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:
Thursday, October 20, 2005
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:
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.
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 ..
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.
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.
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 .. \:
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!
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.
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. \:
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:
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:
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.
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.
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.
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 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:
[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.
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.
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