Tuesday, December 27, 2005

a list of links, and a new preprint on the arXiv.

I've built a new webpage of bookmarks on del.icio.us recently, and many of them are tagged by the term "mathematics."

Have a look at what I've saved, if you like. Here's the link to the mathematical pages, if you like.



I've been out of the game for a while (as a spectator) and have forgotten the current state of affairs of the Isoperimetric Problem in the Heisenberg group. However, there's a new article in preparation with another partial solution; some of the abstract is omitted.
Area-stationary surfaces in the Heisenberg group H^1 [link]
Authors: Manuel Ritoré, César Rosales

We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carath\'eodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero ..

.. we prove our main results where we classify volume-preserving area-stationary surfaces in H^1 with non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C^2 surface in H^1 area-stationary under a volume constraint must be congruent with a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points.

As a consequence, we solve the isoperimetric problem in H^1 assuming C^2 smoothness of the solutions.
I might not understand the hypotheses of "area-stationary surfaces," but it seems plausible that one main difficulty of the Heisenberg Isoperimetric problem -- the symmetry -- may have been resolved. Once I get out of this holiday/post-prelim funk and summon enough will to set aside my thesis work, I'll have to read it carefully ..

If the article is correct, then what remains is regularity of the boundary of the extremal set. Immediately I wonder why the C2-smoothness is used by the authors, other than as a convenience. With more work, can it be done without such an a priori assumption?

I should read the damn paper and be done with it .. if only I could convince myself that
mathematics ≠ thesis work.

Monday, December 26, 2005

Lipschitz thoughts, and the spectre of Rectifiability

In the last few days I've accumulated a good share of stir-craziness. Like a laptop running Windows XP, I feel as if my brain is on "Stand By" mode.
To return to my usual work habit, all I must do is

  1. flip up the screen,
  2. tap a key,
  3. type the magic words (my passphrase)

and after a few process cycles, everything will run smoothly, good as new. However, it's not quite time to work yet, and I'm still in this mental holding pattern of "Stand By" mode.

If now is the time to think about mathematics again, then it's time to think about my thesis problem and to read Hirsch's Differential Topology for clues about the diffeotopy extension problem.

It means looking towards which direction of research is next, and what I must prepare or explore.

It also means addressing the matter of my prelim from 20 December: what exactly happened and why. I'm not ready for that.

So it's not time to think about "work mathematics" quite yet. Let me stay on vacation and think of "play mathematics" instead.



What didn't happen during my prelim were notions of rectifiability.

It was the one topic which caused me the most unease, though the framework is elegant. In Mattila's exposition, the essential message is that there are 7-8 equivalent notions of rectifiability in Euclidean space. But as you'd guess, the proofs are somewhat technical.

Time ran out and neither member of my committee asked about any of that, which surprised me to no end. So I have no idea how well (or unwell) I understand this notion. I've only my own guesses, which are unreliable and sketchy, at best.

But I do want to know, if only because it seems like a topic worth investigating and generalising to the context of metric spaces. Recently L. Ambrosio and B. Kirchheim have written a few articles on this general context (one in Acta) ..

.. and as per my usual tyronism, it amazes me that one can say anything about this and metrics spaces: they seem to me too "floppy" and general and too abstract for me.

Let me explain myself. Perhaps I am too stuck on my visual intuition, and I find it hard to imagine such and such a metric space with such and such a measure. Conformal modulus is still a new tool to me, and possible pathologies of curves and their images don't come easily to my mind.

I've warmed up enough to my advisor's suggestions, and now Lipschitz manifolds seem to me rather interesting and somewhat elusive objects.

In the Euclidean case, rectifiability of sets in Rn is inexorably related to Lipschitz functions and parametrizations.

From what I've been told, if you enforce a strong enough condition of rectifiability [1] on a given set, it will give rise to a bi-Lipschitz parametrization, i.e. a Lipschitz manifold structure.

Now that's interesting, if only because images come easily to mind. It is like some geological means of detecting a topological character.

I envision walking across a field, and picking up a rock. With only the flat of my palm I study its exterior and see how flat it gets at this spot or that. If it's almost flat, in some measure sense, then guess what?

It arises from a rough sort of manifold. Huh.

[1] It's something related to tangent plane approximations, called Reifenberg's condition.

Saturday, December 24, 2005

holiday thoughts, whence running.

so i had succumbed to inward temptation today .. not only did i drink three servings of coffee [1] and eat a half-dozen brownies and several handfuls of roasted peas (an asian goodie), but i thought a little about mathematics today.

i had gone running, the thoughts wouldn't shut off [2], and somehow a theorem of Federer and Fleming came to mind: it states that the Sobolev Embedding Theorem for p = 1 gives rise to an isoperimetric inequality, and vice versa; better yet, the optimal constants are the same. from the pedestrian viewpoint [3] it does give a definitive correspondence between function theory and geometry, which is still not obvious to me.

the thought and a few fancies about Federer-Fleming lasted the three miles i ran .. not very long and not very deep thoughts, but they linger still ..



[1] .. one of these servings arose from breakfast at the diner. there the coffee was weak, but i had 2-3 refills of the stuff. i suppose that it evens out, then.

[2] .. after running cross-country and track in school, i can't not pay attention to minute details when running. listening for biofeedback and watching for race opponents has been ingrained in me over 5-6 years of youth competitive running, and now i can't turn it off: i can't stop thinking or processing sensory data when i run.

[3] .. assuming that this pedestrian had a little knowledge of Sobolev spaces and Hausdorff measures, that is. q:

Tuesday, December 20, 2005

i'll say no more, but ..

well .. i passed.

it was terrible, i looked like a complete idiot, forgot all the mathematics that i should have known since undergrad, couldn't integrate a damn thing, but i passed.

now all i feel is revulsion and hate. funny how the one thing you want is the very thing you can't bear after you have it.

an acquaintance of mine warned me that these latter years were the bleakest for him, pools of depression and gloom and things like that. i didn't realise that he was that right.



EDIT, AS OF LATE TUESDAY NIGHT: thanks to my friends who were around East Hall for their own reasons tonight, listened to me whine and rant, and bantered with me until the evening hours became infeasible.

I feel like a person again .. still a dolt who should have failed his prelim but didn't .. but nevertheless, a dolt who happens to be a person.

Saturday, December 17, 2005

when Death smiles upon you ..

My god. I am so f*cking stupid, but I mean that in a good way.
If it clarifies the situation, I'm laughing at myself .. half-hysterically and half-nervously, of course. But regardless of context, it is laughter.

It took me "far too long" [1] to realise that a particular claim follows ..

.. not from a big theorem, a moderate proposition, or even a simple lemma. It follows from a change of variable, via integration. That's all.

Well, that's embarrassing, but I suppose I should take the advice I've always given my students:

one stupid mistake now means one less stupid mistake later [2].

All righty. Back to work, then.



[1] .. and no, I'm not going to define that precisely. I am embarrassed for a good reason, you know.

[2] The advice was meant to be reassuring, for when my students do poorly on the quizzes I give them.

