Rants after a Bad Day of Reading.

I feel like I'm in a minor rut. I was able to do some actual work today, though I gained no ground or progress. To sum it up, there seem to be three stages to reading a mathematics text:

1. A lot of smiling and nodding and page-turning, because everything makes sense. Usually no paper or pen is involved at this stage.
   
   
2. A lot of frowning and confusion, because what made sense before no longer does. Usually paper and pen have manifested themselves, and in particular, there are many pages covered with hasty diagrams and scrawls .. all of which contribute nothing to one's understanding (but seemed like good ideas at the time).
   
   
3. A lot of groaning and self-incrimination, because everything does, in fact, make sense. Usually by now extraneous ideas are set aside and one peers into the heart of the matter. Furthermore, two sub-cases occur: (a) the result is clear and you wonder why you didn't see it sooner, or (b) you were right all along but your paranoia caused you doubts and in particular, your counter-example is not a counter-example, after all.

I'm currently on Chapter 4 and Stage 2: what fun and joy, I tell you. q:

However, thinking about these various stages of learning and understanding, it makes me wonder how any of us got into the mathematics business. Are we mental masochists? Fanatics? Or is it that we have nothing better to do, and mathematics is the least of all evils?

Don't mind me; those previous questions were meant to be rhetorical. \:

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As alluded in a comment to my previous post, I believe that I entered mathematics because I couldn't leave "well enough alone" and certain problems in analysis and geometry were so tantalizing that I couldn't let them go. Now I'm trying to make a career out of it.

I don't know about other students, but this inclination of mine causes untold frustration when I'm learning something, which at this stage of the game is always abstract theory. I begin to wonder how this or that definition came to be, and why anyone would ever think of such a thing. Textbooks often emanate an illusion that as presented, this theory is fact and for lack of a better term, "God-given." It has been written and taken this form. Understand it so.

It's easy to forget that books are written by people who have specific perspectives and ideas on how to view a certain phenomenon in mathematics, whether it is the structure of a particular space subject to its metric, or the nature of functions in accordance to their regularity. It's equally easy to forget that these ideas and intuitions take different shapes, from how they appear in print and under the full regalia of logical rigor. If there is one meta-lesson I've learned from reading mathematical texts, it is this:

Always draw a picture or work out an example, but be careful not to trust the picture or example too much.

I'm lucky to be in a department where my professors are willing to explain to me the motivations of how these theories were designed: as frameworks to overcome difficulties and issues in solving problems. They patiently draw pictures for me and show me how these matters reduce in simple cases, and my nerves settle.

All of this lead me to wonder: Will I ever be able to do this for myself? Moreover, if I want to be in the business of solving problems, then such skills are key, and I must be able to do this for myself.

I suppose there is no underestimating the power of negative thinking. \:

Looking Back (EDITED)

I was playing hopscotch with mathematicial blogs today and I stumbled upon an article written by a professor at UC Berkeley. It concerns undergraduate education for those math majors who do not proceed to graduate school.

(The article can be found here in PDF format.)

Mathematics without graduate school?
Heresy, you say?

I admit, that was my first inclination and now I fear I've been fully indoctrinated by academia. But that's fine, because I'm not arguing about that today. In fact, I'm not arguing anything at all: just reminiscing.


my education. Most of you already know the story of how I fell into the field of mathematics, so I won't retell it. But as for those years of enrolling in classes, I've realized something.

My early study of mathematics was hardly circumspect. As an undergrad I took exactly one abstract algebra course. I never took topology or number theory or logic / set theory, and if you asked me what Galois theory or representation theory or cohomology was, you'd receive a blank look in return.

The more I think about it, the more I realize that I learned very specific topics for a very specific purpose. I was still dabbling in the prerequisites in my sophomore year of college, stuff like Advanced Calculus I & II, Linear Algebra, ODE, and Abstract Algebra.

It was later that summer that my Linear Algebra prof offered me a research project and threw me a problem. Read this paper and learn what you need to understand it, he suggested. Ask me questions, and read. When you're ready, we'll test the technique on another case. Maybe we'll prove something new.

Forget the vector bundle approach (as if I knew then what it was). I was introduced to Riemannian Geometry from the concrete point of view of matrix theory and inner products, then the far newer subject of sub-Riemannian geometry from a control-theoretic viewpoint. I never realized geodesics and extremal problems would be that interesting.

