Saturday, March 26, 2005

not quite mental wanderlust

Nothing much to report today. I think something is fundamentally wrong with my pattern of working and studying, because once again I feel like I'm sinking into another rut. I spin my pen between my fingers between minutes-long sessions of computations, and the ensuing symbols make little sense. I'm not looking at them anymore; now the pen is on the desk and I'm fixated at the random little ink stains on the sides of my fingers.

The List of Things to Do grows longer and the deadlines closer. Despite the time I spend doing this or that, it seems that none of the items on that List ever gets checked or crossed off. Nothing ever gets resolved.

My mind's wandering and I can feel it. I try to re-focus my attention but it only slips away, again and again.

a metal gate,
a broken latch,
a strong and inconstant wind
.

(counts syllables)

Nope. Not quite a haiku, but it sounded like one. \:



I could ask myself where I'd rather be or what I'd rather be doing right now, but I wouldn't have an answer for that either. I don't feel like travelling and I don't feel like loitering in the Ann Arbor coffeehouses. I could do work, but it doesn't matter either way. I could read a little with the same sentiment.

I feel like a void, empty: no good ideas and no bad ideas. I don't have any ideas or opinions at all. I feel like ashes: the remains which dwell after a fire spreads and consumes, brightly and fiercely, the flames more alive than organisms and kin to the combustion of the sun.

Warm ashes .. I suppose if you touch me, then I'll be warm to the touch.



Writing more won't add any more sense or sentiment, so perhaps I'll stop now. Maybe I'll find some stimulus by wandering just as my mind is wandering .. away from the house and away from the office, towards books and people even if I don't know them very well or at all.

Maybe I'll look up Mother Nature's address, her residence in the color and concrete of Ann Arbor. I'll observe her effects and non-effects in light of industrialisation and post-industrialisation, and ask her her opinions on the matter.

Or maybe I'll just wander aimlessly and thoughtlessly. I might as well work my way up.

Saturday, March 19, 2005

Modern Mathematics and Academia.

This originally began as a reply to something a friend of mine wrote in her LiveJournal. If you're curious and I rememeber rightly, as an undergrad she survived a double major in English Literature and Physics.

Brave woman. \:

from First Post: Does anyone else find grad school to be lessons in how to grow an inferiority complex?

from Second Post: Oughtn't there to be some better way to do this?

My response is below, and full of my own opinions. Please be mericful.




You've got me. I can speak a little, but only about mathematics, and let me start with the present state of affairs.

Formalism. Walking into this subject is a little hazardous, because since the early 1900's mathematicians have sought (with some success) a general-use formalism with which to express their ideas. Modern mathematical arguments are embedded in logic and precise definitions of appropriate concepts to the argument at hand.

It's now what everyone expects and celebrates as a sort of "crown jewel," and serves as a basis for the rigor and 'infallibility' of mathematics. We know exactly why certain facts are true and why certain (theoretical) phenomena occur, because we are very specific about what causes them, what hypotheses we make, and what assumptions we use.

For example, are we in a space where interior angles of triangles add up to 180o? If not, some facts are no longer true (think of a surface of a globe, and draw arcs joining two points of the equator and the north pole: that gives you at least 180 degrees of angles). In such a circumstance, for instance, you can't move vectors the same way and expect to point in the same direction as before (i.e. the problem of parallel transport, from physics).

Comparison. With this in mind, I have to say that it's a lot harder to be a modern mathematician than an 'ancient' one (up to, say, the 1890's). Thinking rigorously is exceptionally hard and takes years of practice, and it's a wonder how newcomers can breach this obstacle and do "good" modern mathematics.

The precision of ideas in the 19th Century wasn't up to par with current logic, but they were extremely intuitive and serve as a basis for modern problems. One example is Riemann's Hypothesis concerning distributions of primes (which are related to roots of a function of 2 variables). Another is an idea of growing popularity: the concept of string theory sponsored by the successors of Einstein, Heisenberg, and the Bohrs. The essential problem lies in how to make the right definitions so that the modern mathematics work.

