a book and its audience.

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

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

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

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

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

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

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

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

---

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

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

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

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

In some mathematics books, it feels more like one timeline of reality or possibility is presented: the prevailing and practical one, up to some consensus. It's not so much rewriting history, because there could be alternate realities; we just choose not to study them.

But enough of philosophy. I'm curious to hear what you guys think: how do and how should authors label the "level" of books they write? Should they account for the factors below, or are there missing factors?

• How specialised the topics are, or how obscure a field.
  
• The treatment of the topics, and amount of technical details.
  
• Better sales, varying by the size of the audience

Or could it be there is no good scale?

"You'll do fine."

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

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

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

---

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

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

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

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

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

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

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

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

on studying, remembering, and what to remember.

After a studying session for my prelim, I've observed a few things:

1. I'm making slow but steady progress. This might be illusion or delusion, but it honestly feels like I'm gaining some degree of intuition from the arguments that I've been reading. I can only hope that my memory serves and I don't forget it all, but perhaps that can be corrected by how I've been processing this information.
   
   
2. I've given up on remembering the technical lemmas which are used to prove big theorems. Instead, I've been trying to remember the big theorems ..
   
   
   (.. which aren't too hard; they are big, after all.)
   
   and what is the "right way" to prove them; this also allows me to build intuition on how to work with certain concepts and properties. If I remember the "right way," then I'll have to collide into what smaller claims I'll need to prove it, and those are the very technical lemmas that I would fumbled to remember. It's the difference between a priori and a posteriori knowledge, I suppose.
   
   Perhaps I am slow-witted and should have thought of this long ago. I've learned it now, though, and it suggests that I might not be a wholly hopeless cause .. (:
   

In particular, I never realised how cool the Area Formula for Lipschitz functions is. From the approach in the Evans-Gariepy text, the idea is to dissect the domain into Borel subsets, where the given Lipschitz function, when restricted to one of these sets, is "essentially" bi-Lipschitz with controlled constant.

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

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

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

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

Oh well. It's something, at least.

a night ago

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

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

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

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

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

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

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

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

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

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

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

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

---

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

gambling on a seminar, and an abuse of notation.

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

(See my previous post, here.)

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

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

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

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

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

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

---

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

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

---

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

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

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

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

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

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

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

highlights from a work session.

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

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

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

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

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

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

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

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

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

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

This just doesn't bode well .. \:

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

GERTRUDE: More matter, with less art.

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

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

holiday procrastination

I still haven't been able to summon up the will to do any work, despite sitting in the coffeehouse with a pleasant brew in hand, pleasant big-band music streaming from the ether called internet, and my book and notebook sitting pleasantly right before me.

It's the perfect time to work, too: it's Friday and working now gives a good excuse for weekend follies [1] later .. yet no prelim reading accomplished and no research done. I think I know why:

Deep down inside me, I want this holiday break to be an actual, proper holiday -- not just the peace and quiet where one settles down, without students underfoot, and gets down to work -- but doing non-academic things and non-work things.

I mean having "fun" which others outside of your building and department would also consider to be fun. For example,

• Renting and watching movies is considered fun, by most persons.
  
  I haven't done this in a while.
  
  
• Building snowmen, then tackling them in a drunken mirth might be considered a little strange, but still fun.
  
  I might have done this before, but I can't quite remember ..
  
  
• Writing on chalkboards and using the Triangle Inequality [2] might not be considered fun by the "regular" stretch of the imagination.
  
  I think I do this at least once a week, and my friend Kevin does this almost every day.

At any rate, maybe I don't need to work this break. My advisor will be away all next week, which means unlimited prelim studying time (and time for a bit of research for when we meet next). But there are talks to give in a few short weeks, and work to do for them.

But as we all know, procrastination needs no motivation. It is merely a state of mind and if one willingly enters that mindset and realm, then that is enough. However, one must reach that realm first, and not sit or stand on the neutral boundary, lollygagging and squandering time that could otherwise be spent on real work or real play. That "frozen" mindset reminds me of this passage from Dante's Inferno, Canto I.

And just as he who, with exhausted breath,
having escaped from sea to shore, turns back
to watch the dangerous waters he has quit,

so did my spirit, still a fugitive,
turn back to look intently at the pass
that never has let any man survive.

