One example of mathematical achievement that has application to the "industry" of medicine is the Radon transform, that enables to produce x-ray tomographies.
The Radon transform of a plane section of your body is the set of information that you obtain after shooting x-rays through all the possible lines contained in that plane, and recording the intensity of the ray that comes out at the other side. More formally, its an integral transform: it's the result of integrating the density of the body tissue along each of the lines.
Johann Radon defined his transform in 1917, and calculated a formula for it's its inverse. That is, he deviced a way to recover from the transformed data the density of tissue in each point of the plane section. The inversion formula involved a lot of calculations that couldn't be handled at that time.
Cormack and Newbold implemented the idea with the aid of computers to handle the calculations. They got a Nobel Price in 1979 because of this.
