| Line 5: |
Line 5: |
| | ==Early life== | | ==Early life== |
| | Riemann was born into an impoverished family in [[Germany]], a devoutly [[Christian]] son of a [[Lutheran]] pastor. He was [[homeschooling|homeschooled]] by his father, and was sent to attend a prominent school (Johanneum at Luneburg) at the age of 16. Riemann quickly became bored with the math class and asked the director for more advanced material. The director gave him most advanced math books (including [[Leonhard Euler]]'s works and Adrien Marie Legendre's Theory of Numbers). Riemann mastered them in merely a few days.<ref>http://www.andrews.edu/~calkins/math/biograph/199899/bioriema.htm</ref> | | Riemann was born into an impoverished family in [[Germany]], a devoutly [[Christian]] son of a [[Lutheran]] pastor. He was [[homeschooling|homeschooled]] by his father, and was sent to attend a prominent school (Johanneum at Luneburg) at the age of 16. Riemann quickly became bored with the math class and asked the director for more advanced material. The director gave him most advanced math books (including [[Leonhard Euler]]'s works and Adrien Marie Legendre's Theory of Numbers). Riemann mastered them in merely a few days.<ref>http://www.andrews.edu/~calkins/math/biograph/199899/bioriema.htm</ref> |
| | + | |
| | + | ==Riemann's Four Great Papers== |
| | + | |
| | + | Riemann wrote four brilliant papers that laid the foundation for modern mathematics:<ref name="Chakra">https://www.quora.com/What-were-Bernhard-Riemanns-contributions-to-mathematics Analysis of Mathematician Hetu Chakra]</ref> |
| | + | |
| | + | ===Complex Analysis=== |
| | + | ''Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse'' (1851). This paper covers the scope of what is commonly taught in a college complex analysis course today, and sets forth the Riemann Mapping Theorem. |
| | + | |
| | + | ===Differential Geometry=== |
| | + | ''Uber die Hypothesen, welche der Geometrie zu Grunde liegen'' (1854). This paper discusses manifold, curvature, the Riemann curvature tensor, and the Riemann metric. This work lays the mathematical foundation for the [[General Theory of Relativity]] a half-century earlier. This paper established Riemannian geometry. |
| | + | |
| | + | ===Algebraic Geometry=== |
| | + | ''Theorie der Abel'schen Functionen'' (1857). Called one of the finest mathematical works ever, this paper discusses Riemann Surfaces and moduli spaces, touches on modern topology, and explains the Riemann-Roch theorem and the Riemann-Hurwitz formula. |
| | + | |
| | + | ===Analytic Number Theory=== |
| | + | ''Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse'' -- in English, "On the Number of Primes Less Than a Given Magnitude" (1859). The [[Riemann hypothesis]], which is the greatest unsolved problem in mathematics today, is in this paper. |
| | | | |
| | ==Riemannian geometry== | | ==Riemannian geometry== |
| − | Riemann created a new type of geometry, which was elliptical. Riemannian geometry then became the basis for advances in [[physics]] in the 20th century. | + | Riemann created a new type of geometry, which became modern differential geometry. Riemannian geometry then became the basis for new theories in [[physics]] in the 20th century. |
| | | | |
| | Most of the theorems of Riemannian geometry are different from their counterparts in Euclidian geometry. For example, the sum of the angles in a triangle in Riemannian geometry is greater than 180 degrees, which is the sum in Euclidian geometry. Also, there are no parallel lines in Riemannian geometry. | | Most of the theorems of Riemannian geometry are different from their counterparts in Euclidian geometry. For example, the sum of the angles in a triangle in Riemannian geometry is greater than 180 degrees, which is the sum in Euclidian geometry. Also, there are no parallel lines in Riemannian geometry. |