One typical way that GPS is invoked as an application of mathematics is through the use of general relativity. Most people have a rough idea of what the GPS system does: there are some (27) satellites flying in the sky, and a GPS device on the surface of the earth determines its position by radio communication with the satellites. It is also pretty clear that this is a hard problem to solve, with or without mathematics. The basic idea is that if your GPS device measures its distance between 3 different satellites, then it knows that it lies on three level sets which must intersect at a point. This is the standard idea of triangulation. Of course measuring distance is hard to do, and relativity comes into play in many different, nontrivial ways, but there is one way in particular that is interesting and easy to explain.
If one uses the euclidean metric to determine the distance (so, straight lines) from the GPS to the satellite, then it will be impossible to determine the location on the earth to a high degree of accuracy. So instead the GPS system uses the kerr metric, that is the lorentz metric that models spacetime outside of a spherically symmetric, rotating body. Naturally this metric gives a different, more accurate distance between the observer on earth and the satellite. The thing that is surprising to people is that the switch from the euclidean to kerr is required to get really accurate gps readings. In other words, without relativity you might not be able to use that iphone app to find your car in the grocery store parking lot.
People are often surprised and interested to learn that the difference differences between relativity and newtonian gravity really are observableand useful. Other standard examples are the precession of the perihelion of mercury (which was a famous unsolved problem before the introduction of GR) and the demonstration that ligth light rays do not travel along straight lines by photographing the sun during an eclipse. This last observation demonstrated, for instance, that the metric on the universe is not the trivial flat one.
