Bernhard Riemann

Born: 1826
Died: 1866
Title: Revolutionary of Geometry

The Man Who Paved the Way for Einstein

Riemann was Gauss's student, extremely introverted and shy, but his ideas were revolutionary. He died at age 39, but in his short life, he revolutionized geometry and laid the mathematical foundation for Einstein's theory of relativity. He showed that space itself could be curved—an idea that seemed absurd but is now fundamental to our understanding of the universe.

Core Contributions - Deep Analysis

Riemannian Geometry: Curved Spaces

Before Riemann, geometry was flat—Euclidean. Riemann asked: What if space itself is curved?

He developed a new geometry where:

• Parallel lines can meet
• Triangles can have angles that don't sum to 180°
• Distance is measured differently

This non-Euclidean geometry became the mathematical foundation of Einstein's General Theory of Relativity, which describes how gravity curves spacetime.

The Riemann Hypothesis: Mathematics' Greatest Unsolved Problem

Riemann proposed a conjecture about the zeros of the zeta function $\zeta(s)$:

All non-trivial zeros of the Riemann zeta function have real part equal to 1/2.

This is considered the most important and difficult unsolved problem in mathematics today. It concerns the ultimate pattern of how prime numbers are distributed in nature. This hypothesis, if proven, would reveal deep secrets about the distribution of prime numbers. It's one of the seven Millennium Prize Problems, with a $1 million prize for its solution.

Riemann Integral

The rigorous definition of integration used in calculus today. Riemann showed how to precisely define the area under a curve, making calculus mathematically rigorous.

The Shy Genius

Riemann was extremely shy and struggled with health problems throughout his life. He was terrified of public speaking—his first lecture at Göttingen was so nervous that only Gauss (his advisor) attended. But his mathematical ideas were revolutionary.

Legacy

Riemann's work transformed mathematics and physics:

• Geometry: Non-Euclidean geometry, Riemannian manifolds
• Analysis: Riemann integral, complex analysis
• Physics: Foundation for general relativity
• Number Theory: Riemann hypothesis connects geometry and primes

His idea that space itself can be curved was so revolutionary that it took decades for physicists to fully appreciate it. Today, Riemannian geometry is essential to:

• General relativity
• String theory
• Differential geometry
• Topology

Riemann showed that mathematics can describe realities that seem impossible—and that those realities might actually exist in our universe.