The Works and Days of Hands.

.. There will be time, there will be time
To prepare a face to meet the faces that you meet;
There will be time to murder and create,
And time for all the works and days of hands
That lift and drop a question on your plate;
Time for you and time for me,
And time yet for a hundred indecisions,
And for a hundred visions and revisions,
Before the taking of a toast and tea.
~ T.S. Eliot, Prufrock

---

My productivity level has struck an all-time low .. or so it feels.

I think I'm strung out (or is it strung over?) on caffeine, partly because of high workload, low productivity, and the fact that it's $2 Latte Wednesday at the corner Espresso Royale Caffe. [1]

At any rate, the outlook doesn't look good. By typical standards, it's still early in the evening, yet the parallel lines on the sheets of legal pads are beginning to burn into my retinas; fluorescent light (the office standard) has slowly become an intense, eye-squinting white and everything underneath its glare takes on this surreal, unrelenting glow.

I feel mildly terrible and generally unmotivated towards the tasks at hand. A few talks and presentations come to mind.

---

• As of the middle of last week I'd been working feverishly to understand a theorem whose proof will be one of my class presentations this semester. Great stuff and fun pictures, despite the technical nature of the theorem.
  
  
• As of the last few days I've maintained a mild paranoia, if only because there seemed a discrepancy between the strict statement of the theorem and the version we've been using freely in class.
  
  
• As of a few hours ago, I've realized that I was reading the wrong theorem. The one my prof suggested was actually a lighter version to the theorem with the supposed discrepancy. In fact, I only needed to read the first five pages of that article instead of, say, the first fifteen.
  

I suppose that's what happens when you don't keep in constant contact with your profs. \:

On a related note, my presentation for that class has been delayed for another week and a half. It never fails: I always choose the wrong times to procrastinate and by symmetry, the wrong times to over-work .. but at least the presentation is done.

Then again, inevitably the week will pass and I will forget everything and fumble through the presentation. Trust me.

---

Later that week is possibly another presentation. Rather, it's a talk: a contributed talk at a five-day conference at Dartmouth where I have to explain how the problem I worked on (with my very kind collaborator, of course) differs from other solved problems in Sub-Riemannian Geometry and the Analysis of PDE, and why it might be of any interest to anyone.

Mind you, the audience would be the movers and shakers of the aforementioned areas of study, which means that they don't have to be merciful.

[ponders this]
[cringes]

I'll just stick to explaining what we actually proved. That's likely a lot safer ..

.. but still, I have a bad feeling about this.

---

Some ideas are beginning to take shape, in my mind. I can't explain what I mean. But it feels like progress.

I think I'm developing a knack for asking (somewhat) interesting questions, if only to myself. These seems a good sort: the sort which requires strategy and work to answer. They come fleetingly and irregularly, but they come.

Now if I could only solve a few .. at best, they live in the pages of my notebook (the one I keep for mathematical speculations) until the day that they are found to be ill-posed or solved, by me or others.

It feels like hope, if only a little hope .. say the varying levels of understanding tragedy, as expounded by Aristotle. A good tragedy should affect men of all levels: for noblemen it should inspire and induce catharsis. For lesser men, it should at least give a sense of shock or pity, however misunderstood.

It's something, at least: something to work with.

---

I'm still advisor-less, and by implication, a coward.

Wonderful.

Unrelated Note: I've discovered during warmer days that I can make it there from my office in about a minute, if I'm going at an all-out sprint and there's no undergrad traffic (say, during the dead time between start and end of classes).

Not what you'd expect ..

I received wind of this from the blog community LJ: Mathematics, and likely you'll see the same message plastered all over the place in days to come. But let me plaster it here.

Bob Dylan gives his opinion, as I've read here @ this music website, nme.com.

BOB DYLAN has launched a withering attack on contemporary rock bands in the programme notes for his latest American tour.

"I know there are groups at the top of the charts that are hailed as the saviours of rock'n'roll and all that, but they are amateurs. They don't know where the music comes from," he wrote, adding, “I wouldn't even think about playing music if I was born in these times... I'd probably turn to something like mathematics. That would interest me. Architecture would interest me. Something like that."

Not the first thing I'd expect from Bob Dylan, and likely it's just a point which he's making on whim. But it is nice to hear: he apparently realizes that mathematics is more than plain computations and calculus, which many of us encounter in the modern world of convenience and indulgence.

(Mind you, I make this inference because he likens maths possibly to architecture, but warn me if I read too much into this quote!)

".. And sο castΙes made of sand / faΙl intο the sea / .. eventuaΙly."

If all your ideas are working, then you're probably not getting enough ideas.
~ Jim AgΙer
  (as told to me by Lιnda Pattοn, after my fifth consecutive idea didn't work)

I suppose that the title of this blog is more fitting than I had originally thought.