Wednesday, December 14, 2005

article post

"Natural Born Mathematicians" - an excerpt:

"We are born with a core sense of cardinal number", says neuropsychologist Brian Butterworth, author of The mathematical brain, reviewed in this issue of Plus. "We understand that sets have a cardinality, that is, that collections have a number associated with them and it doesn't really matter what the members of that set are. Infants, even in the first week of life, notice when the number of things that they're looking at changes.

Read more here. What do you think?

Monday, December 12, 2005

weary, and miles to go ..

Up to one key detail, I think I finally understand what I'm going to talk about tomorrow in Hyperbolic 3-Manifolds class. It took far more hours of turning analytic arguments into geometric reasoning than I'd have liked, and in the end, I was only rarely successful ..

Now all I have to do is write the outline of the talk in such a way so that, sleep-deprived and weary, I have enough information to go on auto-pilot and say what I mean to say. I think I know myself too well; there will be some hitch in the plans, and as a result I won't get any sleep tonight.



On a partially related note, I think my coffee consumption is spiralling out of control. I have half a mind to go "cold turkey" during winter holidays .. that is, after my prelim has come and gone.

It might be a disaster, otherwise. I do want to pass this thing, after all.



On an even more tenuous note, I don't think I understand rectifiability in Rn after all. Sitting in Geometric Measure Theory class today I had trouble following the motivation and intuition, and then I wondered if I adhered too closely to Mattila as the gospel truth.

Too often I forget to be distrustful of books. Illusions shatter: better they do now than next week.

It still feels depressing, though. This was one thing I thought I actually knew, too .. \:



Lately I've forgone the notion of finishing the term strong and well, competent and carefully. It's enough just to finish: do what needs to be done, and deal with the malaise and self-recriminations later.

After all, there will always be self-recriminations. More fuel to the fire, is all.

Sunday, December 11, 2005

quick post: a Mobius Band.

I've been in the office for most of the day, and fool that I was, I brought my laptop with me. As a result, I've been distracted by this Tangled 'Web and have gotten very little done .. and there is so much to do ..

Anyways, the little thing I discovered today: there is an electronica band out there called "Mobius Band."

I'm not kidding; someone took the pun to a new level! [2]

You can listen to a few of their songs via audio-stream on their myspace page. They're not bad, but no news yet about whether there's an umlaut (ö) on their spelling of "Mobius." q:



[1] Having hoped that I would write my entire talk for Hyperbolic 3-Manifolds class in one day, I've now cut my losses and am trying to finish two of five sections, and do the rest after I grade quizzes and team homeworks and when I get home.

Meanwhile, there's the problem of finding time to read about rectifiability of sets in Rn so that I can talk about them competently in 9 days' time (read: my prelim exam).

Then there is that last thing my advisor asked me to prove .. ): You'd figure I'd be smarter than this .. or at least more efficient with time.


[2] I haven't been this startled by a band name since someone told me about "Jedi Mind Tricks," which happens to be a hip-hop rap group!

Saturday, December 10, 2005

a book and its audience.

In the spirit of procrastination, earlier I stopped reading a section from Mattila for a moment, and turned the book over. This line caught my eye:

Essentailly self-contained, this book is suitable for graduate students and researchers in mathematics.

Wow. Then I remembered something else, pulled out Evans-Gariepy, and flipped to their preface. It was as I remembered it.

This book is definitely not for beginners. We explicitly assume our readers are at least fairly conversant with both Lebesgue measure and abstract measure theory. The expository style reflects this expectation. We do not offer lengthy heuristics or motivation, but as compensation have tried to present all the technicalities of the proofs: "God is in the details."

It feels like a pleasant difference from the usual disclaimer of many books I've encountered, which would read something like

This book is suitable for beginning graduate students and advanced undergraduates.

I suppose nothing is wrong with such an assessment, except that it damages my ego and my self-esteem. It's the same philosophy with the terms "clear" or "obvious" or "easy," relative to proofs and arguments you see.

If I didn't know any better, I could swear that either (1) many authors underestimate the content of their books, or (2) I'm not as intelligent as I would like to be. Being that my "beginning" years of graduate school are fast ending, the bar of expectation has been raised and it is now a good question whether I can vault over that bar.



On a more objective and less self-deprecating note, I wonder how authors determine the "level" of the books they write.

Many books do arise out of lectures from a class taught to graduate students, which explains why the reading level is set for graduate students. The exposition is meant to be organized, and the key results are understood to a level which omits most of the mess from thought experiments and previous endeavors.

In short, many good books are polished. It's easy to read such books and forget that the viewpoint is retrospective, and a generation ago, the path of argument could have been rather confusing and unclear. Unless the choice is deliberate, most books and lecturers may never show you the "wrong paths" that were taken to argue this or that, and depending on relevance and clarity, it's purely a judgment call.

It's like how some authors rewrite history to suit a particular image or ideology: to oversimplify, nations win wars because our side has "just cause" and our enemies are "evil."

In some mathematics books, it feels more like one timeline of reality or possibility is presented: the prevailing and practical one, up to some consensus. It's not so much rewriting history, because there could be alternate realities; we just choose not to study them.
But enough of philosophy. I'm curious to hear what you guys think: how do and how should authors label the "level" of books they write? Should they account for the factors below, or are there missing factors?
  • How specialised the topics are, or how obscure a field.
  • The treatment of the topics, and amount of technical details.
  • Better sales, varying by the size of the audience
Or could it be there is no good scale?

Friday, December 09, 2005

"You'll do fine."

O, teach me how I should forget to think!
Romeo, Romeo & Juliet

Everyone's been telling me not to worry so much about my upcoming prelim exam and that I'll do fine. Even my advisor said so during our meeting today ..

.. so I suppose it can't hurt to listen. q:


Thinking about it, worry doesn't really help very much. It didn't help, for example, when I was taking and re-taking my Analysis Quals during my first year.

To this day I think I was overworried about doing well, and after doing poorly once, I plunged into a vicious cycle of worry and fret and failure. I only passed Analysis when I didn't study too hard for it, and at that time I was focusing on the Topology Qual.

It could have been that measure theory and complex analysis had to gestate in my mind for that long (one year) before I could fully process that information and implement it in an exam. Nonetheless, the worry was of very little help.

Maybe my concern is complacency and laziness. One could say that I am a high-inertial person: if there is no push for me to do or act towards something, then I'll likely not.

I can think of countless times when I posited goals or plans one day and forget them after a day or a week. Completely ineffective and intolerable, that! What good is my word if it cannot be trusted, or my promises if they never bear fruit?

If you think me an idealist, I'd have to disagree. I'm not trying to be a moral person, but a consistent one.

I suppose this is really my attempt at making sure that I am not myself, or rather, not the part of myself that is fallible and inconstant. Said otherwise, perhaps I'm trying not to let my humanity get in the way of progress.

As for what that means, I don't know and I don't think I want to know, either. Let me worry simply about doing well on this prelim, and leave the philosophy to January when I have time to ponder it. \:

Sunday, December 04, 2005

on studying, remembering, and what to remember.

