Global and Local

Discrete versus continuous, simple versus complex, global versus local, linear versus nonlinear, deterministic versus stochastic, analytic versus numerical, constructive versus nonconstructive – those contrasts are among the great organizing themes of mathematics. They are forks in the road of mathematical technique – the concepts along one fork are very different from those along the other, even when they give complementary views on the same phenomena. It is hard to find a clear and elementary exposition of any one of those contrasts, but perhaps it is the global/local distinction that is worst served by current theory. A beginning graduate student in mathematics is certainly expected to have a sense of the distinction and to be able to talk coherently about “local minima versus global minimum”, “a local solution to a d.e. that is not extendable to a global solution”, and so on. But there is no article available on the distinction in Wikipedia, the Springer Encyclopedia of Mathematics, or Wolfram Mathworld. (Wikipedia and Mathworld do have very brief articles on “local” in the sense of topological spaces.) This article brings together some mostly familiar examples of the global/local distinction from a range of different areas, as a basis for explaining clearly what the distinction is and why it is central to mathematics.