If I ever get my act together and make a decision, then I can salvage something from today, whether it be great fun or a modicum of work. Maybe both .. who knows? But I have to stop idling and sitting so mentally-transiently, and do something.

[1] .. never mind the fact that I never do anything on weekends anyways, with the occasional exception of being invited to math grad parties, which are far better than they sound, as my non-mathmo flatmate can testify.

[2] .. Roughly speaking, the Triangle Inequality (Δ≠) is your common sense about distances, but written in mathematical symbols: that is, if you're travelling from A to B, and then B to C, then that total distance travelled is no shorter than had you travelled directly from A to C.

(Δ≠)   |A - C| ≤ |A - B| + |B - C|

See? That wasn't so bad, was it?

To the experts, yes: I did use norm-notation without much reason why. q:

laptops in the classroom: an article post.

Here is an article from Slate Magazine about laptops in the classroom, but it's not a techie piece. It instead considers the nature of lectures and professors.

I haven't formed any opinions yet, but certain paragraphs trouble me. This is one paragraph which I've split into parts, and I hope it is not too out-of-context.

In any event, even when multitaskers can't keep track of the professor, it probably doesn't matter much. In lectures at large universities, especially in the humanities and social sciences, class time is usually taken up by the broad outlines of the subject.

The real learning occurs when we bear down and pore over the hundreds of pages assigned every week—the lecture I'm currently tuning out assigns about 3,000 pages of reading over the span of the semester—and when we attend small discussion sections with graduate students who go over what we've read. Any good grade-grubber knows that the trick to doing well on exams is knowing the reading, not what the professor said last week.

If what the author says is true, then what is the point of attending lectures, other than if there is a class attendence policy? Why have lectures at all, if they are obsolete in the learning process?

Is it also possible that, in some cases, the grades don't reflect what the prof discusses in lecture because the prof doesn't grade the exams? Or is that superfluous?

Perhaps the real problem with laptops in lectures isn't the laptops, but professors' over-reliance on the lecture as a learning tool. Earlier this week in Slate, M. Stanley Katz contended that "the most effective learning is active learning … teaching must involve presenting students with problems to solve rather than merely lecturing about those problems."

Amen, professor. You try listening to rambling, jargon-filled disquisitions for 15 hours a week without reading blogs. At least Gawker solicits our contributions.

Wow. "Rambling, jargon-filled disquisitions." Are lectures really that bad, these days?

state of the union

Prelim studying is going all right, I guess.

In a ways it feels like downloading a large (legal) media file: at every instant you can read off the streaming speed from the Download Manager, but you cannot predict for certain how long it will take before the download is finished. If you have a deadline, then you watch the screen and hope, then watch the screen and hope some more.

Similarly, I feel like I'm learning at a good rate, but it remains a mystery whether I can be fully prepared for all the topics on my syllabus. We do what we can and I understand that, but I'm having trouble accepting that it's all I can do.

Earlier this weekend I thought I understood tangent measures (arising from blowups of space [1]) but now I'm not so sure.

It can be troublesome to learn from books, because if one is not very clever (i.e. me) then one adopts the perspective of the author by default. There's nothing inherently wrong with this, but one runs into trouble if

1. .. the book isn't "very good," as measured by difficulties such as omitting details, unclear language, possible errata, and the like.
   
   
2. .. despite the fortune of clarity, the book has an unorthodox perspective, and what one had thought was standard terminology is actually "author-speak."
   
   This becomes a deeper issue if the methods of proof differ from one source to another and if one seeks to generalise arguments from a common starting point (for example, Radon measures in Euclidean space).

So I worry. There might be no "right way" to understand a particular idea, but fortunately or unfortunately, there are some "wrong ways" to gain understanding.

---

Research is .. well, research. It's shifty, ever-changing, and hence unpredictable.

Three hours ago, I thought I had exhausted every possible idea out there and that the problem is insoluble, despite what M. Morse claimed in the 1950s .. [2]

.. two hours ago, I began to draw the same diagrams, only to realize that I drew them wrongly! Now things look more promising, but they'll require a bit of trickery before I can prove what I want to prove ..

.. and for the record, an hour ago I was eating dinner: vegetarian Indian food which made everyone else in the room, who were Indian food-free, salivate. I felt guilty and sinister at the same time. (;

---

Teaching is what it was before: neither good nor bad until you think about it and make judgments.