After a studying session for my prelim, I've observed a few things:
  1. I'm making slow but steady progress. This might be illusion or delusion, but it honestly feels like I'm gaining some degree of intuition from the arguments that I've been reading. I can only hope that my memory serves and I don't forget it all, but perhaps that can be corrected by how I've been processing this information.

  2. I've given up on remembering the technical lemmas which are used to prove big theorems. Instead, I've been trying to remember the big theorems ..

    (.. which aren't too hard; they are big, after all.)

    and what is the "right way" to prove them; this also allows me to build intuition on how to work with certain concepts and properties. If I remember the "right way," then I'll have to collide into what smaller claims I'll need to prove it, and those are the very technical lemmas that I would fumbled to remember. It's the difference between a priori and a posteriori knowledge, I suppose.

    Perhaps I am slow-witted and should have thought of this long ago. I've learned it now, though, and it suggests that I might not be a wholly hopeless cause .. (:
In particular, I never realised how cool the Area Formula for Lipschitz functions is. From the approach in the Evans-Gariepy text, the idea is to dissect the domain into Borel subsets, where the given Lipschitz function, when restricted to one of these sets, is "essentially" bi-Lipschitz with controlled constant.

One proceeds to argue that the Lipschitz images are comparable in measure to images from linear transformations (relative to these Borel subsets), and then we apply this linearity with impunity and without apology.

Fine ideas, if I've quoted them correctly (I've been wrong before), and it seems reasonably general (though the use of linearity worries me).

What strikes me most is the "optimism" or "courage" behind this type of argument: subdividing the domain as necessary, Lipschitz functions behave like bi-Lipschitz functions or linear transformations in the sense that we want them to. How does one learn such bravado, and learn to get away with it?

So let me say that I'm getting better at remembering, if only because I'm learning to reduce to the essentials. This could mean that I'm becoming a better student, but that doesn't mean that I'm becoming a better mathematician.

Oh well. It's something, at least.

Wednesday, November 30, 2005

a night ago

This is from Monday night, while I was still in the office.

I think I had 3-4 cups of coffee today, and as of now (9:30 pm) I'm on the verge of sleep, teetering on my office chair and blinking at too frequent of a rate for a waking person. I could swear that earlier Matti1a was easier to understand, and now I'm debating whether a nap is reasonable, or maybe I should crash early tonight and get a good start at work tomorrow morning.

But crashing "now" means stumbling the 20-minute walk home, which isn't very inviting on a damp, rainy night like this one. I might even wake up on the way home, which would have made the whole endeavor pointless ..

The work's piling up, though I feel I've made a tactile [1] dent in the pile today after a large dose of reading and thinking and scratching out examples. If I could get tangent measures straight in my head, find time to understand the area and coarea formulas, and remember all the Sobolev stuff, then a mid- to late December prelim might actually work out.

One should put emphasis on the word "might." There are several matters to attend to, this December: I'm meant to give a talk in each of my two classes this term, and damn it -- I should have worked on them earlier. Now there's hell to pay and little to no time to negotiate the concepts needed to give a 50-minute exposition on matters that I should know better than I actually will.

Then there's final exam grading for Calc II, on 15 December. It's a one-day commitment, but the last thing I want to consider at this point.

Funny how it seems: after all these years, all I really want is to sit down and be left alone to do one thing at a time. I don't mind giving a talk or two, and I don't mind reading and preparing for an oral exam .. but just not all at once. It feels like too much, the end goal too far, and the light at the end of the tunnel too bright. I can't seem to do a damned thing right anymore.

There seems like so much to do and I don't even know where to begin. This is too much like last fall where teaching took half my life away and three courses' worth of problem sets took away the other half. I was too busy being a GSI and and student to look for or even think about an advisor. Now I feel like I'm too busy being a GSI and a researcher in training to be any sort of student.

You'd figure that with the time constraints at hand, I'd find better uses of my time than e-ranting over the internet. Funny how that goes, too: maybe it's a means of escape from the cold, unrelenting reality that is my non-life and the work that won't ever diminish in bulk.

Well. At least rectifiability seems comprehensible. I'll hit a big theorem tonight, and I might even be able to make sense of it. If my plan isn't fully shot to pieces then I'll read about Ah1f0rs' Measure Conjecture, hyperbolically harmonic functions, and other matters of geometry. That way, I might be able to settle down with my Hyperbolic Manifolds prof and have a decent conversation about what I'm supposed to talk about next week.

Getting close to 10 pm. Damnit. I wonder if my kids will notice that I didn't do any preparations for their Calc II class tomorrow. Oh well: it's separation of variables from ODE, they've done the reading, and they need the practice. I'll let them loose on problems and give my suggestions and point out their errors. There's little else I can do for them, anyways.

Epilogue: the plan did get shot to pieces. I never did read about Ah1f0rs' Conjecture until today, but that seems more comprehensible and less tenuous now.



[1] I would have said tangible, but that's not precise enough. This is the sort of dent that you can't see with the naked eye, but you can feel if you have sensitive nerves on your fingertips.

gambling on a seminar, and an abuse of notation.

After long last (and showing up to an empty lecture hall last week), today was my big chance to learn about K-theory without committing any real effort in doing so.

(See my previous post, here.)

It was a fine talk, and despite being a colloquium for the "non-expert," there was something in it for everyone: the topologists and the geometers, even the analysts, and of course, the algebraists and algebraic geometers ..

(There is a difference, so I've been told.)

My only regret is that I can't remember the exact definitions, but K-theory has a similar flavor to homology and cohomology theories. In fact, it seems to me a little simpler to digest, even though somewhere one requires definitions from algebraic topology and straight algebra

(At some point one requires the notion of the direct limit topology, which never sat well with me from when I learned about rings and modules.)

However, the analyst in me felt vindicated at the mention of Banach and C* algebras. q:

At any rate, this colloquium was for me a success: I actually learned about some of the words in the title and abstract .. though I might have stacked the deck, this time around. After all, the title was: "What is K-Theory and What is it Good for?"



The rest of the day went reasonably well. In particular, my prelim reading has gone wonderfully over the last two days: the lack of interruptions and commitments have helped enormously. I wish I didn't have that many positive things to say from my advisor being away on business, but fortunately or unfortunately I'm making the most of the time that would otherwise be spent on research and attending classes.

I've browsed through all of Chapter 16 of Mattila, thought about the proofs of the theorems, and I must say: rectifiability via approximate tangent planes and Lipschitz images seems more sensible to me than tangent measures and Marstrand's Theorem.



Today I barely prepared for teaching, but in some sense I wanted it that way. Ours was a computational class today, and if I made it too polished then it would look too easy, which is no good when I'm trying to convince my kids to be careful and to practice more problems.

My Calc II kids have now learned the "Separation of Variables" technique for ODE, and after a hitch or two concerning constants of integration (one or two?) and where to apply the initial condition, I gave my derisive opinion on an abuse of notation.

You can probably predict which notational issue I mean. Consider the computation

dy/dx = -x / y  →→  y dy = -x dx  →→  ∫ y dy = - ∫ x dx

Now what the hell does 'dx' mean, without a notion of integration? Nothing: it means nothing. The symbol Δx would mean a small but finite length, but dx? Maybe infinitesmals exist in some non-standard logical system, but they are not for common use!

