James Burke makes few connections from mathematics to all the science and technology(except for in the most indirect ways). I've filled in a little bit throughout this blog. Some more mathematical connections would be Monge's descriptive geometry. This is basically technical drawing - schematics.
Sometimes you want a more functional schematic than a pure drawing. A functional schematic looks nothing like the actual physical entity; it just shows connections whether electrical or mechanical. It shows objects and how they are related. Functional schematics often has codes to the literal, kind of projective geometric, drawings of the object(whether, computer, radio, car, airplane etc).
Schematics isn't exactly mathematics; but, then again, calendars are not either; yet, calendars allowed the Egyptians and Mesopotamians to make agriculture happen. What's more schematics allows techs to fix things they otherwise would have no business touching. I was former Navy; the saying goes, the airplanes, and really everything in it are 'sailer proof'! Modularizing of aircraft components and schematics allows technicians to get a job and do the job and keep the airplanes up pretty continuously for the last hundred years!(not to mention massive amounts of statistical analyses; every nut and bolt has had a good amount of statistical analyses by engineers to make airplanes or anything work as they should).
Monge was much more of a mathematician than this; he's generally credited with starting differential geometry. Differential geometry in the hands of Frederick Gauss and Bernard Riemann at least became pretty powerfull. Differential geometry can determine the overal shape of space by considerations of local curvature. This hints at it's application to General relativity. In the late 1700s, early 1800s, mathematicians created non-euclidean geometry(Euclidean geometry being the plane geometry of today's high schools) just by switching out the fifth postulate. There were three, euclidean geometry with the parrallel postulte, and two others with angles suming to more than ninety degrees or less; one is a kind of spherical geometry, the other is a kind of hyperbolic geometry(or the use of a psuedosphere). These were not differential geometry; but, Bernard Riemann, in one of the things he did was to create a differential geometry which could derive all three!
As for schematics and differential geometry, well, maybe someday human's will need schematics of spacetime to get around the solar system and then interstellar space!