I tried to switch things a little and make my lectures more example-driven and start matters off with what the students know well (or should know well from their last exam). Time will tell if they are just as confused by the material or if it's actually working.

I must admit, the examples were fun: I introduced differential equations today by giving examples of unhindered and constrained population growth, and perhaps as a moment so that they could catch up with theit notetaking, I idly mentioned the famous viewpoint of Thomas Malthus, if only to mess with their heads and get them thinking.

However, one bit worries me, and I call it the "Heisenberg Uncertainty Principle of Calculus Education." It goes something like this:

The more interesting you think the class topics are, the less likely your students will understand what your talking about. Conversely, the more your students understand what you're lecturing about, the more likely you find it boring ..

That's mean of me to say. Any opinions? q:

---

[1] and by that, I don't mean the blowups from algebraic geometry! Perhaps the term "rescaling" is more accurate here.

[2] In one of M. Morse's papers on the Schoenflies Problem (before he studied the problem in terms of conical points with Huebsch), there is a claim which asserts that a smooth analogue of the topological Schoenflies Theorem fails.

Morse cites a paper of Milnor (the one about exotic 7-spheres) and since then, everyone else says the same in a rote manner. I've might have mentioned it before, but my current task is to make sure that Morse's claim is correct.

feeling mathematically inarticulate

There's a term from Victorian England called a monoconvolute.

It's a derogatory term and brought up after the study of phrenology became passé and neurology became its successor. The idea was that the convolutions (wiggles) on the surface of the cortex provide "space" for storing information and memory, and if you had only one convolution on your brain, then you must be a very stupid person.

At any rate, the existence of such a word demonstrates some importance about having a complicated brain for higher cognitive skills and whatnot.

But I believe that the opposite extreme can be troublesome, because I suspect it in myself. Either my brain is too muddled and twisted, or my thoughts require more discipline: something in me is muddled, at least.

In fact, this reminds me of an instance from high school: I was a sophomore taking the usual English class and my teacher was a fine editor but a harsh grader, and used red fine-tipped pens by the box.

There was one essay assignment we had to write one week .. but so that you know, we had essays every week. Monday was our deadline and Wednesday she returned them to us, mires of scarlet and scrawl atop once-cleanly typed white pages. But this one assignment everyone had jaws dropped, grades slashed,

.. in the same way that I grade Calc II quizzes, come to think of it ..

and there was general despondence and "gnashing of teeth," as one friend of mine would say.

I could swear that my essay looked the worst. It might as well been dipped in the red ink, but the strange thing was, there was no grade on it: not on the first page, the second, nor the third. I was able to read the last of the scrawls, which I interpreted as either:

Keep these ideas. Rewrite this.
or
Kaput the ideas. Reunite us.

Deciding to use common sense, I rewrote my essay and brought it back on Friday. Being a curious child, I couldn't help but ask my teacher, "Was it really that bad? Was it really ungrade-able?"

"It wasn't finished, and the language was awkward," she replied, "I wanted to see what your completed thoughts were."

I nodded, handed her the rewrite, and that next Wednesday I received two essays back. My rewrite looked only speckled with red this time, and later I learned that nobody had ever been given a rewrite option from her before. I don't know if she ever did it again.

Suffice to say that English is my native language, but mathematics isn't. It seems like I never quite phrase things properly, and every time I discuss some matter with a fellow maths person, we would discuss the same object or concept, but their viewpoint is much grander and more precise than mine.

When I studied literature and history, I never felt like a plagiarist, but now that I study mathematics, I feel like I steal all the good stuff from other people.

Take my meeting with my advisor today: it was the third session we've sat down (well, figuratively) to hash out this lemma of Milnor's, concerning existence of diffeomorphisms of topological n-spheres when given a smooth homotopy condition (more precisely, a differential isotopy) on maps of (n-1)-spheres.

I still feel as if it shouldn't have taken that long. Had I remembered to tend more to geometric issues, then this could have been put to rest sooner. Unfortunately I still think too formulaically and not very intuitively, and honestly, I use rather terrible notation.

Had I not drawn enough diagrams, my advisor might have thought I spoke nonsense .. which I probably did. Happily and fortunately, he's a forgiving man.

Perhaps there's no real conclusion in all of this .. only that today is an "mathematically inarticulate" day for me. \: