Jean-Baptiste Joseph Fourier

Fourier had an unusual CV even for his generation. Son of a tailor, orphaned at nine, educated at a Benedictine school, radicalised in the Revolution, arrested twice for the wrong politics at the wrong moment, rescued from the guillotine by Robespierre's own fall. He taught at the École Polytechnique, got drafted into Napoleon's Egyptian campaign, and spent three years governing Lower Egypt before returning to France.

Back home, as prefect of Grenoble, he started studying heat flow in solids — a practical problem, not a glamorous one. In 1807 he submitted a paper arguing that the temperature profile in a bar, however it started, could be written as a sum of sinusoids, and that each sinusoid would decay on its own independent timescale. Lagrange and Laplace, two of the great mathematicians of the age, read it and said flatly: not possible. Discontinuous functions can't equal sums of smooth ones. They blocked its publication. Fourier expanded the argument into a book, Théorie analytique de la chaleur (1822), and forced the issue.

He was right, in a sense nobody in 1807 could have made precise. It took Dirichlet, Riemann, Lebesgue and the invention of measure theory to state clearly when his claim holds. But the tool Fourier invented — writing a function as a sum of its modal components — is now the most-used technique in physics. Quantum mechanics, optics, signal processing, MRI, JPEG, seismology, spectroscopy: all run on it.