holiday procrastination

I still haven't been able to summon up the will to do any work, despite sitting in the coffeehouse with a pleasant brew in hand, pleasant big-band music streaming from the ether called internet, and my book and notebook sitting pleasantly right before me.

It's the perfect time to work, too: it's Friday and working now gives a good excuse for weekend follies [1] later .. yet no prelim reading accomplished and no research done. I think I know why:

Deep down inside me, I want this holiday break to be an actual, proper holiday -- not just the peace and quiet where one settles down, without students underfoot, and gets down to work -- but doing non-academic things and non-work things.

I mean having "fun" which others outside of your building and department would also consider to be fun. For example,

• Renting and watching movies is considered fun, by most persons.
  
  I haven't done this in a while.
  
  
• Building snowmen, then tackling them in a drunken mirth might be considered a little strange, but still fun.
  
  I might have done this before, but I can't quite remember ..
  
  
• Writing on chalkboards and using the Triangle Inequality [2] might not be considered fun by the "regular" stretch of the imagination.
  
  I think I do this at least once a week, and my friend Kevin does this almost every day.

At any rate, maybe I don't need to work this break. My advisor will be away all next week, which means unlimited prelim studying time (and time for a bit of research for when we meet next). But there are talks to give in a few short weeks, and work to do for them.

But as we all know, procrastination needs no motivation. It is merely a state of mind and if one willingly enters that mindset and realm, then that is enough. However, one must reach that realm first, and not sit or stand on the neutral boundary, lollygagging and squandering time that could otherwise be spent on real work or real play. That "frozen" mindset reminds me of this passage from Dante's Inferno, Canto I.

And just as he who, with exhausted breath,
having escaped from sea to shore, turns back
to watch the dangerous waters he has quit,

so did my spirit, still a fugitive,
turn back to look intently at the pass
that never has let any man survive.

If I ever get my act together and make a decision, then I can salvage something from today, whether it be great fun or a modicum of work. Maybe both .. who knows? But I have to stop idling and sitting so mentally-transiently, and do something.

[1] .. never mind the fact that I never do anything on weekends anyways, with the occasional exception of being invited to math grad parties, which are far better than they sound, as my non-mathmo flatmate can testify.

[2] .. Roughly speaking, the Triangle Inequality (Δ≠) is your common sense about distances, but written in mathematical symbols: that is, if you're travelling from A to B, and then B to C, then that total distance travelled is no shorter than had you travelled directly from A to C.

(Δ≠)   |A - C| ≤ |A - B| + |B - C|

See? That wasn't so bad, was it?

To the experts, yes: I did use norm-notation without much reason why. q:

laptops in the classroom: an article post.

Here is an article from Slate Magazine about laptops in the classroom, but it's not a techie piece. It instead considers the nature of lectures and professors.

I haven't formed any opinions yet, but certain paragraphs trouble me. This is one paragraph which I've split into parts, and I hope it is not too out-of-context.

In any event, even when multitaskers can't keep track of the professor, it probably doesn't matter much. In lectures at large universities, especially in the humanities and social sciences, class time is usually taken up by the broad outlines of the subject.

The real learning occurs when we bear down and pore over the hundreds of pages assigned every week—the lecture I'm currently tuning out assigns about 3,000 pages of reading over the span of the semester—and when we attend small discussion sections with graduate students who go over what we've read. Any good grade-grubber knows that the trick to doing well on exams is knowing the reading, not what the professor said last week.

If what the author says is true, then what is the point of attending lectures, other than if there is a class attendence policy? Why have lectures at all, if they are obsolete in the learning process?

Is it also possible that, in some cases, the grades don't reflect what the prof discusses in lecture because the prof doesn't grade the exams? Or is that superfluous?

Perhaps the real problem with laptops in lectures isn't the laptops, but professors' over-reliance on the lecture as a learning tool. Earlier this week in Slate, M. Stanley Katz contended that "the most effective learning is active learning … teaching must involve presenting students with problems to solve rather than merely lecturing about those problems."

Amen, professor. You try listening to rambling, jargon-filled disquisitions for 15 hours a week without reading blogs. At least Gawker solicits our contributions.

