Unifying theories in mathematics
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In mathematics, there have been many attempts down the centuries to unify the whole subject. This is in line with the feeling of many (not all) of the greatest mathematicians, who have had an intuitive sense that the whole subject hangs together, whatever the outward appearances of disparity of subject matter and methods might suggest. One approach is to postulate a unifying theory in mathematics; this was particularly favoured in the mathematics of the second half of the nineteenth century, and also in a period around 1960-1980.
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Intellectual fashion
Several attempts to unify an area may occur simultaneously. Which attempts benefit from the most attention is partly a matter of intellectual fashion. One fundamental rift occurred between the Bourbaki group and sympathisers, and the effective opposition. The opposition could point to the list of combinatorics topics as containing matters that Bourbaki ignores. Whatever the contrast between trendy (me-too) research and honest (problem-solving led) original work, the field has had to absorb numerous such theories. They add up to a list of influential concepts. On the other hand, mathematics as a whole is too dynamic internally, and is also often externally so strongly coupled to its applications, for conformity to a unifying theory to be a good indicator of value.
Reference list of major unifying concepts
A short list of these theories might include:
- Cartesian geometry
- Calculus
- Complex analysis
- Galois theory
- Erlangen programme
- Lie group
- Set theory
- Hilbert space
- Recursive function
- Characteristic classes
- Homological algebra
- Homotopy theory
- Grothendieck's schemes
- Langlands philosophy
- Non-commutative geometry.
Postulating a conjecture-led view of unification
Influenced by category theory and in particular the functor concept, one can postulate a concept of a unifying conjecture. That is, a conjecture that one branch of mathematics is isomorphic to another, in a loose or possibly quite precise sense. A calculation in one branch that is difficult might be translated into the other branch and become easier to perform. Here the unification is essentially modelled after the physics concept (cf. GUT).
Since the history is rarely tidy, this can only really be judged on the basis of 'case studies'.
Recent developments in relation with modular theory
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 (in such a way as to preserve the associated L-function). There are difficulties in identifying this with an isomorphism, in any strict sense of the word. Certain curves had been known to be both elliptic curves (of genus 1) and modular curves, before the conjecture was formulated (about 1955). The surprising part of the conjecture was the extension to factors of Jacobians of modular curves of genus >1. It had probably not seemed plausible that there would be 'enough' such rational factors, before the conjecture was enunciated; and in fact the numerical evidence was slight until around 1970, when tables began to confirm it. The case of elliptic curves with complex multiplication was proved by Shimura in 1964. This conjecture stood for decades before being proved in generality.
In fact the Langlands philosophy is much more like a web of unifying conjectures; it really does postulate that the general theory of automorphic forms is regulated by the L-groups introduced by Robert Langlands. His principle of functoriality with respect to the L-group has a very large explanatory value with respect to known types of lifting of automorphic forms (now more broadly studied as automorphic representations). While this theory is in one sense closely linked with the Taniyama-Shimura conjecture, it should be understood that the conjecture actually operates in the opposite direction. It requires the existence of an automorphic form, starting with an object that (very abstractly) lies in a category of motives.
Another significant related point is that the Langlands approach stands apart from the whole development triggered by monstrous moonshine (connections between elliptic modular functions as Fourier series, and the group representations of the Monster group and other sporadic groups). The Langlands philosophy neither foreshadowed nor was able to include this line of research.
Isomorphism conjectures in K-theory
Another case, which so far is less well-developed but covers a wide range of mathematics, is the conjectural basis of some parts of K-theory. The Baum-Connes conjecture, now a long-standing problem, has been joined by others in a group known as the isomorphism conjectures in K-theory. These include the Farrell-Jones conjecture and Bost conjecture.