I can accept the fact that it's a shorthand for applying the Chain Rule, but that doesn't mean I have to like it. Maybe it's my inner analyst or my unconscious desperately demanding some sense of rigor and decorum in an otherwise nonrigorous and watered-down course.

At any rate, the kids seemed to take to this ODE technique. I suppose it's the best I can hope for.

Saturday, November 26, 2005

highlights from a work session.

I've been posting quite a bit, lately. Only time will tell whether the amount of "interesting stuff" I write is an invariant or not; if so, it would probably mean that the more I write, the less interesting it gets.

Anyways, let's go with the "more matter with less art" [1] approach to blog-posting.

mathematical high point of my day: Drawing the picture and realising what it all means.

This is a lesson which, for me, is often forgotten and often relearned.

mathematical low point of my day: Coming up with an example but confused on how to settle it. The cause of this difficulty lies either in my memory or my slow-wittedness, and neither is very appealing to consider.

Abstraction is wonderful: I imagine ideal objects drawn with clean, straight lines. However, one must remember that abstraction is a tool of generalization, and the motivation lies in concrete examples. I suppose that concreteness is also nice, but nice in the sense of mud soccer or preparing raw chicken before cooking and serving it.

personal observation: I'm no good at multi-tasking anymore. I don't see how the rest of the world does it (well).

Tell me to finish two things and I will be confused and accomplish nothing;

Tell me to finish one thing and up to human and circumstantial limitations, I will do it. Then tell me to do the second thing and I'll do it in the same way.

How is it possible that I become so confused, so easily? Has this much schooling changed me and made me incapable of non-iterative processes? Do I over-think everything now?

This just doesn't bode well .. \:

[1] In Shakespeare's "Hamlet" there is a scene where we first encounter Claudius, Gertrude, and Polonius at once. Suffice to say that Polonius is verbose and fails to get to the point, and Gertrude admonishes him for this.

GERTRUDE: More matter, with less art.

POLONIUS: Madam, I swear I use no art at all.
That he is mad, 'tis true: 'tis true 'tis pity;
And pity 'tis 'tis true: a foolish figure;
But farewell it, for I will use no art.


I wonder if that's how my advisor feels about me, during our meetings. q:

Friday, November 25, 2005

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:

Tuesday, November 22, 2005

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?

Monday, November 21, 2005

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.

Thursday, November 17, 2005

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? (;

Tuesday, November 15, 2005

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.

Thursday, November 10, 2005

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 → Rm with Lip(f) = 1, then there are B1, ..., BN in Q, N ≥ N(&epsilon), such that
Hn[f(Q \ (B1 u ... u BN)] < ε
and each f|Bi is bi-Lipschitz with Lip((f|Bi)-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 Nε where for all 1-Lipschitz functions f mapping Q into Rm, there are balls {Bi : i = 1 to N} with N < Nε, where Hn(f(Q \ Ui Bi)) < ε and f|Bi is Nε-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 Hm instead.

Anyways, back to work.

Saturday, November 05, 2005

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.

Tuesday, November 01, 2005

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 ..

Monday, October 31, 2005

teaching today, and on the nature of "mathgradness"

So I finally got around to it, today: I decided to shut up.
Instead of a tiresome lecture about the finer points of sequences and series, I made a worksheet of problems, some from the book and some I wrote myself, and suggested that the students work together and discuss them at their tables.

If I hadn't mentioned it before, the classroom is arranged so that there are square tables. Four students sit at one table, and each takes one side. As a result, two students per table almost always have their backs to me, during class.

You have no idea how much that bothers me .. \:

But it worked out: the students were still rather quiet, but they discussed the problems, and I wandered around and lent a hand when a team had trouble with a particular concept or computation.

Working this way almost makes me a disbeliever of large-scale teaching. I know it bothers me when I have to lecture in front of 30+ students. It's not because of nervousness, but because I have only a small sense of whether they actually understand what I say.

It's easy for me to feel that the lecture is boring and pointless and nobody's getting anything out of my careful method of explaining this or that. If that happens then it's a waste of everyone's time.

What's the point of that?

This also raises another question: how do diligent, thoughtful students become that way? I like to think that I'm not that lazy and confused student in the world ..

(and at the least, people do say that I take good notes)

.. and how do students become maths students? I've seen enough of my friends and familiars to realize there is a rich variety and diversity in personality and manner when it comes to one's approach to academics ..

.. yet there seems an intangible invariant: "mathgradness," as Jo so clearly put. What is this special quality, and what drives us to do maths and keep others from experiencing this coolness?

I'm not sure if that was a rhetorical question. It needn't be. Any takers?

Let me not go into nature and nurture arguments, and I'm happy to know why I like mathematics. Some days I can actually explain why, too. q:

Thursday, October 27, 2005

Thoughts before the Brown Jug ..

(.. and after the weekly meeting with my advisor)

It feels good to be back "in the game." Granted, I'm still doing more reading than anything else. I was suggested a new paper to read, but on a wholly different topic this time: last week it was Hajlasz, Sobolev spaces, and approximations on manifolds and metric spaces, and this week it's Milnor, smooth functions, and manifold theory.

I've learned a few lessons already:
  • Fibre bundles are pretty cool, once you hear about them from someone who understands them.

  • I will never again underestimate how hard the smooth category is. Before I was mistaken: I thought once you had a smooth function, then life was relatively easy.

    Suffice to say, that is far from the truth!
It fascinates me, how long it takes to read mathematics and to understand our lessons well. On the other hand, it makes me wonder how long, on average, it takes for people to write a good article!

So long a time spent, writing something, and so equally long a time spent reading the same something! I could make an analogy to Achilles and the tortoise, but just mentioning that probably gives my meaning already.

One last thought about reading and work: the list piles up and up. Besides Milnor, there is material to read about:
  • the Ahlfors Measure Conjecture, for a student talk for my Hyperbolic manifolds class;

  • Gromov-Hausdorff convergence of metric spaces, for a talk I promised the folks at Student Geometry-Topology Seminar;

  • and certainly not least, my prelim reading!

    The books by Evans-Gariepy and by Mattila are sitting in my bookbag and if I don't start now, I won't know the results and details well enough to talk competently about them!
Never a dull moment. But it is Thursday, and close to Brown Jug time. The reading can wait an hour or two. q:

Tuesday, October 25, 2005

"wait .. where am i?"

I think I misrepresented myself, or gave an inaccurate depiction of what I was actually doing (see last post).

It is true that I am reading a paper of J. Milnor's from 1956 or so, but only part of it [1]; more accurately, I'm reading enough of it to determine whether a certain property of spheres [2] is false in the C1-smooth setting; it is known to be false in the C-smooth setting.

On that note, it's actually been a bit of fun. I'm reviewing some differential topology that I hadn't thought about in ages, and am finally applying some facts that I thought would be useful only for passing my Topology qual.
I've struck upon a regular bit of confusion, though: perhaps I'm bad with the notation, but it doesn't seem easy to tell where a particular function is 'living' (i.e. its domain, and whether I am looking in the chart or on the manifold ..). Hence the title.

