Bernhard Riemann (1826–1866) lived only thirty-nine years and was painfully shy, yet he reshaped mathematics so thoroughly that a startling number of its landmarks carry his name. The most down-to-earth is the one that finally said, rigorously, what an integral is: chop the area under a curve into thin rectangles, add them up, and refine — the Riemann integral.

The geometry of curved space

Asked by his teacher Gauss to lecture on the foundations of geometry, Riemann invented an entire new subject on the spot: Riemannian geometry, the mathematics of curved spaces of any dimension. At the time it looked like pure abstraction. Half a century later Einstein picked it up and found it was exactly the language he needed for general relativity — gravity as the curvature of spacetime. Riemann had built the toolbox decades before anyone knew what it was for.

The greatest unsolved problem in mathematics

In a single short paper on prime numbers, Riemann made a guess about where a certain function's zeros lie — the Riemann Hypothesis. It controls how the primes are scattered among the integers, and more than 160 years later it is still unproven, carrying a one-million-dollar Millennium Prize for whoever cracks it. Not bad for a sickly, retiring man who died of tuberculosis before forty, leaving behind ideas the rest of us are still chasing.

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The full story (with far fewer jokes) is on Wikipedia: Bernhard Riemann — Wikipedia.