Wow. "Rambling, jargon-filled disquisitions." Are lectures really that bad, these days?

state of the union

Prelim studying is going all right, I guess.

In a ways it feels like downloading a large (legal) media file: at every instant you can read off the streaming speed from the Download Manager, but you cannot predict for certain how long it will take before the download is finished. If you have a deadline, then you watch the screen and hope, then watch the screen and hope some more.

Similarly, I feel like I'm learning at a good rate, but it remains a mystery whether I can be fully prepared for all the topics on my syllabus. We do what we can and I understand that, but I'm having trouble accepting that it's all I can do.

Earlier this weekend I thought I understood tangent measures (arising from blowups of space [1]) but now I'm not so sure.

It can be troublesome to learn from books, because if one is not very clever (i.e. me) then one adopts the perspective of the author by default. There's nothing inherently wrong with this, but one runs into trouble if

1. .. the book isn't "very good," as measured by difficulties such as omitting details, unclear language, possible errata, and the like.
   
   
2. .. despite the fortune of clarity, the book has an unorthodox perspective, and what one had thought was standard terminology is actually "author-speak."
   
   This becomes a deeper issue if the methods of proof differ from one source to another and if one seeks to generalise arguments from a common starting point (for example, Radon measures in Euclidean space).

So I worry. There might be no "right way" to understand a particular idea, but fortunately or unfortunately, there are some "wrong ways" to gain understanding.

---

Research is .. well, research. It's shifty, ever-changing, and hence unpredictable.

Three hours ago, I thought I had exhausted every possible idea out there and that the problem is insoluble, despite what M. Morse claimed in the 1950s .. [2]

.. two hours ago, I began to draw the same diagrams, only to realize that I drew them wrongly! Now things look more promising, but they'll require a bit of trickery before I can prove what I want to prove ..

.. and for the record, an hour ago I was eating dinner: vegetarian Indian food which made everyone else in the room, who were Indian food-free, salivate. I felt guilty and sinister at the same time. (;

---

Teaching is what it was before: neither good nor bad until you think about it and make judgments.

I tried to switch things a little and make my lectures more example-driven and start matters off with what the students know well (or should know well from their last exam). Time will tell if they are just as confused by the material or if it's actually working.

I must admit, the examples were fun: I introduced differential equations today by giving examples of unhindered and constrained population growth, and perhaps as a moment so that they could catch up with theit notetaking, I idly mentioned the famous viewpoint of Thomas Malthus, if only to mess with their heads and get them thinking.

However, one bit worries me, and I call it the "Heisenberg Uncertainty Principle of Calculus Education." It goes something like this:

The more interesting you think the class topics are, the less likely your students will understand what your talking about. Conversely, the more your students understand what you're lecturing about, the more likely you find it boring ..

That's mean of me to say. Any opinions? q:

---

[1] and by that, I don't mean the blowups from algebraic geometry! Perhaps the term "rescaling" is more accurate here.

[2] In one of M. Morse's papers on the Schoenflies Problem (before he studied the problem in terms of conical points with Huebsch), there is a claim which asserts that a smooth analogue of the topological Schoenflies Theorem fails.

Morse cites a paper of Milnor (the one about exotic 7-spheres) and since then, everyone else says the same in a rote manner. I've might have mentioned it before, but my current task is to make sure that Morse's claim is correct.

feeling mathematically inarticulate

There's a term from Victorian England called a monoconvolute.

It's a derogatory term and brought up after the study of phrenology became passé and neurology became its successor. The idea was that the convolutions (wiggles) on the surface of the cortex provide "space" for storing information and memory, and if you had only one convolution on your brain, then you must be a very stupid person.

At any rate, the existence of such a word demonstrates some importance about having a complicated brain for higher cognitive skills and whatnot.

But I believe that the opposite extreme can be troublesome, because I suspect it in myself. Either my brain is too muddled and twisted, or my thoughts require more discipline: something in me is muddled, at least.

In fact, this reminds me of an instance from high school: I was a sophomore taking the usual English class and my teacher was a fine editor but a harsh grader, and used red fine-tipped pens by the box.