The technique never worked, and the problem is still open. But we wouldn't know that until three years later.

My prof left on sabbatical but suggested that we can think about it a year from now, but there was much more to know before we could tackle other methods. When I asked him what sort of topics, he told me and I chose my junior and senior year curricula.

Complex Analysis. Real Analysis I & II. Functional Analysis. PDE I and Topics in PDE. The Calculus of Variations. Differential Geometry. Linear Operators. I read some topology on my own, and tried to understand why Sobolev spaces were the right kind of object to study.

As a result I've always thought of myself as an analyst-in-training and a problem solver. Theory and theory-building don't come natural to me, and every so often I catch myself thinking in terms of computations and technical matters. Now in a different school and a different place, the rules are different and I swear I've learned the mathematics that I missed and relearned all the mathematics that I thought I knew.

I don't think that sort of education was a waste. Every time I studied something new I knew what motivated me to do so .. but later the reasons changed, and I became an academic. Learning for learning's sake.


Why Graduate School? I can't remember if there was a distinct moment when I realized that I wanted to go to graduate school in this field. I wonder if there was ever such a moment. I'm tempted to say that "it just happened."

That's all well and good, but for those of us who've gone through the onslaught of forms and applications, deciding to go to graduate school requires a fair bit of thought and effort. There is the general GRE and the mathematics GRE, the jerryrigging of presenting yourself on paper, and the search for faculty who are willing to aggrandize you to other faculty. We ponder what qualifying exams and oral preliminary exams are like, what research is like, what the job market is like ..

.. but before all that, what five more years of schooling are like.

I guess I was just motivated early on. Someone showed me a seemingly simple problem, how complicated it becomes, and how quickly it does so. Perhaps it is my pride in humanity and its capacity for progress which has led me here: some problems are meant to be solved and someone must do so. If it means that I must become a mathematician, then I shall and so be it.

I wouldn't have it any other way.

Recent Events from my "Life"

• Left town for almost a week to St. Louis, where I attended a very nice conference in honor of Al Baernstein's 65th Birthday [1]. It was much like going to a family reunion: I saw some old friends, my advisor from my undergrad days, and my current research collaborator.
  
  
• Returned to Ann Arbor with my collaborator in tow, and played the occasional role of host. Research work ensues, and some spaghetti sticks to the wall, so to speak: we have results!
  
  
• The Ahlfors-Bers Colloquium begins and on one weekend afternoon I made coffee for over fifty people. It didn't end up too bad. Paradoxically I begin to cut down on my coffee consumption.
  
  I also see old friends, and my acquaintances from separate worlds collide.
  
  
• I continue to play an occasional host, and life almost returns to normal: I scramble to read more for my weekly meetings with my thesis advisor, and during my downtime I think of research ideas that are both doable and interesting.
  
  
• the Near Future: my collaborator leaves town, and my share of the writing begins. Meanwhile I struggle with reading and learning new things, and my advisor might forgive me with my continual shortcomings.

Life isn't so bad. It's nice to come back to the daily grind. Too often during conferences I feel as if I'm not being productive, and listening to others expound on their work, paranoia ensues. I fear that I haven't done enough as a mathematician, that I haven't proven myself, and perhaps I never will.

It's fine, now. A day of steady work took care of that. I've a few new ideas and some goals from before, so it will remain a busy summer. Who knows? Maybe by summer's end I'll actually know what I'm doing! q:

[1] Actually, his birthday had been two weeks before the start of the conference.

Photos of Me, in the Office.

First day of summertime .. well, not really. Today I woke up and it was 43^oF with a high of around 54^o. I mean academic summertime: the semester is over and four months of uninterrupted time await me. Plenty of time for studying and reading and maybe a little writing and exercise, if I budget my time well enough.

I've implemented my plan for working like G.H. Hardy, but instead of 9 am to 1 pm, I opted for a later four-hour block: 11 am to 3 pm. I'm happy to say that the time flew by, and unhappy that I didn't get much done.

Rather than tell you about my day, it's easier if I show you.

Anyways, the night is young: plenty of time for more reading.

I need a better camera.

Speculations Concerning Half-Learned Lessons.

I was thinking about a few things I learned and read over the last year: among them were a few seminar talks about the Wasserstein metric space of probability measures, which seems a handy idea; for the Analysis group here at UM, I believe the motivation for learning this area was to understand a result of Lott and Villani concerning analogues of lower curvature bounds in metric measure spaces.

