this is turning into week three of trying to understand a proof that is only two paragraphs long (and maybe an estimate, but not much more). i've been working day after day and overall there's a little progress, but the big picture isn't fully clear to me yet.

yes: it is something important and worth knowing, but i can't quite detect whether i'll be able to extract anything new from the argument or prove anything new or make a useful observation.

so even 'victory' won't really be victory. we will gain nothing, but merely regain the certainty in something already known.

there will be something, maybe another lead after this proof is settled. between the advisor and me, something will turn up; i believe it. there are always questions, even if i cannot think of them as quickly or easily as others can.

life goes on, wherever it's going.

---

october turns into november, and if i'm still on the five-year plan, there's about a year's time left on the clock. a year before i'm supposed to be ready to look for jobs.

i'm still nowhere. i spent between 12 and 14 months on the first thesis problem, which is sunk, and after 2 months i've no real progress on this second one. nothing, except a souped-up example which lends nothing to a good theory.

two months is nothing, and yes, i'm impatient. but i don't know what i'm looking for or where this project will go and honestly, i don't know if i'll succeed. this is a dark time in my life ..

.. yet one day, somehow i'm supposed to look back to now and think about how much free, unhindered time i have and why i was so worried over nothing. if time travel were possible, i think i'd travel to that future and punch my future self for being so nostalgically saccharine and dismissive. smug bastard.

---

i don't feel tired, but it sounds like i am.

in which i dare the darkness.

tonight i remained in the office longer than i'd have liked. it was necessary, though, because i was dabbling in the dark arts known as algebra ..

.. and i couldn't very well permit that stuff follow me home now, could i?

it would taint the very air, lengthen shadows to sinister darkness, spoil the soy milk and make stale the coffee ..

[shudders]

it's not that i detest all things algebra; for instance, i like lie groups and lie algebras. kleinian groups form very interesting illustrations via tessellation. i also like hilbert spaces and linear spaces in general; the weak-star topology would be nothing without the notion of duality, which is a linear notion.

but outright algebra -- modules over rings and left- vs right-actions and (argh) universal properties (bleah) -- they just makes me uneasy ..

.. in the same way that superman hates magic, or how batman hates moving into action without a foolproof plan [1]. let's face it:

give the villains half a chance, and they will lay a trap for our heroes. the heroes will escape, thwart the villains' nefarious plans, and save the day, but there's always that annoying trap towards a dénouement.

anyway, moving along ..

i also kept muttering to myself this mantra: it's not really algebra, and tried to convince myself that this wasn't evil.

"look's there's a norm! it can't be all bad."

"C^*-algebra is just a convenient name. the elements are functions, and remember: functions are our friends .."

"you're in the commutative case. think: L^∞, measures .. happy things .."

it worked, at least for a little while. but eventually i could bear it no longer, jotted down quickly some remarks on my notes, and left to find .. an algebra book from the science library.

but all's not lost: the identifier "Banach" appears twice on the title, and "module" only once.

still, i would never have thought that i'd need this algebra in order to get my work done.

[1] by this i refer to the grim, gritty batman, and not the pansy batman of the '70s tv show or the joel schumacher-directed debacles. give me a christopher nolan-type batman, or even the gothic batman of tim burton.

in my book, batman should be a borderline personality, whose paranoia and meticulous planning keeps the crime of gotham at bay.

fitful day, little done.

i couldn't concentrate today. at most i reread some class notes from 2 1/2 years ago that i don't remember taking. all the better to reread them, i suppose, and better yet, they concern results from a paper of J. Cheeger that i was going to read anyway.

then, absurdly enough, i took notes.

yes, you read that correctly: i took reading notes from my class notes. they weren't hard to follow, but somehow the process of notetaking sharpens my understanding, much like taking notes during research seminars.

indeed, there's a 5% chance that you might actually look at seminar notes again, but the point is to keep busy and listen more attentively; you can't slack off if you're recording the nuances of the talk.

still .. absurd. yes, absurd.
but i couldn't help myself.

---

there's an off-and-on feeling i get, and though i can predict it coming i can never shake it off when it does. it's a feeling of being overwhelmed, but it's not so much stress as being impressed.

take this cheeger paper that i mentioned. borrowing generalisations from the quasi-world theory, he reproves a form of rademacher's theorem which is genuinely new in a modern, more general context ..

