Unifying Theories in Mathematics - Uniting Theories

Uniting Theories

On a less grandiose scale, there are frequent instances in which it appears that sets of results in two different branches of mathematics are similar, and one might ask whether there is a unifying framework which clarifies the connections. We have already noted the example of analytic geometry, and more generally the field of algebraic geometry thoroughly develops the connections between geometric objects (algebraic varieties, or more generally schemes) and algebraic ones (ideals); the touchstone result here is Hilbert's Nullstellensatz which roughly speaking shows that there is a natural one-to-one correspondence between the two types of objects.

One may view other theorems in the same light. For example the fundamental theorem of Galois theory asserts that there is a one-to-one correspondence between extensions of a field and subgroups of the field's Galois group. The Taniyama–Shimura conjecture for elliptic curves (now proven) establishes a one-to-one correspondence between curves defined as modular forms and elliptic curves defined over the rational numbers. A research area sometimes nicknamed Monstrous Moonshine developed connections between modular forms and the finite simple group known as the Monster, starting solely with the surprise observation that in each of them the rather unusual number 196884 would arise very naturally. Another field, known as the Langlands program, likewise starts with apparently haphazard similarities (in this case, between number-theoretical results and representations of certain groups) and looks for constructions from which both sets of results would be corollaries.