Of course, the fun might end soon. There's a sizable section or two which requires some knowledge of homology and cohomology. As you can imagine, algebraic topology isn't my strong point .. \:

Oh well. Que sera, sera, and it's back to work for me!


[1] In particular, it's "On Manifolds Homeomorphic to the 7-Sphere" from the Sept 1956 Annals of Mathematics.

[2] and yes, I am deliberately being vague. q:

Monday, October 24, 2005

Disparate bits ..

I finally finished a write-up of my work from the last month or so. I think it took a year or so off my life. Maybe from now on I should LaTeX the results as I prove them, and save myself those nights of brain-disintegrating toil.

I wonder if my advisor will read it, at some point. Maybe I should have done something else, these last few weekends.



Today was the first day in which nobody came to my Office Hours. It was quiet and pleasant and I got a bit of work done, but strangely enough, I actually missed the student questions a little.

I must be getting soft.



There are some results in mathematics that everyone's heard of and which you never expect to worry about. Take J. Milnor's result on differentiable structures on the sphere; it serves as a moral to those who study manifolds that smooth structures can be very, very strange and eerie algebraic topological invariants may arise out of the ether.

But does anyone ever look it up and read the paper? I never thought I would. But guess what: I'm reading it for Thursday, as a suggestion of my advisor.

I'm not sure how I feel about that. Milnor, eh?



Well, once more into the breach, my friends. Until later.

Thursday, October 20, 2005

the After-Math of my Talk.

Pardon the pun, of course.

So today it was my turn to talk at AnSS (Analysis Study Seminar, where we talk about research that we read, not what we've done) and it went tolerably.

The strange thing was that I wasn't too nervous: having become a regular fixture to that seminar, I did understand that I was amongst friends and nobody was out to get me .. at least then and there. Let me not vouch for the rest of time and space. q:

Instead, I think I was too cavalier, which led me into trouble. Once I nearly gave an incorrect proof, and other times I omitted key details of some nontrivial depth. It was personally embarrassing, but if others picked up on that, they pretended not to notice.

Oh well. It's over .. at last. Now I can get back to work writing up my results, and looking up this smoothness extension stuff that my advisor recommended .. and there is always the books to read for my December prelim, and more pending, the typed-up list of topics!

Who'd have thought that I'd be so happy to return to the daily grind? q:

Sunday, October 09, 2005

grading frustration; time dwindles but "Things to Do" does not.

I finished preparing for Monday and Tuesday lectures already, but to my discredit they are exam review days. Maybe my class should have more exams, so I don't have to prepare as much.

Nah. Too much grading and too often. Nobody would be happy, then.

Speaking of which, grading homework went all right, though I could swear that reading typed non-LaTeX math is hurting my eyes. I've told my students not to type their computations over and over again; it's hard to read and typing lines of calculations wastes their time and mine.

But do they listen? Oh no.

It is like reading the output from a T1-83 calculator, line by line, and it is quickly pissing me off. If my vision gets worse (and I can't afford to have it grow worse) then I swear I'm going to kill somebody.

Never mind. Moving on ..



Perhaps it was an unwise idea to meet my advisor on Tuesday. It's only a difference of two days (as you might remember, Thursdays are our usual meeting days) but the list of "Things to Do" isn't shrinking very quickly and I can't seem to get all of this done:
.. writing up my results,

.. reading parts of papers which I've mentioned before,

.. looking up some semi-classical stuff
    (smooth extensions of maps on spheres)

.. looking ahead to see what I should prove next,

.. desperately looking for time I can spend on prelim reading ..
Argh. The work is going well, but there seems so much of it. It's like being in a relationship with a jealous, high-maintenance partner. She will take all your time away and you have none to spend with friends or family. You may relent to her wishes, but she'll only demand more and more from you.

Huh. That sounds familiar: maths already do that to my life. Better put, maths dominate my existence.

I wouldn't call it a "life," or at the very least, it doesn't seem much of one. When one starts to divide the day into work time and non-work time, then something seems to have gone horribly wrong. For instance, you may start to believe that you are better suited as a machine. Thinking about it now, I'm amazed that I even wrote that, though I don't doubt the conclusion.



Oh well. I'll finish what I can. There are plenty of weeks before end of term, and plenty of terms before I start panicking about defending and graduating. Plenty of time for work.

Thursday, October 06, 2005

another paradoxical Thursday night

Thursday evenings are a periodic paradox in my academic life.

It is the time right after I meet with my thesis advisor, so all of last week's work is done and productivity reaches an absolute low.

More colloquially, I've lived to `fight another day,' or rather, another week. My advisor is not a mean guy and in point of fact, is quite an understanding man. However, there's something about a regular meeting and holding oneself accountable for adequate work: it does not sit well with me.

But I manage. Wednesday nights are often devoid of sleep. Thursdays are when my coffee addiction is at its worst, and when my food intake would lose the approval of good mothers everywhere. More often than not, it's Thursdays when I most often wear my Green Lantern T-shirt [1].

During today's meeting, the worst thing said was that 'I might be too obsessed with charts," which to some degree is true. Surely there are worse fates.



On the other hand, Thursdays are quite liberating. The bulk of the work is done for one week's time, and afterwards is free tiem. It means that I'm free to work leisurely on whatever matters are most fitting.

Say I've been meaning to leaf through Vaisala's book on QC Mappings to review my studies, or browse a preprint that a friend emailed me last week. Maybe I thought about an idea to a problem earlier in the week, and between teaching, classes, and thesis work, I haven't had time to sit and jot out the details.

I can now do any one of those things. For one night I am puissant and capable of exercising my heart's content. It needn't even be mathematical: I can linger around at Borders Books & Music and read comic books all night .. or hopefully some finer literature than that .. but the freedom is there.

Then there is the paradoxical feeling of enthusiasm. The meeting went well, I have a new agenda, and there are new things to do that haven't had a chance to lose their novelty yet. I think of how much more I can get done if I start right away.

This betrays a consistent illusion of mine, I suppose: there's always that glimmer of hope that I've done enough work early in the week so that I can coast a few days before my weekly meeting .. not being lazy, mind you, but work on 'icing on the cake,' that is, those fine little details that add to the aesthetic character of mathematics, but not its content.

Of course, that never happens, but a boy can dream, right? Between being sleepy from lack of sleep and dreaming about a possible future, I wonder where I stand on these strange Thursday nights.



[1] Green Lantern is a super-hero in the DC Comics universe, and you can see a version of him on the cartoon series "Justice League." He wields a Power Ring, one of the most powerful weapons in the galaxy. With it, Green Lantern can create any weapon or object which comes to mind; it materialises and can be used immediately, and the scope is limited only by imagination and willpower.

I wear my Green Lantern shirt on days when I'm in terrible shape and am operating under sheer willpower. Say I've had little/no sleep and food, and an absurd amount of work to do: that's a GL shirt day.