There was one essay assignment we had to write one week .. but so that you know, we had essays every week. Monday was our deadline and Wednesday she returned them to us, mires of scarlet and scrawl atop once-cleanly typed white pages. But this one assignment everyone had jaws dropped, grades slashed,

.. in the same way that I grade Calc II quizzes, come to think of it ..

and there was general despondence and "gnashing of teeth," as one friend of mine would say.

I could swear that my essay looked the worst. It might as well been dipped in the red ink, but the strange thing was, there was no grade on it: not on the first page, the second, nor the third. I was able to read the last of the scrawls, which I interpreted as either:

Keep these ideas. Rewrite this.
or
Kaput the ideas. Reunite us.

Deciding to use common sense, I rewrote my essay and brought it back on Friday. Being a curious child, I couldn't help but ask my teacher, "Was it really that bad? Was it really ungrade-able?"

"It wasn't finished, and the language was awkward," she replied, "I wanted to see what your completed thoughts were."

I nodded, handed her the rewrite, and that next Wednesday I received two essays back. My rewrite looked only speckled with red this time, and later I learned that nobody had ever been given a rewrite option from her before. I don't know if she ever did it again.

Suffice to say that English is my native language, but mathematics isn't. It seems like I never quite phrase things properly, and every time I discuss some matter with a fellow maths person, we would discuss the same object or concept, but their viewpoint is much grander and more precise than mine.

When I studied literature and history, I never felt like a plagiarist, but now that I study mathematics, I feel like I steal all the good stuff from other people.

Take my meeting with my advisor today: it was the third session we've sat down (well, figuratively) to hash out this lemma of Milnor's, concerning existence of diffeomorphisms of topological n-spheres when given a smooth homotopy condition (more precisely, a differential isotopy) on maps of (n-1)-spheres.

I still feel as if it shouldn't have taken that long. Had I remembered to tend more to geometric issues, then this could have been put to rest sooner. Unfortunately I still think too formulaically and not very intuitively, and honestly, I use rather terrible notation.

Had I not drawn enough diagrams, my advisor might have thought I spoke nonsense .. which I probably did. Happily and fortunately, he's a forgiving man.

Perhaps there's no real conclusion in all of this .. only that today is an "mathematically inarticulate" day for me. \:

short post first: seminar talk woes.

Apart from student-run seminars and analysis-related seminars, I have terrible luck with choosing talks to attend. I might look at the title and abstract, think

"Oh! That looks interesting. I'll certainly go to that!"

and as an afterthought, bring my folder of work with me .. just in case.

It is unfortunate that I have little background, and too often it is the case where the speaker has lost me for good, or that the jumps in reasoning are non-obvious (to me, at least; it is a relative term) and I cannot think fast enough to follow the argument.

It then becomes frustrating: I slip into self-driven ignorance, because I cannot summon the nerve to stop the speaker and ask for a reminder or two of what is going on. Then, eyeing my folder, I slip out my notes out or a copy of a research article and hope that the speaker doesn't mind too much.

One of my commentators from my LiveJournal blog has remarked about this so-called "courtesy" of ignoring the speaker when it is inconvenient to do so, and I don't have a good answer to support or defend this practice.

But should you pay the price of time and confusion because of a mistake in choosing the wrong talk to attend?

Maybe my expectations are too high. Every so often I attend a talk, hoping to learn what the words mean in the title.

I attended an Open Problem Seminar once to learn what a Kähler manifold was, but never found it out. I later asked a friend, who told me a casual definition and now I've forgotten what it is.

Another time I attended a Colloquium in order to learn what a Gromov-Witten invariant is, but to this day I still don't know what such invariants are. I suppose they are hard, if only because they sound hard. q:

But I'm going to try again. I think this is really going to work, this time. In 1 1/2 weeks' time there will be the following Colloquium for the UM Maths Dept:

What is K-Theory and What is it Good For?
Paul Frank Baum
Pennsylvania State University

This is a survey talk on K-theory and will consist of four points:

#1. The basic definition of K-theory
#2. A brief history of K-theory
#3. Algebraic versus topological K-theory
#4. The unity of K-theory

This sounds interesting!

