scattered and unproductive.

i don't think i'm being very productive. certain things i do by habit, like waking up and then thinking about mathematics for a few hours, over coffee .. taking notes and pondering problems.

lately i've been confused.

i think i've been able to coax all the 'easy' results from the fusion of these two topics that i've learned .. though i don't think i've learned them very well .. and now the work will become more difficult.

put one way, i would say that the theorems i've proven over the last two months have been pointed observations and casual conclusions (and not all of them mine).

i wouldn't call them all of them corollaries, but i've needed to use a nontrivial fact or two from the works of others. without them, these theorems would still be claims, dead in the water, without proof.

now it won't be so easy.

i have a claim in mind, but i can't prove it, and those facts from before are no longer applicable and not obviously relevant. i think i'll have to invent or discover a small something, but a new something, in order to solve this part of the problem ..

.. but i don't quite know what something it should be. i suppose that if i did, i'd probably settle the problem and have a laugh about it. for now, i'm not laughing.

most of the time, i'm frowning at those pieces of paper.

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my mind feels scattered.

i can't really encapsulate my research as a thesis problem anymore. then again, we've always kept this second "problem" open-ended, like an exploratory mission of sorts. but before, it was easier [1] to say what i was doing ..

.. but now this idea runs this way, and another idea runs the other way. often i don't know which to follow, which is more productive or fruitful, and often i just pick one, hope for the best, and get to work ..

.. and every so often, wondering if "the grass is greener."

i also find myself switching around these sub-projects. for example:

yesterday & today, i thought about a problem which has more to do with the currents of Federer & Fleming on euclidean spaces, rather than the new-fangled theory in the context of metric spaces.

the flat forms of whitney and wolfe also appeared briefly, as those in the know .. well, could probably have known.


two days ago, i was pondering the (first) heisenberg group and what could be said there, in my line of work. i conclude what i always conclude:

does anyone actually understand this space? it's a beautiful geometry, but do we know how it works, analytically?


the days before that i was working on metric spaces which are general enough to do what i'd like them to do. the advisor then asked me about something, which was a good idea.

i'll probably try and work it out tomorrow .. unless i keep with this Federer-Fleming thing until i reach an intractable dead-end, which might take a while, and then there's the rare chance that it could work out ..

but you get the idea: "scattered." (:

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[1] well, "easier" if you narrow down the world to a few dozen people, maybe even a hundred.

i'd like to think that i could explain the rough idea to a mathematician, but then come the "prerequisites," such as

* measure theory & functional analysis, for starters,
* a little differential geometry, for motivation,
* some awareness of the analysis on metric spaces, for relevance,
* some geometric measure theory, for intuition and analogy,

i might be able to gloss over certain things, but it would be hard-going.