Wednesday, October 05, 2005

Reading, reading, reading ..

Time grows limited tonight. Since there is too much to read carefully and understand in one sitting, I might as well tell you what I tried to read .. or will get to, if I ever plan well enough and have enough time to read it all.

But anyways, here is the list.
"Boundary Regularity and the Dirichlet Problem for Harmonic Maps" by R. Schoen and K. Uhlenbeck.

.. as suggested by my advisor. I'm up next for talking at AnSS (Analysis Study Seminar for those @ UM) and I need their results on smooth approximations of Sobolev mappings between smooth manifolds.

Right now I'm encountering trouble understanding how degree, initially a topological notion, applies to Sobolev functions and their Jacobians, which are analytic notions.

"Degree and Sobolev Spaces" by H. Brezis, Y. Li, P. Mironescu, and L. Nirenberg.

.. as a possible means to understand the notion of degree and how it applies to the Schoen-Uhlenbeck counterexample.

Edit (as of October 6th, 1:30 pm): it doesn't help with Schoen and Uhlenbeck. This BLMN paper is quite interesting, but just not what I'm looking for. Maybe I'll have time to read it later in life

"Approximation of Metric Space Valued Sobolev Mappings: Four Counterexamples and a Theorem" by P. Hajlasz.

.. as added material for the aforementioned talk @ ANSS .. next week? The clock is indeed running, but happily Hajlasz writes very well and clearly.

Correction (as of October 6th, 8 pm): I'll be talking @ AnSS in two (2) weeks's time, so now I can read this Hajlasz paper in a more leisurely manner, which is nice. Somehow one never enjoys reading if there is an enforced deadline .. \:

Some typed lecture notes by R. Canary, and to come, "Fundamental Polyhedrons and limit sets of Kleinian groups" by L. Ahlfors.

.. which I've chosen as my student talk for Prof. Canary's hyperbolic 3-manifolds course. Getting away with an analysis talk in a topology seminar is a chance not to be missed! For shame: I even get to talk about harmonic functions on hyperbolic manifolds .. q:

Now if only I can get to the details at some point ..

"Collapsed Riemannian Manifolds with Bounded Sectional Curvature" by X. Rong, and as a reference, parts from A Course on Metric Geometry by D. Burago, Y. Burago, and S. Ivanov

.. as a means to an end: I promised a few fellow students I'd give a joint talk at SGTS (Student Geometry & Topology Seminar) with my cheerful, tall Scottish friend, John. Happily he'll know more about it and I can have him field all the hard questions. q:

Measure Theory and Fine Properties of Functions by L.C. Evans and R. Gariepy, as well as the Mattila book: the exact name escapes me at the moment.

.. as the agreed reading for my Oral Prelim Exam this mid-December. Yikes!
Even when I break up my time for these reading tasks, somehow the whole of my time seems far less than the sum of its parts, and that sum isn't that much to speak of, anyways.

How did I ever get anything done at all, before this point?

Maybe I never did. I don't feel like I know anything anyways, so perhaps I am Achilles in Xeno's paradox, racing just to reach a finite point but never getting there .. \:

Tuesday, September 27, 2005

teaching now and learning, long ago.

I've said this before to my friends, and it doesn't hurt to repeat it here: I'm glad I'm not a student in my own class. I wouldn't be able to stand myself.

Today I must have been unbearable as an instructor. As I went over an example today, even I found it boring and tiresome. That sucks. Calculus mightn't be the most exciting subject in all of academia, but it shouldn't be that boring.

One of these days I'll remember to keep my lectures short and snappy, and send the kids off to work on problems .. maybe even a worksheet of my own design. Better to keep them active and thinking, and doesn't the UM Teaching Staff always say: "Let the students discover the facts for themselves?"

At any rate, I seem to have lost the ability to ask questions that my students are willing to answer. Maybe they don't know the answers because they didn't do the reading, or maybe I'm simply not asking the right questions or questions that make any sense. Maybe they're just confused.

At any rate, class hasn't been going very well, lately.



I'm trying to remember when I first learned differentiation and integration, and it must have been about .. 6 or 7 years ago, when I was about 17 and still in high school.

Christ. I was that young once, wasn't I?

I barely remember learning it, but it didn't seem that hard at the time. I do remember being scared to death of my Calculus teacher, Mr. Bevelander.

Yet at the same time I always felt at ease asking questions. It was a class of 8-10 students (how that happened, I have no idea) and every time someone had a question, we'd ask. If it didn't make any sense after a moment, our teacher would toss us a nub of chalk, and whoever had asked the question would walk over to the board and elaborate on the matter.

It helped tremendously in the cases that one of us did the problem in a completely different way than Bevelander had imagined. In fact, I wouldn't even say that he taught; rather, he put us to work on problems and critiqued our work. It went in a somewhat formulaic manner, but then again, so did plenty of other things in high school.

I remember rather liking chalkboards after that, and it helped to think right in front of it, adding another line of computation as I wondered where to go next or whether this approach was any good or not.



At any rate, teaching is over until Thursday. I'm not complaining.

Research is going well. I figured out the mystery behind that weird-@$$ exponent (I had mentioned it before, here), and it was a mistake -- just not the one I expected. The proof is still the same and much progress remains, but at least it's slightly saner.

Sunday, September 25, 2005

on hope, and the learning process.

Grading upsets me.

I think it's because I am a closet perfectionist, despite my occasional, outwardly relaxed appearance. Details do matter and so does rigor, otherwise I'd be a student philosopher and not a student mathematician. Then again, there are plenty of non-rigorous bits of mathematics done, each and every day. But we're supposed to try for correctness and accurate, logical thinking.

I keep reminding myself that most of my students are about 18 years old, and about 6 years younger than I am. Most of them have just arrived from high school, where mathematics may have been glorified computations, excessive formulae, and few insightful ideas.

They may not have been asked to justify their answers in the way we, as Calculus instructors at UM Mathematics, demand them to justify.

It may be that, aside from logical reasoning when writing essays, they've never been held accountable for this sort of thing. They may not be used to thinking of a generic case, or searching for a pathological situation or worst-case scenario in order to compare with what must "always" be true.

That's a lot of 'may's and it tries my patience. It only convinces me that perhaps I'm not well suited to the life of an educator. I have enough trouble with my own shortcomings, so how do I deal with those of others?

It's not easy to find hope in this line of questioning.



I went to a dinner party last night and to be simplistic about it, the other guests were bourgeoisie-in-training: law students, MBA students, and an Urban Planning and PolySci Ph.D. student each.

There's nothing wrong with these people, mind you, because the world has to keep running. Money is an important commodity to everyone, and I say these people are being exceedingly realistic. I just choose not to be one of them, because the world and its people frustrates me too easily and quickly for my taste; more importantly, I don't have a mind for money.

One of the MBA fellows half-asked/half-told me, "You must be really smart, to study mathematics!"

I replied, half-jokingly, "I wouldn''t say that. Nobody's actually good at mathematics. It's just that mathematicians just never give up. We're just stubborn that way, and we just try and try."