.. but that's not all.

impressive as it is, that theorem is used as an application to deduce a structure theorem: generalised cotangent bundles exist for a class of metric (measure) spaces.

amazing. absolutely amazing.

it's not exactly what i study, but an elephant of a question arises: how in the world do i add to that?!?

under the cool light of calm and reason, one can always think of something, perhaps by identifying weak points and asking questions.

but for me, that takes a while. i have to withdraw and let it sink in me, much like how i am quiet after a good film or concert. sometimes i see or hear or understand and in a single moment it is too much, and i can say or do nothing.

who'd have thought there would be a downside in being able to appreciate something?

teaching and science and other thoughts.

i learned several things after teaching today.

1. i really, really hate British units of measure. units of mass aren't so common, and instead pounds refer to a unit of gravitational force. fine: if that's the physics definition, then i can't and shouldn't argue it.
   
   but it doesn't make any sense. how do you discuss density of solids, then?
   
   you'd figure that the Brits wouldn't mind Newtons (N) as a unit of force. it's named after one of them, after all. everyone remembers what Newton did, but nobody recalls the deeds of, say, Alfred the Great .. who, oddly enough, is the only English king to be named 'Great.'
   
   frustrating.
   
   
2. basic mechanics isn't as simple-minded as i thought. moreover, my students seem much more inclined to physics than, say, volumes of revolution or polar coordinates. they actually seemed lively .. though that could be more my volley of errors that people kept noticing.
   
   as i was fumbling with certain concepts, such as force induced by pressure in a fluid and gravitational force, a few of my more vocal students jumped in with plenty of good pointers. admittedly, i didn't know they had it in them.

again: works and days.

if it were summertime i'd wait a few days and then undergo a post-conference slump. but the fall term remains in full swing and i spent most of today "taking care of business" ..

prep and teaching,
then attending a grad union committee meeting,
then attending Geometric Measure Theory lecture,

then i was nudged out of my office, because my officemate's calc ii students flooded into the floor and chairs of our already crowded hovel in east hall.

retreating to the psychology atrium (which is far quieter than the mathematics atrium) i tried to focus and to think.

but in the end my thoughts were sporadic and ineffectual. coming home after fetching supplies from trader joe's, the mood's been the same.

i can't get any research done today .. the issue isn't even research; the confusion lies in understanding an existing proof of an 8 year-old theorem. it leads to nothing new and possibly something instructive, but at the core it is prerequisite before anything else is done.

this seems to echo conversations from this past weekend with friends and peers:

i've said before that the content of my talk would make a tolerable paper, but i would be the wrong person to write it. it would be an anthology of topics, and most of the results would be previously known and the only novelty would be to see them in one place.

others have said that there is something valuable in making connections, even if the results are known. it would solidify and further the body of knowledge in the field. i don't disagree with them, or more accurately they haven't disagreed with me.

but they weren't listening.

i didn't dismiss the role of such a paper. my point was that i'm the wrong person to write it. the most important quality of a student or a young researcher is creativity and innovation.

i'm not agreeing with g.h. hardy and i'm not saying that 'mathematics is a young man's game,' but in the sociology that is mathematical academe, new researchers have different priorities than established, tenured researchers. my priority is to create or to discover something useful.

i haven't had very many good ideas. in recent history, i have had two good ideas which have worked, but one isn't really worth anything and the other is a computation that any analyst could do. countless bad ideas make the rest.

but i digress.

in sum, i was distracted today by obligations, and i did very little thinking. the object of my thoughts isn't new, but it must precede any future innovation. i'm not clever enough to be impatient, and i'm not as patient as i need to be.

---

maybe tomorrow will be better. the obligations and errands will meet me again tomorrow and greedily snatch away my time.

october soon ends. in a year's time i might have to look for jobs, and i don't know if i'll be ready by then.

how the talk went.

preamble: i have to stop giving talks for the first time at conferences. indeed, i sound like a nervous twit when i do ..


minus signs.

1. i went over my allotted time: a conference sin!
   
   
2. i misread the audience. it was an AMS Special Session, but the same crowd lingered for all the talks, so i wasted two slides giving definitions that everyone already knew.
   
   
3. i should have drawn diagrams on the transparencies themselves, instead of drawing them on-the-fly, on the whiteboard behind the lectern. this happened four times.
   
   
4. i think i misrepresented myself and may have intimidated some of the audience by mentioning milnor's exotic differential structures on the 7-sphere.
   
   
5. i never reached my last two slides, which would have said something about harmonic functions. a great tragedy ..
   

plus signs.

1. at the least, i think i spoke clearly and that everyone followed me.
   