I've heard the term tossed around by algebraic topologists, and it is always a fine thing to learn a new word, though I suppose when most people say that, they mean words like vitriolic or recondite.

At any rate, I remain hopeful this time. The speaker can't possibly avoid saying what K-theory is, right? (;

neuroses of a mathematician-in-training.

Earlier I think I over-exerted myself in terms of how long I could spend in the office while remaining sane. I eventually left for a little sushi, and lazybones that I was I took the bus home.

But this is what I wrote, before my escape.

---

Every so often I hesitate while doing perfectly normal activities in the office or chores at home. When other people are around, I think it looks like I 'zone out' or 'blank out' for a spell, and when I recover then all is well. Still, when it happens, it feels eerie for everyone involved.

For instance, I'm sitting in my office and staring at a white page of paper. I know what I'm going to write (or I think I do, at least) and the pen's already in my hand. But my hand is shaking slightly, you see, and other than that I can't seem to move it for a second or two. I can't write down the function or draw the diagram.

It sounds obsessive-compulsive or neurotic or some standard psychologically-motivated word and I can't help but wonder if something is wrong.

There's trouble with focusing, too, and trouble from distractions.

After teaching I can't return straight to work because the transition to silence and solitude feels too great and I must do something else, like grade quizzes or check my email .. and 15-20 minutes later my brain feels ready for tackling research or coursework or prelim reading or what-have-you.

After office hours I can't remain in my office. There's something about my desk now, which emits some residual state in which my mind rested. It's too computational and unfocused and if I start proving things in that mindset then I will argue myself in circles. I must leave it, leave to a coffeehouse or the second-floor Atrium/study area and work there. Let the experiential residue of teaching to dissipate, and then I can return to 1852 EH and work unhindered.

My god. This sounds like neurotic talk, and it feels like I'm less mentally capable than I used to be. It feels like something in me has changed -- maybe snapped or dissolved, say -- and as a result I've lost some peace of mind.

I don't know. Maybe it's all in my head, which could be relieving .. or troubling, depending on how you choose to see it. I really don't know.

after much objection ..

I will write down carefully the David-Jones Theorem, directly from the Mattila text. So here it is:

For any positive integers m and n and ε > 0, there exists an integer N(ε) such that if Q is the unit cube in Rⁿ, m ≥ n, and if f : Q → R^m with Lip(f) = 1, then there are B₁, ..., B_N in Q, N ≥ N(&epsilon), such that

Hⁿ_∞[f(Q \ (B₁ u ... u B_N)] < ε

and each f|B_i is bi-Lipschitz with Lip((f|B_i)^-1) ≤ N(ε).

Sorry for the confusion and errata, guys. Let me think once or twice before I comment on this theorem again!

plotting my own doom ..

Or as it's more commonly known here, writing a syllabus of topics for my Prelim Exam. A good bit of them I've seen before, but it still seems a scary task to remember it all .. or at least enough to keep my committee happy.

If you're curious, here is a tentative breakdown of the subjects:

1. Preliminaries
   
2. Lips¢hitz Functions and C0nvex Functions
   
3. Sob01ev Spaces and Fine Properties
   
4. Highlights from Ge0metric Mea$ure The0ry

Believe me, you can say a lot about each heading. I only hope that I can, too. Wish me luck!

---

As I was browsing through books for syllabus items, I found this theorem in a book of P. Mattila. The result is credited to P. Jones, which generalises a previous result of G. David.

Maybe I am rather dull and ignorant, but it seems a little unbelievable.

Let Q be the unit cube in Rⁿ and let m ≤ n. Given ε > 0, there is a number N_ε where for all 1-Lipschitz functions f mapping Q into R^m, there are balls {B_i : i = 1 to N} with N < N_ε, where Hⁿ_∞(f(Q \ U_i B_i)) < ε and f|B_i is N_ε-biLipschitz!

Essentially, this means that off a small set set of small image, 1-Lipschitz functions from the unit cube to a higher-dimensional Euclidean space are locally bi-Lipschitz (and probably with very large biLipschitz constant). It's not even clear to me that a 1-Lipschitz function should be locally invertible, much less locally biLipschitz!