(It's not entirely accurate if you believe that some people are either 'naturally' good at maths or have learned to be "good," but in light of how frustrated mathematicians can be as a result of their exertions, there must be a speck of truth in this.)



Maybe that's it, then. Maybe it's just enough to try, and by trying, to reach some modicum of improvement. It's evident that my students are trying, either for a good grade or for a good understanding of calculus. The end doesn't matter so much, but they are trying.

That has to count for something: it did when I was such a student and my teachers had to grade my shoddy work.

Hope always lies in possibility. Perhaps my students will make their mistakes now and when their exam comes, they won't make them again. Perhaps they will learn more than simply how to do well on their exams, and appreciate a little mathematics for its own sake, which belies appreciating the fundamental logic and reasoning for its own sake. Perhaps it's worth the frustration of grading.

Those are still a lot of 'perhaps,' so here's to hope. \:

Friday, September 23, 2005

not so much 'new' as 'old' ..

I dug this out of my desk, and it was written about two weeks ago, when the Fall Term just started and I had a free moment to jot down a few stray thoughts.

For the record, this was the Friday that I held an Office Hour instead of a Math Lab tutoring hour, had tempura at Totoro, and read comic books at Borders Books & Music.




It's been a rather pleasant evening. I didn't do very much today -- unless you count fielding a volley of questions from my eager students at Office Hours. My 1-credit Physics seminar didn't meet today, but I only found out after sitting in a half-empty classroom for fifteen minutes. It wasn't a total waste of time, though.

Other students -- these being Physics grad students -- were also in the room, chatting idly, and I caught more than my share of their Department's gossip. I suppose the scene is much as a Mathematics class would be to the non-mathematics student: academically accessible but ultimately socially insular.

Let me not seek any general meaning from this.



Sitting in my advisor's class reminds me of how different we analysis students are [1], or perhaps how different I am from everyone else. For instance, I seem incapable of saying much during lectures, if I say or contribute anything at all. Somewhere along the way of becoming a maths student I became obsessed with good notetaking.

I use different pen colors and flow tips and everything. I write down as much of the discussion as possible .. what's on the board, which comments the lecturer made, and what comments or questions the audience brings up, and what answers ensue.

I might as well be archiving the experience of that lecture, rather than just its topical contents. It feels like I switch to a Record mode, and while I'm in this mode it seems I can do little else, which is problematic when in my advisor's class. He likes class involvement, you see, and asks plenty of non-rhetorical questions.

He teaches quite well, in fact: motivation, examples, ideas behind the proof .. all the good stuff. It's too bad that my notetaking style is so inconducive to his lecturing style, but happily, this is where the differences between we student analysts can prove valuable.

One of my student peers is really good with questions; some of you know him as Kevin, guardian of the Quasi-World. He takes on questions with detail and delight, and he asks many of his own. My former flatmate Jose' is quick to pick up on clues and is quite clever; he has remarkable intuition. There is Marie, my academic sib [2], who has the gift of clarity: being able to ask the right questions, she sorts out the occasional confusion and is another source of intuition.

Me? I just write it all down.

Maybe in the process of college and the early years of graduate school, I've neglected the ability to process information quickly and to use it effectively. New ideas and concepts take me a great deal of time to work into my thinking. Worse yet, my short-term memory seems to be diminishing and diminishing fast, and I could swear that the time I spend trying ideas and sorting out details takes far longer than I'd like ..

.. say an o(n2) runtime, whatever n is. q:



[1] .. though I prefer the term: "student analyst." It answers two questions in immediate succession:

"What sort of mathematics do you study?"
"What's your academic status?"


[2] In case it wasn't clear, an academic sibling is another student of your thesis advisor. M and I happen to be in the same year in grad school, so more often than not I feel like the evil twin. q:

Wednesday, September 21, 2005

strung out on coffee and work.

I've forgotten one of the unsaid rules of doing mathematics:

Never get emotionally involved with your work.

I've thought about one particular problem for far too long, and now there is too much momentum to stop .. and when I say a problem, I mean a problem. It's a mathematical statement, pure and simple.

I can't tell whether it's true or false.

When I try to prove it, I run into technical difficulties. If you know a little metric topology, the issue involves a lack of control on the diameter of a particular sequence of metric balls. If this is ever settled, then the proof is done.

I can't construct the right counterexample, either. The first things one tries, they don't work. Radial functions f(x) = x * F(|x|)/|x| don't work when F is a power function, and more complicated functions are too complicated and inconclusive.



I still think it's true.

I'm close .. but my estimates are crap, and there's still no proof.

This may as well be a matter of faith. The statement could be false, and maybe I'm missing something.

F*ck. I should leave this alone.



This statement came to mind when I was trying to prove something else, and I don't need this result to continue my research. I should be doing something more productive .. say, working on my Calc II lesson plans or preparing a grading sheet for another week's worth of Team Homeworks ..

.. nnnaaaargh. Gah. Bleah.

I don't want to think about that .. at least not now.



I should get back to work. Tomorrow's another meeting with my advisor, and there is plenty I want to talk about .. that is, if I ever have it ready.

The other aspects of my life are unimportant right now. They may as well be nonexistent, and if I ever stop again and think too hard about my life, I might become depressed again. Then I'd never get any work done.

Soldier on, then. All I need are rock music, calories and fluids, good paper and pens, and a good dose of luck. That's all I ask.

Saturday, September 17, 2005

a rant that turned into a post.

Someone posted on LJ: Mathematics Community about the "Reformed" vs "Traditional" Calculus class formats. I ended up writing this opinion about this previous comment by another community member. My response is below.

If you're not going into higher-level, proof-based mathematics courses, then it can be argued that it isn't necessary to know the rigorous definition of a limit.

An interesting point. If you choose to view courses in a basic Calculus sequence (Calc I, II, possibly III and Diff Eq) as general-ed requirements, and if you will never venture into higher-level pure mathematics (where logical arguments and geometric ideas are more important than computational methods) then perhaps there is little to no need for such rigor as the ε-δ definition of a limit.

What bothers me is whether we demand reductions in other general-ed courses.
Take a freshman writing class - if you're not going to be a Lit or humanities major, then does it mean that you don't have to analyze your readings as deeply? Does it mean that your essays don't have to be as circumspect and readable?

Consider a first course in programming, say in C or in JAVA - does it mean that you don't have to learn to comment your code?
The point of taking courses in certain subjects is to learn what the ideas are, how they are learned, how we can contribute more of these ideas, and if ones reaches a certain point, whether we can improve these approaches to further the area of study.

The ε-δ definition isn't just a description of a limit; that's what a limit is in mathematics. It's not a pretty graph or a table of values, but such methods help us to understand the nature of a limit. A math teacher should emphasize that: if you choose to learn mathematics, then as an important mathematical idea, you should learn this.

If all you need is how to use a limit in practical, day to day life, such as understanding instantaneous velocity or approximating marginal revenue, then that's fine. You don't need to learn it, but if you take a Traditional Calculus course, don't expect a math teacher to cater exclusively to your non-mathematical needs. Its purpose is to teach mathematics and to emphasize mathematical thinking.