This manner of philosophy seems to me quite interesting: one pursues an equivalent condition in a more general context without the a priori tools to do so. In this case, the goal is a condition similar to lower Ricci curvature bounds, and the original condition requires, at the very least, a Riemannian manifold structure. In contrast, a metric measure space needn't even have a manifold structure (consider a metric tree, for instance) and may be a rather 'floppy' object.

Fascinating, how one may be able to speak of 'curvature' things in places where curvature does not exist.

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I was also browsing a book of E. Giusti's (Minimal surfaces and Functions of BV, to be exact) and he, like many others from the school of De Giorgi, formulate functions of bounded variation in a different way from how it is done in Lebesgue theory (cf Royden's Real Analysis, for example). So doing, a BV function is equivalently a function whose distributional derivative is a finite (signed) measure.

This is almost too tempting. Up to renormalization, it seems that we may use the Wasserstein theory to study derivatives of BV functions and apply these results to first-order PDE. And if some hearsay is correct (I remember hearing once that the Wasserstein space has some notion of a "tangent bundle" though it is not a manifold), then perhaps we may consider second-order PDE.

Concerning Cacciopoli sets (sets of rectifiable boundary, which uses this language of BV functions), it would be interesting whether we may write down an evolution equation (PDE) for the parametrization of the boundary. Now this is really a stretch, but what if we could use this framework to study isoperimetric questions?

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Yes, I'm speculating here. It's times like these when I wish I had a better understanding of PDE, the Calculus of Variations, Riemannian Geometry, and another half-dozen subjects which would make such speculations either rigorous or moot.

I'm also fairly certain that these are not original ideas. A quick Google search indicates that these techniques may already be in practice, but where to start looking ..?

(sighs)

F*ck it. I don't know enough .. not enough to try problems in metric analysis and geometry or in these Italian-flavored mathematics. I hate being a kid sometimes, and the only solution seems more work and more lessons, if only toward some better insights into these problems or others.

Oh well. Back to work.

π may be irrational, but it's not that random ..

I'm not feeling very mathematical, today.

I found this article through Slashdot, and thought it was somewhat mathematically interesting. Apparently there are better ways to procure random numbers than to choose from a table of digits of π.

Oddly enough, the discovery was made by physicists, not mathematicians or even statisticians. Go figure.

WEST LAFAYETTE, Ind. – If you wanted a random number, historically you could do worse than to pick a sequence from the string of digits in pi. But Purdue University scientists now say other sources might be better.

Physicists including Purdue's Ephraim Fischbach have completed a study comparing the "randomness" in pi to that produced by 30 software random number generators and one chaos-generating physical machine. After conducting several tests, they have found that while sequences of digits from pi are indeed an acceptable source of randomness – often an important factor in data encryption and in solving certain physics problems – pi's digit string does not always produce randomness as effectively as manufactured generators do.

"We do not believe these results imply anything about a pattern existing in pi's number set," said Fischbach, who is a professor of physics in Purdue's College of Science. "However, it may imply that if your livelihood depends on a reliable source of random numbers, as a cryptographer's might, then some commercially available random number generators might serve you better."

More of the article can be found here.

Not much substance; looking ahead and behind.

Today I cannot think of anything pleasant or insightful to write about my current endeavors in mathematics. A take-home final in Probability takes most of my attention (and this blog post takes the rest), but during the occasional undisturbed moment I think about how I should spend the forthcoming summer days.

I must be more serious and disciplined, for one thing. Having an advisor, I have much more to do than ever, and if I don't set a structure to the weekdays then I fear that I will accomplish nothing. That being said, I need some time in front a desk and without disturbance or distraction .. at least for part of the day.

I thought in the same sporadic manner until I remembered something, and subsequently I decided that for the summer I should follow something like G.H. Hardy's example (see below). I will insist on working diligently for 4 hours each day in the office (perhaps the hottest part of the day, 11 am - 3 pm, if only for convenience) and the rest of the day I will allow for variation and whim.

Perhaps I will do nothing more than those proposed hours, perhaps not. At the very least I will attempt to keep a solid, intense focus during those four hours. I think this will make me more efficient than working idly all day; doing otherwise slows down the pace of my contemplations considerably.

