Unifying conjecture

In mathematics, a unifying conjecture is a claim that one branch of mathematics is isomorphic to another. Thus, a calculation in one branch that is difficult may be translated into the other branch and become easier to perform.

A well-known example is the Taniyama-Shimura conjecture, now the Taniyama-Shimura theorem, which proposed that each elliptic curve over the rational numbers can be translated into a modular form. The two fields, elliptic curves and modular curves, had not been considered as directly related. This conjecture stood for decades before being proved.

It is anticipated that unifying conjectures in general are difficult to discover and hard to prove, so they are expected to retain the title 'conjecture' for a relatively long time.

Unifying conjecture tends also to be referenced often even in finished mathematics - statements such as "assuming Taniyama-Shimura..." were not uncommon in the mathematical literature long before the theorem was proved. Thus a conjecture that is described as 'unifying' in this sense may get more attention.

The difference between a 'unifying' and a merely 'useful' conjecture tends to be social: the implication is that one is 'unifying' several subcultures within mathematics, different groups of people as opposed to different sets of theorems.