Mathematics are frustrating, and in particular, analysis is frustrating. If anything, being a student of mathematics has made me more paranoid and uncertain than anything else. Your seemingly good ideas were flawed by the diagrams you drew or the impossible assumptions you've made. You second-guess and third-guess .. nth-guess the ideas that as a child, you would run to Mom to tell, in rushed and excited breaths.

Occasionally, there is that child-like delight. As with any discipline of thought or practice, what motivates me is that spark of an idea which may be the solution to your woes (well, here I meant mathematical woes, and your current problem, at that). When it fails, it's only a little more frustration atop a larger, ever-growing pile of cold-pressed fury .. but when it works .. (:

I feel like I'm on top of the world, and I can take on anyone and anything. I feel like leaping into the air and cheering, and the negative feelings go away for an instant. Maybe I'm not such a dummy after all, I suppose, Maybe there are ideas in my head after all .. good ideas!

Then you proceed with the idea and carry it through. The cycle of ups and downs continues.

---

Those failed ideas remind me of making sand castles, and getting angry every time the tide comes back and wipes those grainy turrets and gates away. I have a habit of fuming and sulking, I'm afraid, despite the fact that the sea is without cause, motivation, or memory.

It takes me a while to realize: well, there's still a lot of sand, left .. might as well make another one. But when I do and when my hands get dirty again, that last sand cast1e doesn't matter at all.

Wait 'til everyone gets a hold of this one, I think, as I grin and reach for the blue plastic shovel ..

---

I have another idea again. Maybe it will work, but maybe not. It can't hurt to try again.

Wish me luck!

ps. The title is a song lyric from "Castlεs Made of Saηd" by Jιmi Hendrιx. Great song, that.

"Look Before You Leap."

Words to live by, those.

This past week I spent more than a fair share of time studying an approach to one particular problem, only to discover that it is nigh-impossible to implement it. It's likely that the approach cannot possibly work, because it would lead to a rather unbreakable contradiction. Several papers suggest this, in fact, and one might even contain a proof as to why.

Damn it. I should have known that it was too good to be true. For those of you who knew what I was trying to do, you don't have to say "I told you so," because I see what you mean, now. Fool that I am, that it took me this long to realize it.

Nothing new or interesting occurs, as of late. No progress with securing an advisor, and apparently, no good ideas on how to solve anything - not even ideas or inclinations on what would make good problems to study.

Argh ... how does anyone ever solve, or even learn anything at all?

Something like a Socratic dialogue ..

It feels like I'm standing still .. or rather, sitting still. Anyways, you get the idea .. or wait, you may not. In fact, that's the point of this post: it's meant to convey the idea to you. Right. Yes.

"So what's this idea all about?" you may ask.

"Nothing big," I reply casually. "It's just that I've come to a point in my academic life that I should do something. In fact, I feel the urge to do something, to make progress towards some end."

"You're talking in riddles again," you point out, "and you're being vague."

"Fine, fine," I retort, "Lately I've realized the limitations of being a student who only sits in lectures, solves problem sets, and takes exams. It's a passive sort of learning, in that one must be given topics to ponder and ideas to entertain in a reasonably digestable form. Despite the frustration of understanding this matter or that, the efforts are temporary and the work not so motivating, because you have no inherent tie to it. It's just coursework that your instructor has assigned to you, or it's a topic decreed by a syllabus."

"Everyone feels that way at some point," you agree, "but what's the alternative? What's your alternative?"

"I'd like to come up with ideas of my own .. or learn ways of coming up with ideas. I think I see now why the structure of the graduate program has developed the way it has, here in the Mathematics Department. I'm not saying that there's nothing left to learn by sitting in classes, but I'm not growing as a mathematician if I'm not thinking for myself and letting others suggest to me what to think about and what is 'interesting.'

"I feel mentally restless, but it's a good sort of restlessness. It's the same sort which appears during the last week of holiday, when you've had enough good cheer and merriment and are ready to take on the rigors of the daily grind again.

"The course of the graduate program urges me to go and seek a thesis advisor, so that I may proceed to pursue academic research and write a thesis. I think I agree to that now."

"Well, were you ethically opposed to it, before?" you ask in jest .. at least I think it's in jest.

"Of course not, but mine was a neutral sort of opinion, simply because I hadn't thought much about it. Now I have, and I agree with the philosophy. I agree with it as an avenue for intellectual growth, but also as a realistic end.

"Regardless of whether I can learn all that I mean to learn and do the best that I can hope for, at the end of the day there remains a concrete objective. I have to take my oral prelims, I should do a bit of research, and I must write a thesis. Those goals cannot be argued, and taking this bare, concrete perspective, one could say that the rest is icing on the cake, so to speak.

