TITLE: Examples of mathematics motivated by technological considerations
QUESTION [33 upvotes]: I would like examples of technological advances that were made possible only by the creation of new mathematics.  I'm talking about technology that was desired in some period of history but for which the mathematics of the time were not enough.
It would be great if I could have an example in which the technology needed was associated with an industrial problem, and it would be even better if the technology developed had a significant impact in society.

REPLY [2 votes]: One of the most important tools used in Computer Science is the Fast Fourier Transform, which basically made much of image processing possible - described by Gilbert Strang as "the most important numerical algorithm of our lifetime". Note however that unknown to the inventors James Cooley and John Tukey it seems that Gauss actually had already discovered the algorithm (see this paper for more details) but not published it! 
Much of modern medical imaging is only possible due to mathematical advances - for a flavour here are a few example projects from the University of Eindhoven:

Lie Group Theory & Differential Geometry for Medical Image Analysis, Remco Duits
Riemann-Finsler Geometry for Brain Connectivity and Tractography, Luc Florack & Andrea Fuster
Differential Geometry in Complex Medical Imaging & Relativity Theory, Andrea Fuster