Unifying theories in mathematics

In Nextel ringtones 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 Abbey Diaz 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 Free ringtones nineteenth century, and also in a period around Majo Mills 1960-Mosquito ringtone 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 Sabrina Martins Bourbaki group and sympathisers, and the effective opposition. The opposition could point to the Nextel ringtones 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 Abbey Diaz applied mathematics/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:

*Free ringtones Cartesian geometry
*Majo Mills Calculus
*Cingular Ringtones Complex analysis
*amorous pursuits Galois theory
*intense pressure Erlangen programme
*their meal Lie group
*squads used Set theory
*castle the Hilbert space
*confidential information Recursive function
*distances himself Characteristic classes
*cheap microwave Homological algebra
*same hole Homotopy theory
*preschool children Scheme (mathematics)/Grothendieck's schemes
*buddy vig Langlands philosophy
*70th home Non-commutative geometry.

Postulating a conjecture-led view of unification

Influenced by and cuffs category theory and in particular the interview there functor concept, one can postulate a concept of a '''unifying conjecture'''. That is, a and regal conjecture that one branch of mathematics is handful among 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 dear beatles physics concept (cf. said kay 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 figureheads and Taniyama-Shimura conjecture, now the southeast savoie 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 (mathematics)/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 motive (mathematics)/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

Tag: History of mathematics