (a manifold) and a generalization of curves and surfaces.Riemann also proposed the existence of higher dimensions. Although it was not well understood when it first came out, Riemann’s mathematics were to change the way people saw the world around them.
Riemann began to look beyond two or three dimensional geometry and extended geometry into the area of multi-dimensional space (n-dimensions). Riemann’s theory was used by Einstein for Einstein’s general theory of relativity. It is also extremely important in areas of mathematics such as geometry and number theory.
Riemann’s metric was a method to describe how much a mathematical space (manifold) was bent or curved. To do this, Riemann developed the concept of a tensor, a collection of numbers placed at every point in space. A specific number of these tensors would be able to describe how distorted or curved a manifold was. For example, ten tensors (ten numbers) would be needed to describe the properties of a manifold in four dimensions regardless of its distortion.
Riemann’s Zeta Function and the Riemann Hypothesis
Riemann work with zeta functions also had a major impact on the field. Riemann determined that he could use a zeta function to build a three-dimensional landscape. He also realized that his zeta function could help to understand the distribution of prime numbers.
Riemann realized that in specific spots the surface of the three-dimensional landscape dropped down to a height of zero. These spots were termed the zeroes. Riemann was able to show that for some reason, the first ten zeroes formed a straight line through the three-dimensional landscape.
Riemann then recognized that the zeroes were related to the distribution of prime numbers. Riemann developed a hypothesis (the Riemann Hypothesis) that all the zeroes would be distributed on this same line. This hypothesis has not been proven and Riemann died before he could work on it. Unfortunately, many of Riemann’s papers were lost at the time of his death so it is impossible to tell if Riemann was close to proving his hypothesis. This is considered to be a “fundamental question” in mathematics and a prize of one million dollars has been offered to anyone who solves it.
Marriage and Death
Riemann married in June 1862 to Elise Koch, a woman he had met through his sister. Shortly after getting married, Riemann developed pleurisy (an inflammation in the lining around the lungs). Riemann had never been in good health and in fact, his mother, brother and three sisters all died fairly young. He ended up with tuberculosis, which he could have had for a long time given the deaths of his other family members.
In order to cure himself, he decided to take a leave of absence from the university and went to Italy which had a better climate for people suffering from tuberculosis. Moving to a better climate was one of the usual treatments for the disease.
Riemann spent the winter of 1862 in Sicily and then he traveled throughout Italy during the spring. He also visited a number of mathematicians he knew from his work at the university. He went back to Gottingen in the summer of 1863 but his health quickly got worse so he had to return to Italy. He spent a little over a year in northern Italy (August 1864 to October 1865) and then he returned to Gottingen once again to spend the winter there. Once the winter was over, he again went to Italy in June 1866.
Riemann went to Selasca on Lake Maggiore and it was here that he died. The day before his death, Riemann was sitting under a tree working on a paper about natural philosophy. The paper was never finished. Riemann died on July 20, 1866 and was buried at Biganzole cemetery. He was thirty-nine years old.