A marriage for the ages

Abstract

We begin our journey with a surprising analogy between mathematics and the Hawaiian Islands. Each can be thought of as an archipelago. In mathematics, as in the Hawaiian Islands, some islands are large — areas such as classical algebra, geometry, topology, number theory, and so on. There are smaller ones, too, sayMartingalemethods in statistics. In both math and the Hawaiian Islands, the islands are not actually separate but have connections that are not always obvious. From the viewpoint of someone in an airplane looking down, Hawaii looks like a collection of land masses separate from each other. But in reality the airplane passenger is just seeing the tops of a mostly submerged mountain range, and the whole mountain range isn’t seen because the water obscures the view. If we could pull a plug and let the water drain away, we would gradually see the whole mountain range come into view, and we’d discover that the islands are in fact connected. The mathematical analog of water is ignorance, as it clouds our ability to see thewhole. Pulling the plug in this case corresponds to decreasing the level of ignorance, and as that happens, mathematicians discover connections — often unsuspected — between areas of math that were previously assumed to be separate. Discovering connections between seemingly unrelated ideas has often been a basic feature of significant mathematics. For example, oldfashioned geometry and old-fashioned algebra each developed separately