Here is an excerpt from a short bio on G.H. Hardy. I wonder how much of it is true; at the very least I don't think I will take up his interest in cricket, unless it helps me to understand maximal functions.

There was only one passion in Hardy's life other than mathematics and that was cricket. In fact for most of his life his day, at least during the cricket season, would consist of breakfast during which he read The Times studying the cricket scores with great interest. After breakfast he would work on his own mathematical researches from 9 o'clock till 1 o'clock. Then, after a light lunch, he would walk down to the university cricket ground to watch a game. In the late afternoon he would walk slowly back to his rooms in College. There he took dinner, which he followed with a glass of wine. When cricket was not in season, it was the Australian cricket scores he would read in The Times and he would play real tennis in the afternoons.

Hardy was known for his eccentricities. He could not endure having his photograph taken and only five snapshots are known to exist. He also hated mirrors and his first action on entering any hotel room was to cover any mirror with a towel. He always played an amusing game of trying to fool God (which is also rather strange since he claimed all his life not be believe in God). For example, during a trip to Denmark he sent back a postcard claiming that he had proved the Riemann hypothesis. He reasoned that God would not allow the boat to sink on the return journey and give him the same fame that Fermat had achieved with his "last theorem".

Another example of his trying to fool God was when he went to cricket matches he would take what he called his "anti-God battery". This consisted of thick sweaters, an umbrella, mathematical papers to referee, student examination scripts etc. His theory was that God would think that he expected rain to come so that he could then get on with his work. Since Hardy thought that God would then have the sun shine all day to spite him, he would be able to enjoy the cricket in perfect sunshine.

Lessons, and some Phenomena of Note.

Today I met with my advisor, and oddly enough, it went smoothly. For once, I don't think I blanked out at something obvious, but I do not think this phenomenon will last. After all, there's no reason to believe that my level of understanding in mathematics should be, say, a nondecreasing function. (;

It still feels strange to say that I have an advisor. It almost feels like I lucked out, though by no means is it easy. There is indeed much to do and much to learn, and from the looks of things I fear it will be a long time before I'm ready to attack research problems in this area of geometric analysis.

On a side note, today I did learn about two separate and interesting phenomena in analysis. I apologize for any forthcoming lack of rigor.

Measure Theory

In the context of collections of pairwise-disjoint balls { B } whose union has finite measure, then their dilations { λB } for a scale λ ≥ 1 cannot have "very much overlap." In particular, one may form a (weighted) counting function from characteristic functions of these balls, and
its exponential integral must be finite.

Looking at it intuitively, my prof described it as having infinitely many leaves on the ground in autumn (which may overlap in some way), yet poking a pointy stick into the ground always gives you only a finite number of them.

Partial Differential Equations

One may study the Poisson equation Δu = f for a given p-integrable source function f, and it is a fact that u must lie in W^2,p, the Sobolev space of second order weak derivatives. This is amazing, because the Laplacian operator Δ only uses the unmixed second derivatives (d / d x_i)² and says nothing about the mixed partials!

Moreover, the approach is itself interesting (and a little off the wall): one argues this by using a substantial fact about Sobolev spaces of differential forms -- in particular, a function (or in general, a k-form) can be expressed as the sum of the exterior differential and codifferential of two k+1 forms of a k-1 form and a codifferential of a k+1 form. Now using a characterization of Sobolev spaces of differential forms, this immediately gives the right regularity for u.

Amazing: to use differential forms in a case where none are seen (PDE, no less) and they grant you the regularity that you might heuristically assume.

Simply amazing.

Love those Vector Fields ..

Edit (as of 18 April 2005): It's been brought to my attention that AIM stands for Applied and Interdisciplinary Mathematics. Apologies to all the AIM people out there.

Also, don't be fooled by the number of comments. One of them is mine.

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Yesterday there was a seminar talk hosted by the Differential Equations people, but also invited were the AIM (Applied and Industrial Mathematics) folk and the Several Complex Variables crowd: a rare nexus of areas. In retrospect the talk wasn't applied at all, but being that many of the AIM folk do study differential equations, perhaps this isn't such an interesting intersection of interests, after all.

At any rate, the speaker addressed the question of (local) hypoellipticity of a particular differential operator (the sum of "squares" of prescribed vector fields) and recalled a result of Hoermander [1] and the bracket-generating condition on the vector fields. Solid, terrific stuff! This sort of study, among others, vindicates PDE from what numerical methods and applied ends have done to its image; it makes for interesting geometry and analysis, which I've ranted about, since my beginnings.