   
2. i did get a few curious questions and many informative comments and suggestions. analysts are such nice people!
   

now it's a matter of writing a research article based on what i talked about. oh, joy of joys ..

disasters happen and are waiting to happen.

EDIT (FROM 22 OCT 2006): as it happened, on friday night, i was informed that I wouldn't be talking last on sunday morning, but on saturday morning instead.

life again throws curve balls.

---

it's a day or two to go before the AMS Conference @ Cincinnati, OH. by fortune of fate, i'm talking last on sunday, which is fine; someone has to be last. i've already written the talk and need only print out the slides ..

.. but i have a bad feeling about this.

i'll be talking about the Thesis Problem that Was (read: Died) and haven't gotten to writing up the preprint yet. i simply have no time. how does anyone ever have time to work on new research, write up results, teach a section of 31 students, and attend classes and seminars?!?

small favors: i've never been so happy as to have a "Preliminary Report" category before.

---

you'd think that with two days off this week (UM has a 'Fall Break'), i'd be living it up and coasting and enjoying life .. but i'm not.

monday and tuesday off meant an obsessive few days of research: equivalently, i ended up over-thinking the details that don't matter and overlooking the details that do.

today what i thought was a proof .. wasn't.

i realised this only as i was explaining it to the advisor, who was the one to point it out. so i looked foolish again. you'd think that i'd be used to the feeling by now .. but i'm not.

---

i'm losing an obscene amount of time to teaching .. or at least, it saps my mental energy. what i thought was a good worksheet confused my class more than it educated, and tomorrow is a lesson on

"yes, this is how you do the worksheet;
no, you probably won't see something this complicated on an exam .. i hope."

i hate hedging my bets. it doesn't mean that i'd rather write my own exams, but .. i hate having to keep marching on pace because i can't let my students learn at their own pace.

uniformity might be great for convergence of functions, but it's terrible for students and learning.

on a side note, i think i inadverently made one of my students cry. this is exactly why i should never make judgment calls, because nobody should EVER trust my judgment.

argh.
conference talk. teaching. planning. packing.

i must get to it, because sometimes we make promises that we must keep, despite time and circumstance.

something from computer science.

i like reading essays from a particular author/programmer/"startupper" by the name of paul graham.

irrationally, i distrust capitalists and the rich, but any hacker that can use the word "isomorphic" colloquially and correctly is all right in my book; in his essays, paul graham has, so he is all right.

it could also be that he writes essays that i want to believe, like "why nerds are unpopular" (an uplifting piece) and "what you'll wish you'd known" (something like a speech).

anyways, here are two excerpts from his essay, "the hundred-year language" :

Any programming language can be divided into two parts: some set of fundamental operators that play the role of axioms, and the rest of the language, which could in principle be written in terms of these fundamental operators.

I think the fundamental operators are the most important factor in a language's long term survival. The rest you can change. It's like the rule that in buying a house you should consider location first of all. Everything else you can fix later, but you can't fix the location.

I think it's important not just that the axioms be well chosen, but that there be few of them. Mathematicians have always felt this way about axioms-- the fewer, the better-- and I think they're onto something.

so paul knows axioms! dandy! but this second bit seems to me a little insightful.

Languages evolve slowly because they're not really technologies. Languages are notation. A program is a formal description of the problem you want a computer to solve for you. So the rate of evolution in programming languages is more like the rate of evolution in mathematical notation than, say, transportation or communications. Mathematical notation does evolve, but not with the giant leaps you see in technology.

i can't resist another excerpt. as you can imagine, i didn't finish the essay before posting this .. but this is the last, i promise.

it's worth reading because as mathematicians, numbers are numbers, and what paul describes is something like ordinals from set theory. but from a programmer's perspective .. mmmmrrrppgh.

There are more shocking prospects even than that. The Lisp that McCarthy described in 1960, for example, didn't have numbers. Logically, you don't need to have a separate notion of numbers, because you can represent them as lists: the integer n could be represented as a list of n elements. You can do math this way. It's just unbearably inefficient.

teaching, an exam, and the toll on research.

last night was exam grading until 1am or so.

that in itself mightn't have been so bad .. digging ditches would have been worse, for example .. but it was what came before:

• insomnia, then prepping at 8am, and teaching at 10am,
  
  
• urgently thinking about research for a few short hours in the noise and disturbance of east hall,
  
  
• lecture at 2pm, seminar(s) at 3pm, and then office hours until 5:30,
  

and then you know the rest: exam proctoring and grading. i acknowledge that plenty of people out there have long days, every day, and they have my respect.

it's just that i don't cope so well .. not anymore, at least. i don't play so well with others either.

