this would have been more relevant had i written it on thursday or friday, but life is imperfect.
anyways, my meeting with the advisor went reasonably well, despite the fact that i had no new research results to share after a month's time. in that discussion, we identified a necessary step or two in order to push the problem further.
in fact, they are computations; if they work out, then the problem proceeds, and if they don't, the thesis problem is killed.
"so it's like a two-hurdle race, then?" i asked him.
the advisor nodded.
wow. saying it in this way makes it sound like high stakes, but then again, we were in the same circumstance a year ago: we needed a particular extension theorem, and it condensed to a computation and eventually, to an unnaturally looking critical exponent of integrability.
i just hadn't thought of it as risk, last year. put another way, it's strange that i think of the current problem in terms of risk now. i suppose it's the human tendency of rationalising the past, and hoping that our efforts have not been in vain.
computations don't mean the same thing to me as they used to mean. i take the firm belief that if what you study has enough structure to make sense of computations, then you're already quite fortunate.
think, for a moment: for those mathmos out there, think about the structure of the objects that you regularly study, as well as other objects or concepts that you've seen, at some point in your studies.
how often are these structures "nice enough" that you can jot down relations between the relevant objects in a simple way? it must mean that there are enough deep but understood notions in the theory, which account for otherwise intractable difficulties and mysteries.
as an example, i like to think of the gauss-bonnet formula(e) for (Riemannian) surfaces. one can relate notions of geometry and topology quantitatively with a handful of symbols:
it's something that i've told my non-mathmo friends on occasion, but it's hard to say whether they can appreciate what this means. for example, i might tell them that i study functions between manifolds (which is not a total lie) and they might ask:
them: "so what do the formulas look like for these functions?"
me: "i don't know, but i can draw a picture for you."
them: "wait. a picture?"
me: "it's one of a few ways i can understand what's going on."
at a naive level, i think it suggests to the non-mathematician how "abstract" the work of mathematicians can be.
anyways, back to work. it might be a computation, but it's not necessarily an easy one!
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