terminology.

the more i think about it, the more i suppose that my study seminar talk, last week, seemed like a 'comedy of errors.' i distinctly remember discussing the usual notion of a cubical subdivision of the plane, and then someone asked,

"so really, you're talking about squares, right?"

this comment dumbfounded me, for about a minute. squares? but then i finally realised his point:

"oh, right!" i answered, "yes, squares. i was talking about 2-dimensional cubes."

i also remember that it made the audience laugh, but that wasn't my intention. so let me explain.

---

to the non-mathematician: if you spoke with a mathematician about shapes in two dimensions, you might soon be puzzled at his/her use of the word "circle." instead, s/he may often say the word "ball."

what's the difference, you may ask? a circle is a boundary; given a center point C and a radius r > 0, it consists of the points which are exactly distance r away from the point C. in set notation,

{ P ε R² : |P - C| = r }.

on the other hand, the boundary of a (2-dimensional) ball is a circle. it consists of the points which are distance r or less from the center point C. again, in set notation,

{ P ε R² : |P - C| ≤ r }.

why the difference, you may ask? a circle is a 1-dimensional object, whereas a 2-dimensional ball is .. er, 2-dimensional. also, a circle has a hole; a ball doesn't.

to the mathematician: yes, you understood the difference already, but you may ask: what does this have to do with cubes and squares?

i didn't realise it until now, but i reserve the keyword "square" for when i'm talking about boundaries of "cubes."

to me, a square in the plane is a boundary, consisting of four congruent line segments at right angles. the 2-dimensional object is a cube.

so in the language of the SATs [1] of yesteryear, this is how i use words:

circle : ball :: square : cube.

[1] to my non-american friends: SAT was once an abbreviation for 'scholastic aptitude test,' but now it means 'scholastic achievement test.' it's a standardised exam for high school students, so that colleges and universities have a criterion to reject the students whom they don't want.

inspiration and trepidation.

there was a seminar .. actually, a colloquium .. today, before lunch, but h. furstenburg spoke. it was the sort of talk in which one consistently asks oneself,

wow, really?
you can do that?

i've mentioned my idiosyncratic responses to good talks before, but this was really something.

so many disparate bits of mathematics .. just enough to be recognisable, but deep enough to cause wonder, even trepidation:

actions of topological groups on measure spaces,
measure theory and weak-* topologies,
affine group representations and irreducible ones (whatever those are)
harmonic functions and möbius transformations,
i.i.d. random variables and martingales,
hints at margulis-type rigidity theorems for appropriate types of boundary.

i felt like i saw the enormity of mathematics, and supposed that perhaps universalism did not die with poincaré, that it may still live on.

that troubled me. it wasn't until i ate a sandwich and drank a coffee that i felt somewhat like my old self.

small world, this.

Mathematician suggests extra dimensions are time-like

In a recent study, mathematician George Sparling of the University of Pittsburgh examines a fundamental question pondered since the time of Pythagoras, and still vexing scientists today: what is the nature of space and time? After analyzing different perspectives, Sparling offers an alternative idea: space-time may have six dimensions, with the extra two being time-like.

[link]

strange coincidences, in this world: as an undergraduate, i sat through two courses of differential geometry that george sparling taught.

he's a nice guy.
i hope his theory works out.

a conversation, between nerds.

me: .. but what really convinces me to ride to whole foods is this.
[fishes out plastic container]

me: it's the only place around that i can get fresh squid.

[a little squid juice spills onto the table.]
[a few drips reach the carpet.]

me: crap!
[wipes droplets back on table]

me: we've got to clean this right away. it's going to smell terrible ..

luis: you know, there's not much.

me: but we should, anyway.
[rushes to the kitchen]

luis: there's not much, and the smell is nonlinear, you know.

me: [from the kitchen] no, but it is continuous!

about a talk, and some philosophical blather.

so today i gave part i of a three-part talk, about the paper of alberti, csornyei, and preiss ..

[link, via cvgmt @ pisa]

.. and though they are error-prone and paltry, my notes have been through the scanner and are available in pdf form. write me if you want a copy.



as for the talk itself, others have told me that went well: only one person walked out, but i think he had his reasons.

myself, i think it was a fair talk, but so many things could have been better said. such things irk me .. for i am not a frustrated over-analyst for nothing!

but then it came to mind, that i employ the same bias that others tell me that they employ: that they have different rules and expectations for themselves than from other people.

this in itself is not interesting -- especially if many adopt this bias -- but i mean to say that, in addition to it being frequent, it makes sense to have separate conduct for self and for others.

the key point is this: the state of mind when observing one person in some task is closer in form to the state of mind when observing another person at a similar task than the state of mind when you are doing such a task yourself.

the observer's mind and the actor's mind are inherently and fundamentally different states, because the relevant information and assignments of value which comes to the observer's mind are different than that of the active participant.

let me not blather armchair-philosophically too much more. really, this is an excuse for why i can say that it was a fair but sketchy talk, without contradicting others who think it was a good talk.

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.

---

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.

---

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.

