earlier this morning; fatigue and balance.

i woke up at 7:30 today and i couldn't fall back asleep. half-conscious, i sat on my bed and decided there were too many things to do in a regular day's time.

so i got up, made myself semi-presentable to the world, and foraged for caffeine. i think the morning manager at the local caribou coffee thinks i'm a nuisance,

a regular who thrifts and spends no more than a few dollars for a muffin and a coffee,

who lingers about for the free coffee refills,

who talks very little to anyone and stares through the windows at nothing in particular,

who scrawls strange symbols and diagrams all over paper, and somehow in obscenely neat lines.

as it happens, i'm not the only mathematical regular there but i think they don't like me very much. i'm hardly the model customer.

on weekends i see there afsanah, a grad student in algebraic geometry, who i think is writing up her thesis.

two or three students have also made that same coffeehouse their watering hole, and i think they are undergrads but not from um. they seem more .. down to earth and unconcerned in their ways. they remind me of eager kids who'd i find in the pittsburgh mathematics library in thackeray hall, just doing their thing and working out a theory or two.

every time i see them they're discussing point-set stuff from the munkres text and seem to take it very seriously .. seriously enough so that i can hear their arguments about limit points and closures from a few tables down.

they're enthusiastic: much more so than i ever remember feeling. i leave them alone and let them have their topology; i have plenty of work and they are happy by themselves.

maybe i'm getting tired and cynical. i was about to say old, too, but that's not quite right; i know plenty of people past me in age but they show none of it. so i'll reserve the word old for when i look like it.

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i am tired, though. it's been a long week and work urged me, more so than usual. near noontime i felt like i've reached a peak of my mental powers [1], that i was acutely aware of the nature of the problem, its constraints, and its freedoms.

my thoughts flowed, and today's research meeting with the advisor was wonderfully productive. it's silly to "keep score," but after last week's debacle [2] it was nice to be able to say something of some accuracy and substance ..

.. even though i needed an overly strong, simplifying hypothesis. it was all i could do, being a tyro in univalent harmonic theory and knowing only basic techniques. but there is a little promise left: a little example i computed without much thought is now offering motivation for a reasonable compromise of a corollary [3].

what can i say? my advisor's optimism is infectious.

that went all well and good, but the issue with "peaks" is that they don't arise naturally for me. as in the case of social gatherings and when taking on roles of leadership, it's a matter of harnessing and directing enough energy and willpower toward some effective end.

but that energy must come from somewhere, and as in the example from descartes, where there is a peak ..

.. there is always a valley, and maybe a rut below that. anyways, it's been a long day, and i've burnt too bright for that many hours at at time.

until the next post.

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[1] this is not to say that i've reached the epitome of my career, but that i hit a local maximum. there will be more, and also there will be local minima as well. last week was such a latter case, for example.

[2] i meant to post about it, but didn't allow myself the time. the short of it: i gave a false (i.e. non-) proof of a lemma, and the fault rested on a basic inequality - which i thought was true by concavity - but is woefully untrue in general.

[3] as a general rule, i seem incapable of proving theorems, but every so often, i manage a lemma and at best, i prove corollaries to the great theorems of others. it is one of my axioms, along with "i do not dance" and "i am morally opposed to my own birthday."