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.

Topology Realised?

Immediately after reading this article, I immediately desired to see photos of this phenomenon. I wondered whether it could demonstrate a deformation, or even better, a deformation retract from topology.

And then I wondered if this plastic only demonstrates phenomenon concerning elasticity. Must it behave like a homeomorphism? Can it change the genus of a plastic surface, or kill injectivity?

Curiouser and curiouser ..

Yet another Grad School Rant.

This originally began as a response to someone else's blog (the person in question has been accepted into graduate school and has finalized a decision), but it quickly mutated into one of my long rants. Specifically, I replied to the blue font below. Be warned; again I'm speaking loosely and my opinions are splattered over a wide canvas. (;

This is the beginning of the rest of my life and it scares me to focus so hard on one thing, and it scares me to be able to make statements like "well, I won't be done until I'm over 30" and have it be utterly reasonable and accurate.

I'm contentedly convinced that I don't know much of anything anyway and that this will be a recurring theme as I pass through my twenties. It wouldn't matter if I were a student or not; I still won't be certain of very much, so I might as well be uncertain while studying subjects that few others will ever have a chance to study.

That's a fact about graduate school which warrants some inquiry. Most people are able to be students only for a short time in their lives, and this is fine for those people who are happier pursuing other things. There's nothing wrong with that; it is simply a matter of preference.

But if you're the person who is curious about certain questions, and if you have a driving need to understand the heart of matters in a formalized and systematic matter (as in academia), then graduate school is one opportunity to practice this in your area of interest. This not a freely given opportunity, for that matter; it has to be sought out. There are applications and fees, plenty of hoops to jump, and only a few available positions. Those who gain the chance to study do so with a bit of luck.

Concerning time, if you are such a person then for all intents and purposes you will never be done, because there will always be more questions to raise and issues to ponder. In that light, being a student is simply a matter of coping with that fact.

I like to view it optimistically, just as how it feels to walk into a Barnes & Noble or a Borders bookstore: seeing the sheer volume of books and printed matter and media, and knowing that more comes every year, month, and day, you will never reach the end of what you would like to have read in a lifetime. But that's immensely relieving, because if you could reach the end, then there would be nothing else to learn.

That possibility scares me more than anything else: attaining an intellectual limit. I wouldn't be "me" anymore .. no more curiosity, no more long moments of contemplation, no more bouts of frustration from trying to understand something. I'd even miss those. \: [1]

I'm both lucky and happy to be a student, so I'll deal with the repercussions. Some days I'm convinced that I've been dealing with them for most of my adult life.

To make a long story short [2], I don't believe that the end of graduate school marks a time that I will be "done." It's not the pinnacle or paragon of learning, and from observing post-docs and faculty, it seems like only the beginning.

The end of grad school does seem the end to one stage of life, and the meanwhile can be long and unforgiving. It could be that after a year or two years, you find out it's not for you. Alternatively, it could be that you've found your true calling and this is now the means to attain it. Either way, being accepted into graduate school means that you get a chance to try; it's more than what many people get.

Good luck, and be happy. This is a beginning, and in it is all the potential than you can imagine.

[1] On that note, omniscience is one of the last things I would ever ask for. I suppose that means that I wouldn't ever understand the mind of God and I don't care to. At any rate I'm still convinced that gods of any sort don't exist, so the last point is moot.

[2] Ironically, this was meant as a comment to a short post on a friend's blog. In a matter of speaking, this is the summary of my having made a short story long.

surprisingly calm and collected

Currently I'm stuck on a problem, which is in fact an exercise from a book. Provided that it is a good book (which this one is) then exercises from the text do serve a purpose: solving them, your mind raises issues that it wouldn't have otherwise pondered or considered.

As of late I believe that I'm terribly single-minded by nature. It is hard enough for me to read and follow an argument, let alone keep in mind these subtleties and concerns. I suppose, then, that exercises are good for me; they keep my skills of comprehension sharp, and if they are dull then they sharpen them.

But this can be the very worst situation of all: to be stuck on such an exercise. I know very well that there is a solution, and in this particular case it is likely something that is very easily understood after you see it, and just as likely I will curse myself for not having seen it sooner.

As you can imagine, this happens far too often for my liking. \:

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I'll admit that I'm frustrated, but it's far from the worst bouts of frustration that I've encountered. I'm handling this one reasonably well, in that I'm not taking it personally. I know I'm not an idiot for not knowing how to solve one exercise, though I do feel like one.

If only because of this, I'll say that it's not a total loss and that I'm learning something. Odd, though: usually I'm more temperamental than this, so strange as it seems, I remain wary of my own motives on the matter.

What's really going on here?

Looking into the forthcoming months, the answer becomes clear. It's a matter of sanity. I'm pacing myself.

This isn't going to be the last exercise for a while. There are more and likely many more: I'm only on Chapter 2 of 15. If I start upsetting myself and lashing out in anger and hate, then (1) I might as well prepare myself for a coronary by summer's end, and (2) become a Lord of the Sith and start killing all those goody-two-shoes Jedi in their stupid tan robes and their oh look at me 'cause I'm so calm and centered routine ..

[remembers that this is real life]

Er .. let's just consider that first possibility, then. (:

There's a long road ahead, and I've still remember my lessons .. if only from my days of distance running. If anything a journey, however long, is simply a matter of patience and pace.

Let's hope I remember that in the months to come.

A work of fiction, I trust.

Surfing through the web out of boredom and laziness, I stumbled onto this article. It must be some sort of joke. After all, what are the odds that a Christian community rallies against a teacher named Scopes? (;

Excerpt 1: Two weeks ago parent volunteer Holly R. Thanthow took a break from passing out Jack Chick tracts in the school's courtyard to visit her son's fourth period class taught by newcomer John Scopes. What she saw there shocked her to the core.

"He asked them to find the area under a curve- without using cubits at all. I raised religious objections since the lesson completely goes against Genesis 7:20, but he said my son had to do the work anyway. When I asked Mr. Scopes to refrain from teaching from his so-called 'math textbook' he flat out refused. I know, I couldn't believe it myself."

Excerpt 2: Matters came to a head a few days later when Scopes reportedly presented an equation and asked the students to 'solve for the unknown,' a direct affront to parents who had spent years telling their children about the unknowable vastness of God only to have some big-city joker tell them that they can simply solve it with his demonic number magick.

Anyways, back to work. I've a class presentation due Monday, and with any luck I can understand both the proofs and the motivations by then.