Edit: perhaps thinking of $ard's Lemm@ gives plenty of motivation. Now that I think about it, I'm not as impressed. Since n ≥ m, then the Hausdorff content given by Hⁿ_∞ doesn't detect the image that well, since in some ways, the image is an "m-dimensional set." I'd be much more impressed if the theorem used H^m_∞ instead.

Anyways, back to work.

Another Friday night at the office, and why.

The day winds down and I'm more awake now than I was sixteen hours ago, and that would likely be true even if I hadn't bought that cup of coffee an hour or so ago from Espresso Royale Caffe. It's the curse of years of late nights; some I spent doing things of importance and some I spent doing nothing in particular.

Regardless of event and memory, my nocturnal inclinations remain. It's why I'm still in the office, trying half-heartedly to finish a little more work. I don't think this is futile for a few reasons:

1. The weekend flies, and it flies quickly. It's unwise to fit all that you couldn't accomplish during the 5-day work-week and stuff it into a 2-day weekend.
   
   But even weekends have schedules: I'd have to fit some time for grading team homeworks and time for my own studies. Saturday afternoons are basketball and Sunday evenings are BSG nights at a friend's house, and this Saturday morning is a meeting I should attend and I probably will.
   
   Just to make sure there is time for fun, sometimes it's best to do a little work early at unorthodox times.
   
   
2. I've been social enough. I mean it for this particular week. A few of us UM grad students met with the GFT speaker, a grad student from Finland (visiting by way of Cinti), and thought to make her welcome and do a few fun things.
   
   It was nice: I can't remember the last time I went out for coffee with friends .. at least when it wasn't at a conference .. and had long conversations not about mathematics and not about complaining about math grad life. I've forgotten how pleasant such occasions can be.
   
   But that's what they are: occasions. Work beckons loudly and often to me, so it comes first before other matters; whether or not that is wise and correct, I'll debate it when I have less on my plate.
   
   
3. It's quiet, for once. As much as I like my office-mates and my friends and colleagues in the department, their number is great and during the workday it can be a cacophony of movement, conversation, and katzenjammer.
   
   In the late evening it's nice to have the solitary peace and quiet to think.
   
   
4. I'm relaxed. I may be working, but I'm doing so in a leisurely way. I can read without rushing and enjoy those pauses to reflect over a few paragraphs and what this or that really means. It almost feels like leisure reading, but instead of reading Haruki Murakami novels, I'm reading about geometric measure theory.
   
   The best circumstance might be the freedom of not working, but the second-best circumstance is the freedom of working without a deadline and at your own pace.

So that's why I'm working, and it seems perfectly reasonable to me. I just wish people would stop bothering me about it.

Telepathy! (that is, they read my mind)

I received this forward from my department. Talking with others, one suspects that this is a common feeling .. but this common? I didn't realize it had a proper terminology! This better not be in D$M-IV .. \:

Either the world is crazy, or that a sub-world is crazy and we are slowly accepting it .. q:

---

From: [NAME/EMAIL ADDRESS OMITTED]
Sent: Tuesday, November 01, 2005 9:53 AM
To: [NAME/EMAIL ADDRESS OMITTED]
Subject: First Year Grad Students: THE IMP0STER $YNDR0ME

First Year Graduate Students, have you ever had these thoughts?

What am I doing here? I am not smart enough to be in graduate school. Admissions must have made a mistake!

If so, you are definitely not alone. Your feelings are part of a common phenomenon that many graduate students experience over the course of their careers.....

THE IMP0STER $YNDR0ME

The Imp0ster $yndr0me may be undermining your success by making you feel as if you don't belong in graduate school or don't have the capability to do well. DON'T LET THIS BE THE CASE! Develop strategies to deal with your feelings of insecurity, fear and uncertainty.

Join us for ... CONFRONTING THE IMP0STER $YNDR0ME

with [NAME OMITTED], a C09nitive Behavi0ral Therapist from the U of M Psych01ogical Clinic on Thursday, November 10th from 12-2pm in Rackham's Assembly Hall ..