Wednesday, January 04, 2006

Deformations of mathematics

The review mentioned yesterday is finished. As I had forgotten the word limit, it had to be hacked back, so I've made the original available here.

I noticed this paper on Tropical Geometry today on the ArXiv. The idea of tropical mathematics is that many of the constructions of ordinary mathematics usually carried out over the reals or complex numbers can be profitably transferred to various tropical semirings. The term 'tropical' was chosen supposedly to commemorate a Brazilian mathematician, Imre Simon. An example of one of these semirings is R È {-¥}, where "x + y" = max{x, y} and "x.y" = x + y. A good introduction to tropical mathematics is this paper by Litvinov, where he tells us:
Idempotent mathematics can be treated as a result of a dequantization of the traditional mathematics over numerical fields as the Planck constant h tends to zero taking imaginary values... In other words, idempotent mathematics is an asymptotic version of the traditional mathematics over the fields of real and complex numbers.

The basic paradigm is expressed in terms of an idempotent correspondence principle. This principle is closely related to the well-known correspondence principle of N. Bohr in quantum theory. Actually, there exists a heuristic correspondence between important, interesting, and useful constructions and results of the traditional mathematics over fields and analogous constructions and results over idempotent semirings and semifields (i.e., semirings and semifields with idempotent addition).

A systematic and consistent application of the idempotent correspondence principle leads to a variety of results, often quite unexpected. As a result, in parallel with the traditional mathematics over fields, its “shadow,” the idempotent mathematics, appears. This “shadow” stands approximately in the same relation to the traditional mathematics as does classical physics to quantum theory, see Fig. 1. In many respects idempotent mathematics is simpler than the traditional one. However the transition from traditional concepts and results to their idempotent analogs is often nontrivial.
Another example of the deformation of large tracts of mathematics also invokes the quantum language. This historical paper gives a detailed account of this 'q-disease'. This Week's Finds aficionados will recall the series on q-mathematics (weeks 183-188).

Vladimir Arnold has made some fascinating suggestions concerning systematic transformations of blocks of mathematics in his 'Polymathematics'. (See also another account of these ideas in Lecture 2 of the Toronto Lectures on this page.) Arnold suggests that there is a way of thinking systematically about the mysterious relations between apparently diverse fields so frequently noted by mathematicians via various informal processes:
The informal complexification, quaternionization, symplectization, contactization etc., described below, are acting not on such small things, as points, functions, varieties, categories or functors, but on the whole of mathematics. I have successfully used these ideas many times as a method to guess new results. I hope therefore that in the future this method of the multiplication of mathematics will be as standard, as is now the transition from finite-dimensional linear algebra to the theory of integral equations and to functional analysis.
And
The main dream (or conjecture) is that all these trinities are united by some rectangular "commutative diagrams". I mean the existence of some "functorial" constructions connecting different trinities. (Arnold lecture 2: 10)