- 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:
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.
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:
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:
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!
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!
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.
[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:
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.
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:
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:
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 .. \:
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