Uncertainity Principle(s)

Most people know a formulation of the uncertainity principle in quantum mechanics (by Heisenberg), where it basically gives a bound (in terms of the standard deviations of ~) for simultaneous measurements of complementary properties (i.e. measurement operators with non-negative commutator).

There are a number of other places where the principce turns up. It comes from an application of the cauchy schwarz inequality to the fourier transformation. Here's a set of (german) slides from a lecture. Recently Terry Tao posted on a a formulation of the principle due to Mongomery, which has applications in analytic number theory.