---

purely empirically, an 'exam week' isn't that much more work: an additional office hour than usual, and a change in prep routine. it's the stress and concern that changes things.

students care about mathematics and class for the first time in weeks because they finally encounter something that affects their immediate existence (that being a grade). some are nervous, and some are close to panic.

i know i'm not worried about how my students do; they'll do as well as they've studied and prepared themselves. i know i'm not worried about how well i've taught because, apart from a stray lesson or two, it doesn't matter: a student will choose to learn or not to learn, to learn well or not to learn well.

but by some sort of osmosis, it infects me and i worry a little. i find myself less capable of concentrating and less effective at research.

it's like upping the pace on a road run, where you can feel your leg turnover change to something quicker. there's a slight twinge and it feels unnatural.

you know you can maintain it for as long as you need, but when you switch back to your usual pace, that twinge will change to a soreness that won't go away for a while.

so today i returned to my usual pace.

---

today was also my meeting with the advisor, and it was fine. but it wasn't .. optimal. despite the hours of thinking i tried to fit in ..

(amidst the bits of stress from exam routine and between writing shifts, last weekend)

.. i walked into the advisor's office, empty-handed: no results, no remarks, no descriptions of any notable obstructions. i ran out of time and had nothing to say .. or rather, nothing new.

i hate it when that happens. i hate wasting people's time: the advisor's time.

these are parts of papers that i should read, these are claims and exercises i should do, and i should be trying to come up with my own good ideas. but like last week and the week before that, i ran out of time and i couldn't ..

.. and about the ideas, well, often i'm just not that brilliant. the odds are bad, but i try.

i try because someday, that will be my only option. as the saying goes, we are all students and we will always be students, but someday i won't be a graduate student and i won't have an advisor and i'll have to stand 'on my own two feet.'

quite simply, i try to be a mathematician, a researcher .. but more often than not, my endeavors come to nought and i end up just a student again.

".. but some things must be done."

in the last 24 hours i've spent my waking hours writing. rather, i've been LaTeXing.

it's a talk for an upcoming conference in two weeks; the rationale is that if i can convince myself to write a talk, then i might be further motivated to write a research article on the same subject. there are a few slides left, and i think the final form of it will be tolerable and not too boring.

i've even coded a few figures on some slides, so thanks to the graphicsx standard package, the audience will see a picture or two.

more, it will be a friendly audience and i think i will enjoy giving the talk. it's been a while since i've felt that way.

as i've told a friend of mine yesterday, i'm not looking forward to writing this paper, but some things must be done.

it feels to me like writing a eulogy, because the content will be salvage from my first thesis problem. as you may recall, that problem is dead or left for dead.

never mind the difference; the effect is the same.

to me, writing this paper is reliving that inevitable end.

very little of the inquiry is my original contribution; of a year spent on this project, there were less than two months where i was working on original ideas and not investigating the past work of others.

i explored many areas and ideas and learned a few things, but that's not the point; the point was to do or accomplish something, and specifically, something new and preferably interesting.

i can count two original "theorems" from that inquiry, and one of them is a computation anyone could do. the other is more an observation than a theorem, and there is that same "everyman" feeling to it.

i can't seem to articulate what i mean. i know that this first problem is not one of my failures, or a failure at all.

but i admit: identifying the flaw in the method seems one of the rare times when my contribution actually mattered. i did that. inherently negative as it may be, it is an accomplishment ..

.. and look what it did. now there's no more problem. lovely, wonderful, and bloody marvelous.

but no worries. there's a second problem, with mysteries at every turn.

it's all frontier land. there are few blazed trails, and i can choose not to take them. it's terra incognita, where you may define "success" in any terms you want.

but that theorem i proved .. remember that one? loosely speaking, it asserts that a subclass of concrete examples are not worth studying with this theory.

some days i feel like i inquire, if only to wait for another disaster, and for another problem to fall dead.

why do i have a bad feeling about this?
all of this ..?

before the meeting and after; thoughts on salvage and writing.

i wrote this most of this, last night; new text from tonight begins at the symbol '#.' an epilogue follows the symbol '##.'

i suppose it could have been too much to ask for: that is, three consecutive weeks of good research ..

two weeks ago, i made a conjecture and it might still be true. however, it is certain that it remains a conjecture tonight.

at the time, i could only sketch an argument for a rather special case, but it was too simple-minded. i would be embarrassed even to state it as a lemma, because it is too transparent; upon reading the hypothesis in the lemma, it would be obvious how the proof would go. [1]

there's no art in that: no boldness. but when i tried to generalise, there was a gap.

one week ago, i filled the gap and managed a weak version of the conjecture; it's true for a wider class of examples, but remains unknown in full generality.

it's of little interest, and more a toy than anything else. i suppose that one of its few strengths lies in its concreteness. to some degree, the weakness of transparency is gone, but as usual this proof reduces to that first special case.

i dare say, though, that the reduction is not so transparent. you might even call it clever, though it is n. weaver's cleverness and not mine.

