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.

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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. \:

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[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 ..

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.

clock is running; feeling fine.

I finally stumbled into my advisor's office today in order to hold myself accountable for laziness. Evidently he thought I was still on vacation .. which I was, "from a certain point of view." [1]

I haven't spent any substantial amounts of time doing any mathematics and specifically, I can't remember the last full day I spent either at the office or at some table, reasoning and computing away. Then again, I find that summertime urges me to work in short but frequent blocks of time. An 8-hour day is impossible to implement with my (lack of) behavioral discipline.

But I digress.

Late last week I had set a few goals and they remain the same. Now I have a tentative timeline, that is: a means to measure how much time I have left to work before my next advisor meeting, which also induces a working barometer for my work ethic and how often I should "crack the whip," so to speak.

In greater brevity [2], now that I know how long I must toil, I can toil to my heart's content!

This viewpoint, of course, has strange philosophical tendencies: it could mean that, were I a Christian, my version of Hell would be partially Kafka-esque, and kin to the Trial in fact. It would involve some sort of uncertain perpetuity and torment ever unpredictable, and it would probably involve being swarmed by starving, sharp-toothed rats ..

.. but this little mania forms a tale for another blog and another day. Today I am content with my academic progress, and let's leave it at that!

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[1] One of these days I have to reduce my nerdy habits. In particular, I should refrain from inserting Star Wars movie quotes to real-life conversations!

The specific quote comes from the beginning of Return of the Jedi. It is after Yoda dies and becomes one with the Force, and while Luke has a talk with Obi-Wan (Ben) Kenobi about Vader vs. Anakin Skywalker.

[2] That sounds suspiciously like an oxymoron to me: "greater brevity." Thoughts, anyone?

Old Habits Do Die Hard ..

Take, for instance, the way I understand mathematics. Like many, I seem incapable of understanding a particular concept unless I write it down, and like many analysts, it often helps me substantially if I draw a nice, descriptive picture: that way I may accrue some geometric intuition of the situation at hand.

Unfortunately, I also seem incapable of being satisfied with my own work. Regularly I reread my old work notes, think of them as sheer crap, and feel the urge to rewrite them. As a result, I've become accustomed to judging my work based on how long (I think) it will last [1] will last until I dispose of it.

I blame this on my habit of keeping a written journal, and years of half-assed creative writing. Editing and re-editing seem to me second nature if I am working on a matter of some substance, and I can never leave well enough alone.

It's the same with mathematics, such as my idiosyncracies with completing problem sets and doing research. I work and work, trying this idea and that idea, replacing this idea with a better one, but having a poor memory and little mental discipline, after a day full of work, I might be confused at what I was actually proving and mean to continue for the next day. The notes might seem too ragged, too specific to what I had in mind yesterday and not today, and I must summarise from scratch before moving on.

If anything, this reminds me of an anecdote about Archimedes: In the afternoon he would go to the beach, prove theorems by drawing in the sand, and then leave at sunset to pursue other matters. But each day he would arrive to see his diagrams washed away by the sea, and he'd have to reprove them quickly before being able to reach any new results.

One can only conclude that Archimedes must have had an excellent memory of proof, and amazing self-control; I know I never get any work done when I go to the beach. q:

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At any rate, today is a writing day: I'm tired of half-forgetting which details I've proven and which ones I haven't. I've jotted down an outline of main ideas, and now I'm working out the gory details to each main idea, page by page. Maybe through this, my advisor might understand my ramblings when I see him next.

Here's hoping.

[1] To me, there is a substantial difference between work I've done and the paper on which I've written it. Due to errors or a better idea or technique, I'll often rewrite something: it's similar, improved work, but on paper it is truly a different manifestation. If ever I throw out the actual mathematics -- the latest version of the work -- then I have disposed of it.

feeling somewhat like Superman ..

.. in that I, too, may be solar-powered. I don't know what it is about my office in East Hall, but there I seem incapable of developing new thoughts. It is a place better suited for checking details of thoughts that I obtain elsewhere, say at a coffeehouse .. but even that is pushing it.

I was at Amer's Deli just earlier and had a very pleasant time, working out a cohesive path of argument for Lemma 1 of (at least 2). The sun shone through a window on my right and the table, a dull black, was slightly warm to the touch. From the speakers the Amer's staff played the 'Californication' album by the Red Hot Chili Peppers at a reasonable volume: loud enough to catch the catchy beat but not loud enough to dominate one's thoughts.

The whole ambience made for a pleasant time and a motivated work session. It might have helped that I've pondered these mathematical thoughts before, and today I rehashed them and realized how silly I am .. again. With any luck I may have enough results -- rigorous ones, this time -- to discuss with my advisor next week, and still have time to finish my packing and moving into my new apartment.

A guy can dream, right?

I've been sidetracked recently with personal and personable concerns: moving residences, meeting and giving suggestions to the incoming first years, sorting out thoughts and concerns about my life and its future.

