Johan Jensen (1859–1925) was a Danish mathematician who proved one of the most-used inequalities in all of maths — and did it in his spare time, because his day job was running a telephone company. He never held a university post. He was, quite literally, an amateur who beat the professionals.
Jensen's inequality is a deceptively simple statement about "curvy" functions: for a convex
function, the average of the outputs is at least the output of the average. That one line shows
up constantly — in probability, economics, information theory and machine learning — often as
the crucial step that makes a hard proof suddenly work. It pairs naturally with
Jensen was self-taught in the higher reaches of mathematics and made his living as an engineer and manager at the Copenhagen Telephone Company, eventually rising to head of its technical department. He did all his celebrated research on the side, as a devoted hobby. It is a cheering reminder that you do not need a fancy title to make a permanent mark — Jensen's inequality has outlived every phone he ever installed.