---

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.

a passerby encounter.

i saw one of my former students sitting in the undergraduate mathematics lounge on the second floor today, around the time that the weekly undergraduate mathematics talk would be starting. he was among a crowd of young-looking people, all facing the single blackboard in that room.

for the sake of self-delusion, i'm going to assume that he stuck around to hear the talk (which was about quaternions).

if anything, it makes me proud, because:

1. maybe my teaching didn't harm him too much, and it added to his having an open mind to learn about quaternions;
   
   
2. he's smart enough to earn an otherwise-free slice of pizza or two ..
   (or three, if he played his cards right)
   
   .. and having heard me drone on and on about calculus for four months of the fall, has learned to tolerate an hour's worth of spoken mathematics.
   

at any rate, good for him. q:

cripes! foiled again. [EDITED]

argh.
it happened again .. well, sort of.

two weeks ago, i had written about how two of my ideas came too late; a trio of other researchers had similar but more elegant ideas before i did, and they can be found in this paper.

so i had made a claim about a certain class of metric 2-currents (also around two (or more) weeks ago) and thought i had a proof. it wasn't until last week that i realised that the method of proof

1. does not suit the claim;
   i never used one hypothesis.
   
   
2. is invalid, because of an immediate counter-example, and i mean, really immediate: a smooth curve in the plane, endowed with Hausdorff 1-measure, will do it.
   
   well, at least i found it before meeting the advisor;
   small favors, at least. \:

in point of fact, it was this exact claim which caused my mathematics block this morning.

it wasn't until this afternoon that i finally browsed through the latter pages of that paper. [1] in section 8d, they state a version of that claim as a proposition.

so yes, it happened again.

the frustrating part is that the result has been announced, but the paper (a different one from above) which contains the proof has not yet appeared.

so the idea remains theirs, and i am still stuck with how to prove it.

i mean, it could be worse; the claim could have been false. but it would be nice to point at something and say "that's my idea."

at the rate this is going, my thesis is going to be one big corollary. \:

EDIT (as of afternoon, 4 apr 2007): never mind. i was wrong. the proposition that i refer to does not include my case of interest. my case doesn't generalise theirs, either.

now i still don't know how to prove my claim, and what is known doesn't immediately help. so there's plenty of uncertainty left, and plenty of work to do.

[1] and of all the reasons why, it's because my postdoc friend reminded me to submit an abstract for next week's study seminar talk. in an attempt to hold myself accountable to my promises, i looked more carefully at all the sections of the paper.

mathematician's block, & archival procrastination.

i can't seem to concentrate on research today. i know the obstructions (which shape the goals) and i recall some known facts proven by better and wiser mathematicians, but i cannot seem to "get my hands dirty" and see what i can prove.

it could be because i haven't gotten into a good groove, yet. yesterday i was studying something distinctly different ..

(in particular, i was reviewing a few old arguments of mine that i had meant to show the advisor, at some point, but other things came up; it happens)

.. and i couldn't get into a good groove, either. maybe there's something to be said for consistency and comfort of thought?

[shrugs]
oh well.

---

anyways, apparently i've been living under a rock for the last week or two. there are an obscene number of interesting preprints posted on the arXiv, and i found it hard to organize what i will read later and what i will say i will read much later (but will probably never get to read).

so i have done this: i've saved them under my del.icio.us web bookmarks. this is a wonderful (& free!) service. think of the convenience of taking your bookmarks with you ..

("favorites" for those of you who use internet explorer)

.. except you don't have to import/export them, carry them on a usb flash drive. oh no. you can just look them up, as if they formed a webpage for you .. which, in fact, they do.

so here are my mathematical bookmarks, as found on del.icio.us: {link}


ps. if anyone likes photos of sunsets and horizons, i have also developed a collection on del.icio.us, here: {link}

pride and humility.

i'm back from a weekend grad student conference in syracuse. let it be said that the analysts of syracuse know hospitality and entertain well!

i felt a little proud of myself mathematically.

1. i didn't say anything deep,but i was glad to have discussed some questions about p-harmonic systems with tomasz, a student of tadeusz iwaniec. in fact my understanding of that particular theory is superficial; what little i understand, however, does fascinate and tempt me.
   
   if i didn't have to complete research for a thesis, i'd sorely wish to work on such problems. this is not to say that thesis work is going poorly -- the opposite, really -- but i am too tempted by greener grass on the mathematical lawns of others!
   
   
2. my understanding of elliptic complexes is equally poor, but some recent work of derek (another student of iwaniec) reminds me of a theorem of d. jerison. i told him this, and he actually wrote it down. he knows his research better than i would, of course, but i hope this reference will help him.
   

it is a fine thing to feel useful .. hence my rare bit of pride.

---

that diminished, however, after i tried to warm up to work again. having lost my copy of a particular paper, i browsed through an electronic copy and started reading a paper (to appear) which the advisor has written.

it's 92 pages long, and having reached page 52, i feel humbled again and slightly depressed. its breadth is astounding, and today, like some days, i feel as if the advisor knows everything and i can essentially say nothing of consequence.

if someone asked me what a mathematician often feels, i would say, in decreasing order:

• frustration: every good mathematician knows a few claims that (s)he cannot (yet) prove.
  
  
• humility: to each mathematician, there is always a wiser or cleverer someone, who has produced work which that mathematician can appreciate, but does not fully understand.
  
  
• add almost everything else here, but ..
  
  
• joy: because, every so often, one proves something at long last, or stumbles onto a proof which is a pleasant surprise.

anyways, it's time to get myself back together, so that i can accomplish something by tomorrow!