At some point I should stop this dilly-dally of my mathematical/academic goals and set things right: hash out a Prelim Exam list of topics, register for that third Ganelon drumming course for the Fall Term, keep my advisor up to date with my progress (or lack thereof), and begin a discussion of a Student Geometry/Analysis Problem group ..

which is a fine idea but I cannot claim credit for it: it was previous suggested by my friends and colleagues Kevin W. and Ben S., where students of analytical or geometric mind share and solve small mathematical research problems ..

Anyways, enough rambling. Work beckons.

stuck in reality and unimaginative

I'm already feeling guilty for not working on my mathematics for over two days.

Never mind the fact that I make little progress when I am in Ann Arbor, either in the office, at home, or at the coffeehouse(s). At least I'm putting in the effort, you know? Now I just feel lazy and unoriginal, despite the additional fact that I don't prove any exceedingly original or surprising results, much less use any original techniques or methods of proof.

I also feel ground, or rather, firmly stuck in reality. I observe matter and motion, sufficiently macroscopic to suit Newton's Laws, and I listen to small scheduling conflicts which are more stress from the complexity of planning for six people than any real chore: nothing truly worth worrying about.

As a result I feel like my imagination has shut down temporarily; it's gone on holiday just as my physical body has, and I miss visualizing in my mind's eye how Lipschitz manifolds should fit together .. specifically n-spheres at the moment, but you know what I mean .. well, probably not, but you get my point.

Strangely, "visualizing" probably isn't the best word, here. I can't actually visualize a Sobolev space, though I can imagine the standard diagram of smooth manifolds, their charts, and transition map arrows. I suppose that is precisely why I feel this laziness: not being able to see the result easily, I still want to understand a few Sobolev space results easily, which means I should sit my @$$ down and grind out the damned computations.

travelling

I'm at my parents for my biannual visit.

Apparently you can steal wireless internet, even in suburbia. I guess people haven't yet read this article and secured their personal WiFi networks. Oh well: good for me, bad for them. q:

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Oddly enough I was able to get some work done on the flight to New York: to my credit I scribbled on the back of my on-line e-ticket printouts (which were on 8.5" x 11" paper), so no observing busybodies registered this as mathematics work. It also helped that the people sitting next to me didn't speak much English, and as a result they didn't feel confident enough to ask.

Small favors, I guess. At any rate I've been silly: the proof works just as I think it does, and the little details do work out in the first step.

Now just N + 1 more steps to go .. \:

mathematical language, legal language?

A friend of mine made the recent comment (on someone else's private journal; I won't quote it here in detail) that a "pure" subject is suspect to her, and she believes that all fields of study are linked together in intricate ways. Her opinion make sense ..

.. but I wonder whether "pure mathematics" falls into that category or not. Most of the time? Some of the time?

Off and on I've discussed matters with scientists (or rather, grad students in the hards sciences) and on less occasions, folks in the social sciences and humanities. As pure mathematicians we get our share of flak for not doing much interdisciplinary work. Maybe they think that we mathematicians are arrogant, that we don't need them, but they need us. Maybe I'm imagining things.

There are the occasional hotspots in research: dynamical systems are useful for scientific models and more recently, computational biology. Algebra and combinatorics become useful in encryption and search algorithms. The nascent theory of wavelets has grown in parallel to needs of image and signal processing. I'm confident I could find more examples of this sort.

However, I believe that most of pure mathematics is not immediately applicable and to the rest of the world, "current worthless." Maybe different areas will have untold worldly uses in the future, but maybe they won't. I get the impression that past and present administrations of American government fund mathematical research because of hope towards the former, not because of any noble purpose on their part.

If our mathematics helps the rest of the world, fine; if not, then as Archimedes said, don't disturb our circles.

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Me, I like to think of mathematics as logic, or even philosophy, applied to very specific situations, often to geometry or the structure and interaction of numbers and their abstractions. It's a convenient way of saying that it's the study of abstract objects using abstract tools.

Many modern problems have a tendency towards abstraction and take a long time to ponder; perhaps this is the fault of logical formalism and rigor, which adds apparent volume and mass to a little monster, so we fear it more and make much of it.

If you ask a mathematician about a problem, (s)he may tell you the full regalia of the issue and possibly give a few definitions. If you ask a mathematician about his/her ideas, then more likely than not, you'll get a concrete example and possibly heuristical reasons and intuition in the form of pictures and diagrams.

So why the logical formalism? Why the rigor and fearsome-looking definitions? Why a formulaic outline when assessing and/or solving a problem whose words are hard to understand?

It's the price we pay for consistency and accuracy, and it plays a similar role as legal language. In some ways, legal documents and mathematical papers are similar. For example, the first section is full of definitions. In a union contract, how do you define "Employee?" and on some apartment leases, what exactly is a "tenant?"

If you believe your method is correct and it accurately addresses the problem at hand, then you must "fill out the legal paperwork" .. in short, write out precisely what you mean in a consistent way that can be checked logically.

On the other hand, when reading mathematical work it takes a careful mind, often a mathematician's, to decipher the full strength (or weakness) of a new theorem. Is it argued correctly, any logical mistakes? Does it address even the pathological cases, if the definition allows such possibilities? For example, it's a bit like hiring a lawyer for when you're closing on a house: the last thing you want is to hear that sinkholes aren't covered by a "good condition" policy.