# this past week i've been unable to concentrate. i don't know why. an idea i had this past weekend now makes no sense and i can't remember the motivation. i think i fooled myself, and it wouldn't be the first time.

i hate it when this happens. tomorrow i'll stop by the advisor's office and tell him that i have no results. the most frustating part is that i have essentially nothing to say, because it's not even worth mentioning what didn't work, because i know why and it didn't teach me anything.

the meeting will be fine; i know that.

i now know the advisor well enough that that hour and half-hour will be worthwhile, indispensible time. what bothers me is that the advisor is disturbingly clever and wise, and if i don't stop him he'll prove results which should be my task to prove. he can't help it; he became faculty @ um for many good reasons.

advising and guiding are fine things, but there is no honesty in my standing in another's shadow, because one day i will move away and i must face the sun myself.

the way is difficult and we may choose our ways, but some choices are meant to be formal.

---

## perhaps i shouldn't call it a conjecture. the word "conjecture" is too grand and suggests something that is inherently worth pursuing. maybe i should have said "claim."

the meeting did go well. the advisor and i philosophised, and we pondered the direction this inquiry goes. the once-conjecture/now-claim is fine, but ultimately it was meant to follow a path of counterexamples and derive a necessary condition for the geometric function theory that i study.

if i prove it, then it is a little something. but i haven't proven it and at the moment i have no more ideas, so what stands is not theory, but a souped-up class of counterexamples based on another's existing work. even if i prove it, the result won't fully resolve the issues at hand.

there are deeper issues and more relevant connections to the existing theory of the analysis on metric spaces. in particular,

• there is the theory of upper gradients through rich geometries of curves;
  
• there is p. hajlasz's theory of sobolev spaces via lipschitz-like moduli of continuity;
  
• and there another nascent, but existing notion. as yet, there is no rich theory.
  

to paraphrase aristotle, even taxonomy is no easy feat. as mathematicians, we go further and attempt an understanding between elements of this taxonomy, and assess its relevance.

this is among the goals at hand. what does it mean, a sobolev space? let us have a metric and a measure; can we have calculus? are there many types, and are they so different?

now i am too philosophical. let me prove things before i say more.

---

this post grows long, but let me say: i had a thought, a few days ago. it was an old realisation, made acute and sharp by the passage of time and an unexpected reminder. the future nears, a conference quickly nears, and the job hunt will soon come; a year's time.

that first thesis problem is dead; i've left it for dead, but there is something to salvage. i want it to mean something, that a year of my work mean something. in a little while i will give a talk about it (a so-called preliminary report), and the advisor still thinks it a good idea that i write a paper about it.

it is good to have publications, if only for a strong c.v. to date, i have only one and that is a joint work. another can only help.

still .. why is it so hard to get to writing? i've heard a saying once that a mathematical paper is never finished, but given up. to write this, i will have to give up the possibility of progress for a while; i would write what is, and not what could be or is likely to be.

i think i am too used to uncertainty, and cannot bear it. it's silly but true.

[1] and yes, when i say "obvious" i do mean obvious. however, i didn't say that it's trivial .. though i dare say it's close.

an "immortal solution?"

a few weeks ago i remember the title of a preprint from the arXiv concerning things called "immortal solutions." i didn't think much of the title at the time, and in all honestly, i thought it was a catchy name to make the paper more readable.

but now the term is called an "eternal solution" and has made news, at least on ars mathematica. this work of p. smith contributes to the analysis of the Navier-Stokes equations (see wikipedia entry) which is one of "the biggest games in town." the clay mathematics institute, for example, formulates a problem concerning these PDE.

as i've read secondhand, the theory uses viscosity solutions and c. sormani @ cuny has written a little about it. if she vouches for it, i'll listen.

this world is too small. i've bumped into prof. sormani once, years ago at a conference at CUNY Graduate Center, though i don't think we'd remember one another.

some days i wish i worked in PDE.