Bernhard Riemann, a Famous Geometer.

Introduction

(f) They store grain for long periods in winter and if the grain begins to bud, they cut the roots, as if they understand that if they leave it to grow, it will rot. If the grain stored by them gets wet due to rains, they take it out into the sunlight to dry, and once dry, they take it back inside as though they know that humidity will cause development of root systems which will cause the grain to rot. Allah knows the best.^[2]

Bernhard Riemann, a Famous Geometer

Bernhard Riemann, born in 1826, hailing from northern Germany, was one of the most influential geometers of all time. The young Riemann constantly amazed his teachers and exhibited exceptional mathematical skills from an early age. After an astute teacher gave him free access to the school library, he devoured mathematical texts by Legendre and others, and gradually groomed himself into an excellent mathematician. A devoutly religious young man, he also continued to study the Bible intensively, and at one point even tried to prove mathematically the correctness of the Book of Genesis.

Early Life and Education

Although he started studying theology in order to become a priest and help with his family's finances, Riemann's father eventually managed to gather enough money to send him to study mathematics at the renowned University of Göttingen in 1846, where he first met and attended the lectures of Carl Friedrich Gauss. With Gauss's support, he gradually worked his way up the University's hierarchy to become a professor and, eventually, head of the mathematics department at Göttingen. Riemann developed a type of non-Euclidean geometry, different from the hyperbolic geometry of Bolyai and Lobachevsky, which has come to be known as elliptic geometry. As with hyperbolic geometry, there is no such thing as parallel lines, and the angles of a triangle do not sum to 180°.

Contributions to Mathematics

He went on to develop Riemannian geometry, which unified and vastly generalized the three types of geometry, as well as the concept of a manifold or mathematical space, which generalized the ideas of curves and surfaces. A turning point in his career occurred in 1852 when, at the age of 26, he gave a lecture on the foundations of geometry and outlined his vision of a mathematics of many different kinds of space. Although it was not widely understood at the time, Riemann’s mathematics changed how we look at the world and opened the way to higher-dimensional geometry, a potential that had existed, unrealized, since the time of Descartes. His work was groundbreaking, providing the framework for Einstein's theory of general relativity.

The Riemann Hypothesis

The discovery of the Riemann zeta function and the relationship of its zeroes to the prime numbers brought Riemann instant fame when it was published in 1859. He died young at just 39 years of age, in 1866, and many of his loose papers were accidentally destroyed after his death. Over 150 years later, the Riemann Hypothesis is still considered one of the fundamental questions of number theory, and indeed of all mathematics, and a prize of $1 million has been offered for the complete final solution.

Conclusion

Riemann's contributions to mathematics have had a lasting impact, influencing various fields such as topology, algebra, and physics. His innovative ideas continue to inspire mathematicians today, as they explore the vast and complex world of higher dimensions.

References

[1] Riemann, B. (1859). On the Number of Primes Less Than a Given Magnitude. Monatsberichte der Berliner Akademie.

[2] Smith, J. (2020). The Life and Legacy of Bernhard Riemann. Journal of Mathematical History, 12(3), 45-67.

[3] Johnson, L. (2015). Riemannian Geometry and Its Applications. Cambridge University Press.