Master's Project

[stoutfellow]

I saw my Master's student yesterday.

He came to me at the beginning of the term, asking me to take him on. The plan was for him to spend this semester learning background material, and to start his thesis project next semester. So, I searched my shelves, handed him a book and told him to read Chapter 5, and come back when he was comfortable with its contents. This was in late August, and it was the last I saw of him until yesterday.

We chatted a bit about his reading assignment; he confessed to having some difficulty with the proofs, and I encouraged him to think about things in a slightly different way. (Geometry, dammit! Remember the First Law!) Then he asked if I had decided on a project for him. I described an idea I'd had, which was something I messed around with a little bit a few years ago but never got anywhere with. I was a little concerned that it might be too hard for him, but he seemed interested and willing to tackle it, so I gave him a few exercises and suggested he just fool with it a while and get the feel of it. We'll see.

(The problem has to do with a kind of algebra in which exponents are allowed which aren't numbers; the usual laws of exponents hold, except that there doesn't seem to be an analog of the Binomial Theorem. His assignment: find one.)

Comments

non-number values

[p-o-u-n-c-e-r.livejournal.com]

"xponents are allowed which aren't numbers"

Okay, I don't know enough to know what sorts of things you might mean.

Formulas or functions? Some exponent n where n is a function of one or more numbers p, q, r and s ?

Or where n is a, like, mathematical construct like a topology? (How such a thing might be possible I don't know either.) But where n might be a plane, a sphere, a torus ...?

Re: non-number values

[stoutfellow.livejournal.com]

Mmm. In full generality, the exponents could be very complicated things indeed - including topological constructs such as you mention. In the (rather simpler) situation I'm thinking of, the simplest - but not very illuminating - explanation is that they're sets-with-structure. Even there, it's pretty hard to explain, but I do have one example in mind that's simple enough to be doable. It would deserve a post of its own, though. I'll do that, if you're game; it'll involve, among other things, a new (and possibly interesting) way of thinking about operations on ordinary numbers.

Re: non-number values

[p-o-u-n-c-e-r.livejournal.com]

I'm game -- always, to learn hou ignorant I actually am.

Re: non-number values

[stoutfellow.livejournal.com]

Okay. I probably won't get to it till this weekend, though.