Sunday 7 November 2004

A Familiar Ring To It

Although it's not my primary, er, field, this one's for the Pure Mathematics Groupies out there. It's a pdf of a fairly abstruse paper on Trees and Rings - mathematical entities which along with Groups and Fields are to do with permutations of finite elements, like how many different poker hands there are, how many different legal chess games are possible, how many anagrams there are for a set of letters, and so on (to grossly over-simplify).

It's a fairly advanced paper, far beyond my skills (which have rusted after 20 years of disuse), so I could only understand about 10% of it without consulting old textbooks. Even then, I'd have to become familiar with advances and terminology in maths that have only become current in the last two decades. For most people, it would appear as mystifying as, say, Derrida at his worst. Just read the Abstract (below) to see what I mean.

Abstract. We develop the theory of “branch algebras”, which are infinite dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees.
In particular, we construct an algebra over the field of two elements, that is finitely generated, prime, infinite-dimensional but with all proper quotients finite, has a recursive presentation, is graded, and has Gelfand-Kirillov dimension 2.
Alas, some of the concepts really do require their own special terminology: the object is not to Blind with Science, or Baffle with BS, it's to communicate very exact and abstruse ideas to people who have the right background.

The interesting thing about it is the first line :
Rings are powerful tools, and those arising from groups have been studied in great length [32, 39, 40]. The first author’s long-awaited monograph on the topic should prove illuminating [5].
The references,
[5] Bilbo Baggins, There and Back Again. . . A Hobbit’s Tale by Bilbo Baggins, in preparation.
[32] Donald S. Passman, The algebraic structure of group rings, Wiley-Interscience [John Wiley & Sons], New York, 1977, ISBN 0-471-02272-1, Pure and Applied Mathematics.
[39] John Ronald Reuel Tolkien, The Lord of the Rings, Houghton Mifflin Company, 2002.
[40] _________, The Hobbit , Houghton Mifflin Company, 1999.
And, of course, the co-authors.
Seen via Yet Another Weird SF Fan.

No comments: