Applications of mathematics
Sending a man to the Moon (and back).
Hilbert once remarked half-jokingly that catching a fly on the Moon would be the most important technological achievement. "Why? "Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind." (Quoted from Hilbert-Courant by Constance Reid, Springer, 1986, p. 92).
The task obviously required solving plenty of scientific and technological problems. But the key breakthrough that made it all possible was Richard Arenstorf's discovery of a stable 8-shaped orbit between the Earth and the Moon. This involved the development of a numerical algorithm for solving the restricted three-body problem which is just a special non-linear second order ODE (see also my answer to the previous MO question).
Another orbit, also mapped by Arenstorf, was later used in the dramatic rescue of the Apollo 13 crew.
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 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 differences between relativity and newtonian gravity really are observable. 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 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.
Here are some examples
to quote my favorite one:
In 1998, mathematics was suddenly in the news. Thomas Hales of the University of Pittsburgh, Pennsylvania, had proved the Kepler conjecture, showing that the way greengrocers stack oranges is the most efficient way to pack spheres. A problem that had been open since 1611 was finally solved! On the television a greengrocer said: “I think that it's a waste of time and taxpayers' money.” I have been mentally arguing with that greengrocer ever since: today the mathematics of sphere packing enables modern communication, being at the heart of the study of channel coding and error-correction codes.
In 1611, Johannes Kepler suggested that the greengrocer's stacking was the most efficient, but he was not able to give a proof. It turned out to be a very difficult problem. Even the simpler question of the best way to pack circles was only proved in 1940 by László Fejes Tóth. Also in the seventeenth century, Isaac Newton and David Gregory argued over the kissing problem: how many spheres can touch a given sphere with no overlaps? In two dimensions it is easy to prove that the answer is 6. Newton thought that 12 was the maximum in 3 dimensions. It is, but only in 1953 did Kurt Schütte and Bartel van der Waerden give a proof.
The kissing number in 4 dimensions was proved to be 24 by Oleg Musin in 2003. In 5 dimensions we can say only that it lies between 40 and 44. Yet we do know that the answer in 8 dimensions is 240, proved back in 1979 by Andrew Odlyzko of the University of Minnesota, Minneapolis. The same paper had an even stranger result: the answer in 24 dimensions is 196,560. These proofs are simpler than the result for three dimensions, and relate to two incredibly dense packings of spheres, called the E8 lattice in 8-dimensions and the Leech lattice in 24 dimensions.
This is all quite magical, but is it useful? In the 1960s an engineer called Gordon Lang believed so. Lang was designing the systems for modems and was busy harvesting all the mathematics he could find.
He needed to send a signal over a noisy channel, such as a phone line. The natural way is to choose a collection of tones for signals. But the sound received may not be the same as the one sent. To solve this, he described the sounds by a list of numbers. It was then simple to find which of the signals that might have been sent was closest to the signal received. The signals can then be considered as spheres, with wiggle room for noise. To maximize the information that can be sent, these 'spheres' must be packed as tightly as possible.
In the 1970s, Lang developed a modem with 8-dimensional signals, using E8 packing. This helped to open up the Internet, as data could be sent over the phone, instead of relying on specifically designed cables. Not everyone was thrilled. Donald Coxeter, who had helped Lang understand the mathematics, said he was “appalled that his beautiful theories had been sullied in this way”.