"I think I'm ready."

"That's nice," you say.

"Thanks," I reply.

---

ps. I finally wrote to one of my profs. With any luck, we'll discuss matters of advising soon and with some more luck, perhaps this prof will be conducive to working with me.

Exorcising the Inner Demons .. Part 3 of 3.

A few nights ago I found out that my idea concerning a research problem doesn't work. Worse yet, it's the sort of error which occurs late at night when you're excited and really want it to work .. but in the light of day it's painfully obvious that it can't possibly work and you become ashamed that you even thought of it.

But it would have been such a nice idea .. you would have just computed this and piggy-back'ed on that paper .. there would still be much to do, but at least it would then be tenable ..

Bah. Argh. Hmph.

Back to square one, I guess.

It feels like something fundamental is missing from the context of the problem .. just a nagging feeling that I can't seem to shake. The best way I can describe this are those moments when you remember something and it's on the tip of your tongue, but you can't remember the exact word.

It's something like that, except that I'm not remembering but discovering .. more like solving a crime. A mechanism is missing, I can feel it and imagine it somewhat, but I don't know what it is.

Curses! I should be finishing the particulars to my talk, or planning out how to approach a potential advisor .. something. As much as I love pursuing this problem, I have to set it aside because other things must come first.

Business before pleasure, and all that.

This is what I mean: in my mind this is not thesis work, but an idle curiosity of some small importance. It should be set aside if I want to get anything done as a graduate student.

Exorcising the Inner Demons .. Part 2 of 3.

Tomorrow's Part II of my talk. I really hope that I don't overtime again; I can only imagine what a pain in the neck it causes the people who bother listening to me.

Nothing gives the same scale of paranoia as giving a mathematical talk, I'd wager. Irrationally I believe that some heckler will inevitably find a rather deep error, and then this huge Vaudeville-style cane will come out of nowhere and drag me away from the blackboard, while the audience looks on ang laughs.

[re-reads]

My god. I actually wrote that. Er .. moving on ..

I must say, though, that this paranoia can be rather helpful. I've been far more careful with my reading and perspective on proofs, though my explanations still feel painfully vague .. but they are getting less vague!

I'm improving my understanding of the Be11aiche and Gr0mov articles, if only through the help of reading Montgomery. Differential geometry is a tricky subject; sometimes it feels like the study of "What do do if you're not in Rⁿ" .. (;

In particular, reading and rereading these articles feels like retracing footsteps of intellectual giants. You can tell the improvements made over time, and the increasing sophistication and abstraction.¹ It's not as severe from Newton's theories to Einstein's, but it does give a sense of history, and gives hope that something little you can do might lead to a greater whole someday.

At least some good comes out of this. Better yet, maybe the audience actually finds this interesting ..

[chuckles]

.. well, a guy can dream .. (;

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¹ More and more I sense that how one learns is extremely important, and by that, I'll emphasize the order of learning and the level of sophistication. To imagine a C-C space, I have to imagine a Carnot group (the Heisenberg group in particular) and to understand that, I imagine a Riemannian manifold, and before that, a surface in space with arrows on it.

Abstraction is hard. I wonder how anyone, most of all the algebraic geometers, get by with it.

Exorcising the Inner Demons .. Part 1 of 3.

I should quickly secure a thesis advisor, and soon. I would have secured an advisor by now, were it not for the fact that I am a gutless coward.

Argh.

More and more I'm inclined to believe that it's not how you ask that is important, but that you ask at all -- though it still seems better that I talk to said person regulary, then ask at a later day. It would be somewhat strange to ask "Will you be my advisor?" unexpectedly and out of the blue. Ideally, one asks in hopes of a "Yes" answer, and hence it is better to ask when confident of such an answer.

But what am I supposed to talk about? I don't know anything -- anything interesting or deep, or have any good ideas, at any rate -- so what can I say?

---

Part of me worries -- not only about obtaining a thesis advisor, though that is troublesome enough -- but what it means to be someone's student.

Does that mean that I must set aside the idle interests and curiosities that I've collected over the years, and focus all my energies into the coming Prelim Exam and to do research solely towards my dissertation? What if I want to attend a conference, if only to learn something that's not strongly related to my thesis topic?

It's silly to think that one must give up one's curiosities, because that is what makes us mathematicians. But then I remember: after this term I'll have three years left. That's not a very long time to get it all done .. and what if my first thesis topic doesn't pan out, and I have to start again? The curiosities will be the first to go, if ever there is a problem with time.

Perhaps I'm just afraid of making choices: from now on I'll work with this professor and I will study these things. Those other ideas are very interesting, but I don't have time to study that now. There's work to be done first.

I don't know. Maybe I'm just being silly or paranoid or both.