To summarize, the speaker recalls a key step in Hoermander's celebrated proof, in particular the use of sub-elliptic estimates: they allow control of derivatives to induce the hypoellipticity property. However, he soon demonstrates an example where a set of vector fields do not give sub-elliptic estimates, but through which hypoellipticity occurs anyway!

I was ready to be pummelled by an abstract construction, but surprisingly enough the example was simple to state: one takes the Heisenberg left-invariant vector fields and gives them a "contact" factor (a power of z or its conjugate, in terms of complex numbers) and supposed it does the job. The outcome looks like a strange "composition" of Grushin-type and Heisenberg-type vector fields, kin to what one would expect in the articles of Montgomery and/or Sussmann in the Sub-Riemannian Geometry book.

A strange coincidence.

In short I was floored by this talk, and excited, besides! Just when you think that everything's been done with Hoermander vector fields, someone breaks the dam and a new world of questions flood forth.

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It did feel a little weird: looking around, I didn't see many others taking notes, as I was. I was also sitting next to a prof of some distinction, and I could swear that he was glancing at what I was writing [2]; so for a while I did my best to write only intelligent and insightful things on the page.

But that only lasted so long, before I resorted to jotting down ideas that I understood and reminders on what I didn't and what I'd look up at later times. When the talk ended and the announcement came for dinner with the speaker (not immediately, but in two hours' time), I thought a little about it before I realised something.

I was the only student left in the room.

Then I pictured myself later, in a hypothetical evening, the only student and silent amongst profs sharing pleasant conversation and insightful ideas. I imagined this going on for hours, until inevitably someone would ask me why I came to dinner. I wondered if I could explain myself properly, or more generally if I'd be any good at being such company.

Shuffling my papers, I decided against it. Besides, my fellow second-years would likely be planning an Ultimate Frisbee game soon enough, and so I went. Despite our love for certain things, we go where we feel most comfortable .. [3]

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[1] "Hoermander" is spelled with an 'o-umlaut' instead of an 'oe,' but I don't know my ASCII codes that well. \:

[2] I'm still getting used to writing on legal pads, and I find it easier to use them when the pad is upside down (the short side with the free page edges is on top). The prof might have been staring because of that.

[3] .. even if it means that my team lost 3-8 and I couldn't break away from my defender the whole time, for example.

On asking and answering questions.

Today my mathematical studies move slowly. I just feel less mentally capable than on most days (on which I blame an exceedingly late Friday night). The best I can do is remember those lessons after meeting with my advisor.

I walked in two weeks ago to ask about an exercise from his book, and two meetings later we've discussed enough facts and made enough observations to solve it. (Rather, having written the book, he referred to useful facts and I did my best to think quickly and understand the logic and line of thought. The remainder of the time involved my struggling to explain something and his patient waiting.)

Looking back through these notes, I'm amazed. It doesn't appear easy to write down this question in its present form, because it is so condensed. It's done to the point where it can be stated with simple ideas, and without a hint to the level of sophistication needed to solve it.

There is a good deal to remember .. the toil from honest lessons and facts that I wouldn't have thought to check .. and today I've written them down. It gives the day and night purpose at least, even if I have no strong mental capacity for new and creative things.

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At the very least this experience agrees with the popular belief that it is far easier to ask a question than to answer one. One can even reduce this difference to how language works: take subsets of words that everyone knows, say

{who, what, where, when}
{run, make, eat, sleep, sit}
{sky, wind, dirt, water, apple, food}

One selection of words gives: "Who eats (an) apple?"

It's a very simple question, and to answer it, we search the catalog of our memory as to what creatures like apples. It takes some thought, but the process is mechanical and a computer can do the same.

Another selection is: "What makes (the) sky?"

Does this mean the colors, how many, and how they are mixed? Does this refer to physical chemistry and the interactions of gases? Or is this a philosophical question, concerning the origins of the earth, and ourselves?

Sometimes it is a matter of interpreting what a question means, and it is a separate process to understand what can be used to answer it. The second primitive question is not so primitive: parsing it, it's not very clear what is involved in the answer or in this form, if it can be answered.

Every so often I wonder how we know anything at all.
It's a confusing world, and maths are confusing too.