Maths roundup

[pozorvlak]

michiexile has written a really nice introduction to his work for the layman. There's some commonality between what he and I do (if I'm understanding his work right), in that we're both concerned with structures that don't quite satisfy equations, but do up to some kind of transformation. So, situations in which a+(b+c) isn't equal to (a+b)+c, but you can easily transform one into the other. I suspect his #haskell username refers to this - "syzygy" is used to mean a relation between relations between relations between... The fundamental difference is that I'm interested in situations where the things you're "adding" lie on some sort of network (to be precise, they're objects in a category, which is just a funky directed graph), whereas he's interested in situations where they live in some kind of geometric space, and the transformations are paths between points. But there's a lot of overlap, most of which I don't understand, and we use some of the same tools to attack our problems.

[I wrote an explanation of what I do here, if you're interested]

The other day I was looking through my email archives for something, and found the following in an email to Duncan:

BTW, it occurred to me the other day that the security technique called Bernstein chaining can be considered an example of continuation passing style. Don't you love it when two things that you don't quite understand turn out to be only one thing, reducing the number of problems in the world by one? :)

To which my reaction was, roughly, "Is it really? Gosh. Er, what exactly is Bernstein chaining again, younger self?"

In further serendipity news, my office-mate showed me a paper called something like "Geometric presentations of the Thompson groups" and, despite the irrelevant-looking title, it looks like it may actually have serious bearing on my problem. It seems that every set of formal algebraic laws (such as associativity) has associated to it a "geometry group" (which, in the case of associativity, is Thompson's group F, and in the case of associativity + commutativity is Thompson's group V). This totally needs following up. Any successful categorification of an algebraic theory will give rise to groups, namely the automorphism groups of the operators in the theory. It looks like the geometry groups aren't exactly what I want (what with F being vast, and the automorphism group of any tree of tensor products in a monoidal category being trivial), but maybe I can do something like (free group)/(geometry group). Or, more likely, (free groupoid)/(geometry groupoid), whatever taking quotients of groupoids means. Anyway, I need to read the paper properly now.

Another thing I need to read up on is term-rewriting. I was dimly aware that there was a well-developed theory of "ways of rewriting words in a given formal language", and that this might have some bearing on what I'm doing, but it seems like the connection may be closer than I'd thought. In particular, the Knuth-Bendix algorithm looks interesting. I also find it suggestive that two of the big names in string-rewriting (Thue and Huet) have names that are not only anagrams of each other, but actually cyclic permutations! And has anyone ever seen them together in the same room? :-)

Comments

[pozorvlak.livejournal.com]

:-) No worries. The roast & wine sounds good - wormwood_pearl and her flatmates and I had a roast & wine last night, but there was less wine, and it was a Quorn roast. Surprisingly OK, but not as good as yer actual dead flesh.

[mi-guida.livejournal.com]

There wasn't going to be so much wine, but we carried on chatting after food and so got through it all. Oh well.

I'm somewhat less drunk now after downing hot chocolate and throwing half of it over the constit notes I was trying to type up.

"Why didn't you revise for the test?"

"The ink ran on my notes after I threw hot chocolate over them while drunk after a brilliant weekend"

...maybe not. Oh well.