An allergy? You get an allergy, you’re a scientist
Apologies for the post title, which is probably only comprehensible if (a) you remember a certain advert for British Telecom, and (b) you tolerate really, really lame puns…
So, in response to a polite challenge/request towards the tail of this comment thread, here is my attempt at describing something I’ve been working on in my research, in at most 500 words, using as little technical jargon as possible. It was written in an hour or so while procrastinating over the actual work itself, and hasn’t been subbed in any way, so is no doubt far from best possible in style or in selection.
As I will point out in a subsequent post, when I give the “version for mathematicians”, most of what follows is of course a lie in the strictest sense; however, they are untruths not intended to deceive, but to serve as analogies. Which I hope you get, even if you’re not a scientist.
Mathematics often deals with symmetries of an object or construct: that is, the reversible mappings from that object to itself which preserve the given structure of interest. Thus, at least in the context of Euclidean geometry, the symmetries of a circle are the rotations.
If we drop the word “reversible”, and consider *partial* symmetries, then matters become more complicated, and potentially more interesting. In special infinite-dimensional settings, certain partial symmetries, though not themselves symmetries, may be regarded as shadows of symmetries in yet higher dimensions. More precisely, given our object X and a partial symmetry S, it may be possible to find a larger object Y in which X embeds, and a “genuine” symmetry T of Y, such that T takes X to itself and coincides with S on X. One then hopes to understand S by examining T.
A partial symmetry S of this kind is in some sense “residually reversible” (non-standard terminology); it may not be reversible, but we can make it so by passing to a larger ambient object. Such a partial symmetry cannot “destroy information” or “collapse structure” (because if it did, this property would pass to the putative extension T, and this is forbidden since T is reversible). The prototypical example arises if we take X to be a collection of strings, which may have infinite length, but which have a fixed starting point (one may think of the decimal expansions of numbers between 0 and 1, as a very loose model). Then our “residually reversible” mapping S is simply the operation “shift everything one place to the right” (which in our decimal-expansion model corresponds to “divide by 10”). The required “dilation” of S and X to T and Y is obtained by “enlarging X to the left” (which in our decimal-expansion model means “consider all positive numbers, not just those between 0 and 1”).
One of my current research projects examines certain restricted settings where one can state and prove results of the following form: any partial symmetry which is residually reversible is in fact a full, “genuine” symmetry. Crudely speaking, there are no “one-sided shift operators” of the sort described in the previous paragraph: or, put another way, any partial symmetry which is not a full symmetry must either “destroy structure”, or at least “crumple it” in some way that loses information. The interest in doing this is because, in some sense, residually reversible partial symmetries are usually easy to find in non-commutative, infinite dimensional settings (that is: we have infinite degrees of freedom; and the order in which we do things matters). The examples I am looking at are highly non-commutative, yet do not admit these kinds of partial symmetry; and this is evidence for a certain rigidity in the objects being considered, which is in my view still not fully understood, and which merits further attention.
That was apparently just over 480 words, according to the text editor. Bonus points for anyone who can infer from this what the actual piece of research is on, even vaguely. I don’t think there’s enough information above, but you never know.
(I should point out that while I find 500 words or fewer (ahem) an interesting challenge, I’d be leery of depending on it as a criterion for judging the worth of a project, topic or discipline, largely for reasons similar to those laid out in these comments. My own speculative guess is that while bad examples may betray flaws in the proposal or project, “good” ones are largely just an advertising sell. But then, as someone who is personally more interested when the current Charles Simonyi chair for PUoS talks about zeta functions and algebraic curves, than when he talks about symmetries in hyperspace, I’m obviously not the target audience.)
 In case my intended frivolity above is taken too seriously: I am immensely cheered that du Sautoy has taken up the Simonyi chair, and find his efforts to engage lay readers of various ages a Very Good Thing. Even if he is a Gooner.
update 31-03-09: small typo fixed (there was an `S’ that should have been a `T’)