exciting times.

at the moment i am drowsy and tired, from afternoon basketball, an indonesian music production @ hill auditorium (@ U of M), and a good, homemade meal of:

curried couscous with vegetables,
freshly fried falafel,
a crisp white wine,
a coconut rice palau for dessert
(my venezuelan flatmate made some for easter, today)

but lately and mathematically, these have been exciting times.

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it feels like i'm close to the theorems that i want to prove. they aren't much, but they'll serve some purpose.

namely, they may clarify (and make more concrete) some of the abstract nature of n. weaver's construction, regarding (co)tangent modules on metric measure spaces (mms, for short).

the advisor and i have thought about the most immediate cases: euclidean n-space. to put it simply, the theorems in mind really show that the weaver theory is what you'd expect it to be. [1]

it's know that there is a key relationship between weaver's construction and j. cheeger's construction of a cotangent bundle on mms's.

the strange bit is that the weaver theory may also have ties to this recent theory of 'currents on metric spaces,' as discussed initially by ambrosio and kirchheim, and studied, among others, by lang.

thus far the contribution is one-sided; the language of metric currents gives us the results we want about (measure theoretic) derivations on euclidean n-space ..

.. whereas such derivations, at the moment, provide but an additional example of metric currents in a reasonably general and abstract setting. it would be nice to see results in that other direction, though.

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i feel like i'm learning a lot, now.

it's one thing to talk about results in papers that everyone else is talking about, but lately i've become more 'honest,' in some sense: i'm opening the ambrosio-kirchheim Acta paper more often, and delving for ideas and motivations ..

(admittedly, for more metric current stuff,
in order to transform into weaver stuff q:)

.. and, most likely inspired by the zeal of a visiting postdoc, i've begun browsing more often through cheeger's GAFA paper and s. keith's thesis.

i'm learning and perhaps, soon enough, i may be able to say something .. perhaps not quite interesting, but also, not boring.

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the theorems i mention above, i can't tell if they're obvious or not, and some days of the week i feel like i'm only making corollaries to the big, powerful theorems of others. from experience, i know that it's unwise to ask:

"how much of this work is really mine?"

i could say (and actually, have said) that

"if you've read or browsed through the same papers and books that i have, sat through the same courses that i have, interacted and been taught by the same advisor as i have, then you would have gotten the same ideas." [2]

i honestly don't know how unique the mathematically human experience is. i know how stupid and foolish i can be -- again, from experience -- but that's it. in fact, if i've ever had any good ideas, it's because i've thought so foolishly, so often, that eventually i exhaust all the possible foolish notions ..

.. and am left with a few good ones.

but i don't want to dwell on these questions. there's not much time left in grad school and i still want to say something interesting.

maybe i can and will,
maybe i can and won't,
or maybe i can't ..

but i want to try, because for once i feel like maybe i can, and certainly, i should.

[1] and yes, i am deliberately being vague and cagey; after all, this is a work in progress. if you want to know more, feel free to write me ..

.. or if you're feeling really generous, invite me to your school or institution and i'll happily give a talk about it. (:

[2] disturbingly enough, i've said a similar thing once, but it pertained to dating a girl in high school. when she asked me who i was, i think i gave her a list of books and music albums and friends i lost over time, and said those things were precisely who i was ..

.. that i was the sum of my experiences, no more and no less. i think i weirded her out.