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Perhaps a problem with mathematics and interdisciplinary work is, oddly enough, a language barrier. Another problem is that mathematicians are attached to their own problems, just like any other type of academian. Combining the two, sometimes it seems like a one-sided relationship with other disciplines.

A mathematician might be helpful to a think tank of physicists or economists or biologists, but only after (s)he processes the situation and question in a way (s)he can understand, and then asks enough questions so that (s)he understands what can be done and what cannot: maybe it's an issue of conservation of matter, or that negative profit is bad. If this is successful, then (s)he might suggest an idea, and with any luck (s)he can explain the rationale apart from any tendency towards excessive terminology. Maybe it helps, who knows?

I remain wary of a think tank of mathematicians discussing a common (open) problem to a physicist, economist, or biologist. It may take too long to explain, and even with a good dose of patience, it may be too many new concepts to juggle -- and this is not an insult to non-mathematicians, but a matter of technical details. It's a lot to ask someone to think about something away from their area of expertise, and with many issues to consider.

However, I do see the value in mathematicians asking non-mathematicians how they think about their own problems, because it could help the mathematician learn how to think about the mathematical problem in a new way. It could be as simple as analogy: the notion of extremal length, also called modulus of curve families, is mysterious unless you describe it in terms of electro-magnetism and current through a condenser. It doesn't proves your theorems, but it helps you write your proofs.

But there remains the problem: how did we know to think of extremal length in terms of electromagnetic current? Someone else has to answer that one, though my guess is complicated and needs some explanation [1].

I don't claim to know the answers. If non-mathematicians had a tried-and-true method that could help mathematicians with their own research problems, then great! Sign me up! But every time I explain what I'm studying to the non-mathematician, we mutually give up and it leads to little added progress, though by explaining I might understand the issues better.

When from experience others are of little help, we tend to ask less often and continue on by ourselves. After a while you forget to ask, and expect confusion.

What can I say?

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[1] I think it has to do with the relation of extremal length and a notion called "capacity" which, physically speaking, describes the least kinetic energy under a given arrangement of position. If this relationship is true, then there may be an optimal arrangement of the magnetic field induced by the current, and the resulting field lines may give some comparison, as curves, to the given curve family.

Conservation of Paranoia / Planning Ahead.

So I met with my advisor yesterday for the first time in .. wait, six weeks?

I'm still inclined towards disbelief, but between my advisor's one month holiday with his family and the travelling-planning issues for a week before and after, I suppose it must be right. Wow.

At any rate, we met and I showed sufficient progress, but apparently I was working on the wrong metric space. Instead of Euclidean space Rⁿ, the results were meant for its 1-point compactification, or if one wishes concreteness, on the n-sphere Sⁿ in Rⁿ⁺¹.

Thus for more than a week, I was worried about a issue that is, in fact, a non-issue: the (local) Sobolev regularity of the inversion map x → |x|^-2 x is problematic near the origin in Rⁿ, but the equivalent mapping is a diffeomorphism on Sⁿ, so I was worried over nothing ..

.. well, sort of.
Something else is now suspect.

For those mathematically-inclined, I'm trying to prove an extension theorem for a particular type of homeomorphism on particular types of domains. In my efforts to resolve the (non)issue, I began a systematic examination in how "large" a sub-domain can the old and new homeomorphisms agree in value.

This in turn led to the study of the functions' behaviors for points approaching infinity, and rather interesting limits, at that! Apparently these same limits might cause trouble for another function in a different part of the proof. It would have been fine were we in Rⁿ, but it may go wrong in Sⁿ.

I suppose some difficulties never go away, do they? (;

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On a less technical note, I gave my advisor a preprint of an article of my co-author's and mine, and he didn't seem too displeased by it. Then again, he didn't get much time to read it during our meeting, so I suppose we'll wait and see ..

We also talked a little about the future, and my previous guess now seems true: after this term I'll have gotten all my course requirements completed (both cognate and distributional), and tentatively my oral Prelim Exam is set at the end of the Fall 2005 Term.

If I pass, it would mean that I'll be a Ph.D. candidate by January. Zounds! "Candidate" has such a nice ring to it, too .. q:

a month passes ..

.. and my advisor has returned from .. where he was, earlier.

  (I never asked where he went, and he never mentioned it)

Again I am accountable for my mathematical deeds. Now all I need to do is find him and allow myself to be held accountable.

I guess it hasn't been too unproductive of a month .. but there are few results to speak of. Earlier in June, co-author and I ran into some technical troubles in the final stage of our proof ..

  (for those keeping score at home, this is the second project, not the first)

.. so now the project has been put back on indefinite hold. It only adds to my deep-rooted suspicion that as much as I like geometric problems, I'm likely little good at them and should stick to more analytical topics, such as PDE.

I did have an idea to circumvent that technical difficulty, the other day, but I suppose it has to wait until the weekend. Thesis work comes first.