Maybe what is needed is "Calculus I for Scientists and Engineers" or "Calculus I for Economists and Sociologists" or "Calculus I for Mathematicians," and this notion of Reformed Calculus is a step towards that direction. Maybe it is "watered down," but if planned correctly, a Reformed Calculus class teaches you what you need in order to work in your own field. It doesn't teach you what mathematics is like, because that is not its purpose.

I say: if you choose to allow Reformed Calculus classes, then you should also keep a few alternative Traditional Calculus classes. It's a specialization of needs: students are different and their studies demand different approaches. The coursework should reflect that.

Wednesday, September 14, 2005

embracing the mundane ..

EDIT (AS OF 15 SEPT 2005, 5:30 PM): About the "More Mathematical" part: never mind. I made a mistake while trying to find a mistake, so there are no old mistakes, only new ones.

Despite my worries, which were silly and without much point, the advisor meeting went reasonably well. I get the feeling I should write up some results, and prove new things to further the theory.


Aside from my last post (and before that, an article post, which is negligible), I haven't posted very regularly on this weblog. More than anything, my posts seem to occur in blitzes, and I have a hypothesis on why: it's because my life, even my mathematical one [1], is hardly exciting. Perhaps it takes about a fortnight before something noteworthy happens.

So how do I combat this temporal affliction? I can think of one simple answer, right away: don't wait for the noteworthy stuff; just write a lot of trivial posts! I will embrace and extoll the mundane! It's not like my weblog is really good entertainment, anyways. q:

So today I will tell two bits of news, and one is more mathematical than the other:
More Mathematical

I think I made a computational mistake in the last proof I showed to my advisor, which is both bad and good. It's bad because .. well, errors are inherently bad in academia and in particular, mathematics: they detract from the foundation of rigor and fact that comprises our body of knowledge.

However, it's good because the three of us (a visiting prof was also observing our discussion) were kept mystified at this strange exponent from my Lebesgue theory argument. If you know about this sort of thing - Hölder conjugates and Sobolev conjugates, for instance - then they often take a certain form. This exponent of mine was really out there.

Of course, I have to check again that my mistake is, in fact, a mistake, and that I didn't make a mistake while searching for a mistake.

Zounds! This could be worse than compiling computer programs!

Less Mathematical (but somewhat pertinent)

I was surfing through the Slashdot website and reading idly, I chanced upon this new facet of Google: Google Blog Search!

Impressed, I did the first thing I could think of [2]: I ran a search under my own name. This website was the 38th hit, which humbled me .. and for the record, the first 37 hits are from a UK blog: blog.fatality.co.uk.

So I have a British weblog doppelganger. Huh. From reading a few random posts in May 2005, his also embraces the mundane .. (;


[1] Some would argue that my mathematical life is my only life, which could be justified: during a semester I might spend north of eight (8) hours in my office in the Department for six (6) days of the week. The parties I attend are mathmo parties, and aside from a few friends and former housemates, I know very few non-mathmos in Ann Arbor. (;

[2] I suppose this betrays a latent sense of vanity in me.

Tuesday, September 13, 2005

Fall Term Life.

Teaching is taking more time than I had thought.

Students actually show up during my office hours, and yesterday when I had no office hours, I was barraged with questions about Team Homework [1] after class. Some of them even followed me to my office, and so did the questions.

Thinking about it, if not for the quiz I administered, the same thing could have happened today. The results aren't going to be terribly pretty, I fear.



There's nothing like a steady job to make you long for those uninterrupted workdays during the summer. Never mind the fact that I barely accomplished anything ..

(proved a few computational lemmas, but hardly anything to shout at)

.. but to be honest, I was living under an illusion: that when fall came, life would be better. I'd become a more responsible person, manage my time more effectively, and it would be more likely that I'd accomplish my daily and weekly goals.

Oddly enough, almost all of those came true ..
.. except the "life would be better" part.

I'm accomplishing much, but it leaves me drained and I crash at 1 am on most nights, if not before. Mathematics doesn't get me eager and excited, as before. I get my work done, but just barely: there seem like so many little things I should be doing for research and thesis work that it's hard enough just to get started .. and at some point, I need to remember to talk with my advisor about a syllabus for my Prelim Exam, which would hopefully happen this December.

Thinking about it now, I'm not sure what "better" means, or should mean. I envisioned myself happy, but let's face it: that's never going to happen. I'm too good at being pessimistic to leave things well enough alone.

Maybe it's enough to be productive and have work to do, and in that way, have a purpose. Those younger days of summer were bits of fun, but they were also empty and unfulfilled .. and a bit lonely, since everyone would be leaving at different times for home or for conferences.

It could be worse. At least I don't have written homework anymore. \:



[1] Over here we do have cooperative mathematics assignments for PreCalc, Calc I, and Calc II; they're much like lab reports and often involve word problems and a fair bit of frustration -- for the students AND the instructors ..

Saturday, September 10, 2005

something to read if you're bored

I typically try to avoid article posts on this weblog, but I thought these few paragraphs were interesting, if only as a stimulus towards thought. It's from an article in the electronic version of the magazine, the Economist.

What, if anything, can be done? Techno-utopians believe that higher education is ripe for revolution. The university, they say, is a hopelessly antiquated institution, wedded to outdated practices such as tenure and lectures, and incapable of serving a new world of mass audiences and just-in-time information. “Thirty years from now the big university campuses will be relics,” says Peter Drucker, a veteran management guru. “I consider the American research university of the past 40 years to be a failure.” Fortunately, in his view, help is on the way in the form of internet tuition and for-profit universities.

Cultural conservatives, on the other hand, believe that the best way forward is backward. The two ruling principles of modern higher-education policy—democracy and utility—are “degradations of the academic dogma”, to borrow a phrase from the late Robert Nisbet, another sociologist. They think it is foolish to waste higher education on people who would rather study “Seinfeld” than Socrates, and disingenuous to confuse the pursuit of truth with the pursuit of profit.

The conservative argument falls at the first hurdle: practicality. Higher education is rapidly going the way of secondary education: it is becoming a universal aspiration. The techno-utopian position is superficially more attractive. The internet will surely influence teaching, and for-profit companies are bound to shake up a moribund marketplace. But there are limits.


Apart from suggesting ideas and shooting them down, this article does try to isolate an important issue. Surprisingly enough, the author has good things to say about American university education:

The problem for policymakers is how to create a system of higher education that balances the twin demands of excellence and mass access, that makes room for global elite universities while also catering for large numbers of average students, that exploits the opportunities provided by new technology while also recognising that education requires a human touch.

As it happens, we already possess a successful model of how to organise higher education: America's. That country has almost a monopoly on the world's best universities (see table 1), but also provides access to higher education for the bulk of those who deserve it. The success of American higher education is not just a result of money (though that helps); it is the result of organisation. American universities are much less dependent on the state than are their competitors abroad. They derive their income from a wide variety of sources, from fee-paying students to nostalgic alumni, from hard-headed businessmen to generous philanthropists. And they come in a wide variety of shapes and sizes, from Princeton and Yale to Kalamazoo community college.