One key element to observe is that until 1904 or so, most mathematicians were universalists, in that they knew a little something about every area of maths (some say that Henri Poincare was the last universalist) and moreover, had interests in chemistry, mechanics and physics, and even small-scale economic models. In a way their ideas had to be intuitive, because they were being stimulated by so much of Nature.

In short, going into graduate school in mathematics is like learning a conceptual language to extreme fluency, yet preserving the curiosity which is present only in "newborns," who have no such language.

The best problems are the most intuitive ones, and many current research problems can't be understood by the layperson and survive as an artifact of the edifice of academia. They're interesting, but to a select few people who can understand them.

If I may consider myself a mathematician, the odd thing is that we mathematicians still think and work intuitively. We believe that a certain solution may work because of some heuristic (which is really a convenient rhetoric) but we won't know until we prove it: that we demonstrate its validity in a sufficiently rigorous argument, paying special attention to assumptions and smaller arguments which are individually 'infallible.'

That is one problem of modern mathematics, graduate school, and academia: learning how to argue with rigor and precision, and learning how to find good ideas which make good problems. Along the way there are hoops to jump and hurdles to pass, in the forms of exams and course-work, but I believe this is the crux of matters.

Wednesday, March 16, 2005

The end of my academic orphanage.

I now have an advisor, and by implication, I can't possibly be a complete coward (as it took some guts to ask him this). It feels like I've relieved myself of a great burden, and now I have a specific goal towards which to work ..

[ponders the annoyance of avoiding dangling propositions]

.. but as I've began telling the other graduate students this, they exclaim: "All right! It's over, for you!"

Then I blink, and reply, "The end? It's only the beginning. I haven't DONE anything yet."

At any rate, one less hurdle to jump, though that doesn't mean that more have appeared .. q:

Tuesday, March 15, 2005

Back from Dartmouth ..

[This was meant to be a reply to a comment from the previous post, but it seems to take a life of its own ..]

I'm back, after long last. Earlier this evening I was walking home to my apartment when I realized: I'm in Ann Arbor, where my apartment is. I'm in a place where I have a home, and indeed I can walk home. Ah, the small joys of life!

Aside from travelling problems, the conference went rather well. Nobody booed during my talk, and a few profs gave me suggestions and possible directions to extend this project.

With that note, it was good to see a few profs from this field. There weren't very many students attending, so I felt a little like everyone's 'mathematical nephew.' Most everyone was friendly and supportive, or at the very least, hospitable.

Moreover and mainly, it did me good to hear talks about C-C spaces; I feel that I have a sufficient (read: working) knowledge of the background of this area. Now I can think seriously about how to attack certain problems, and I've filled a few pages of my journal of mathematical problems already!



I think I should conclude the advisor search. Now it is a matter of asking .. well, maybe a little more talking, and then asking. I still have a habit of forgetting inform people of my whereabouts and plans.

Something urges me to keep focused and study other areas of study, but it's hard to leave C-C Spaces alone. The questions and the problems at hand seem so immediate ..

Wednesday, March 09, 2005

words of a weary traveler ..

Argh .. still stranded in Philadelphia (since yesterday afternoon, because of a paranoid response to weather delays) but feeling better.

I never expected it would be this much trouble to travel to a conference, and here in the States. Even travelling to Finland for the Jyvaskyla Summer School was easier than this, and I don't even speak any Finnish!

Tomorrow is my talk: a 30-minute affair, and as was suggested to me, likely few people will be listening (because it is the last talk before lunchtime). I have a feeling it will run short, so perhaps the audience will thank me for giving them an extended lunchtime.

[shrugs]

I'm making a stand: no more travelling this semester. I'm tired and I can't keep it up. My life only grows more and more complicated because of it, and I need to get my act together, secure an advisor, and get on track towards research and my oral Comprehensive exam.

There. Fine.

God, another half-hour before boarding, and the plane isn't even here yet. I've said it before and I'll say it again: why do I get a bad feeling about this conference?