Unifying theories in mathematics |
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.
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:
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.
See also
philosophy of mathematics, foundations of mathematics
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