Pas du tout ..

"La plus de change,
la plus de meme chose.."
~ Marcel Proust

Ten months or so have passed since I last visited New York. In late February 2004 it was a mathematics conference at CUNY Graduate Center, which also made for an opportunity to see companions in the NYC area.

So here I am again, doing the same .. almost. Perhaps it's not wise to mix work with pleasure and down-time.

Following the advice of friends far wiser than me, I'm setting down the pen and chalk for a little while. I'm taking a break from thinking about maths until the Winter Term @ UM begins anew in January 2005, though I prefer to think of it as a little "hibernation." (;

(Disclaimer: this does in fact contradict my previous considerations, in view of my post on 21 December '04. No one, however, has ever accused me of being an absolutely consistent entity.)

It will be hard not to sling the backpack of papers and legal pads onto one shoulder as I make way to the coffeehouse. Perhaps two matters of circumstance will save me:

1. My companions over this break (being my siblings) and I have many things to catch up on, which wil provide a good distraction;
   
   
2. Other than Starbucks, there are very few 'good' coffeehouses in the suburban sprawl of Long Island, so the question is moot.

The battle for self-control continues.

A Fine Paper (not mine!)

I was having printer troubles a few days ago in Ann Arbor. As I scowled at the page of feebly printed text (the toner cartridge was low), a friend and fellow second-year looked over and shook his head.
"You might want to try taking out the cartridge and shaking it around before putting it back. Sometimes that lets me print for another 30 pages or so."

Squinting at the lightly printed words, he gave me a quizzical look.

"You know that our break is only ten days long, right?" he asked.

I nodded cautiously.

"Do you really want to spend it by reading math papers and ~possibly~ getting confused and/or irritated by said papers?"

"Oh, no. I know one of the authors of this paper," I replied, waving the printout in the air. "Very good mind, his! It should make for good reading."

"Your time off, your decision." he said, giving up.

"Well, it can't be any more painful than my relatives asking me: 'when are you going to get a real job?'" I added.

He grins. "Good point."

---

It is a fine paper, and you can read it here: [link].

If anything, it demonstrates clarity of thought. It discusses very well the motivation and connection between the 'classical' theory of QR (quasiregular) mappings and this more substantial subclass, called QRG (quasiregular gradient) mappings. Their example of a radial stretch is also rather illuminating.

Now that I think about it, this QRG is a very nice and natural object to study, as well as a startling intersection of many ideas. Within it, there is:

• this aforementioned Theory of QR/QC Mappings;
  
  
• a nod to Differential Topology, because these maps are complex-valued gradients (to be understood in the weak sense), and hence an analogue to exact 0-forms;
  
  
• an undercurrent of Elliptic PDE, which arrives purely from the characterization of the primitives of these gradients;
  
  
• even a little Fourier Analysis, although it was introduced in the examination of homogeneous QRG mappings and the testing of a conjectured optimal Holder exponent.
  

Perhaps my only disparaging remark about this paper is that the authors make a conjecture in Section 4, but in their Concluding Remarks they admit that the conjecture has been solved .. by one of the authors, no less!

I suppose that this is not so much a technical fault but a personal idiosyncracy of one of the authors. After all, who can fault someone for answering a question?

At any rate, this paper is good mathematics. I only wish I can write so well .. or rather, come up with such good results!

Goals before the New Year ..

This is the end /
My only friend, the end
Of our elaborate plans, the end
Of everything that stands, the end
No safety or surprise, the end
~ "The End" by the Doors

Everything's winding down.

After a week's stay in Ann Arbor my collaborator and I parted ways for the holidays; he left this morning to see family in Wisconsin. I'll be doing the same tomorrow, but instead to New York. For the second time today it occurs to me that I haven't been on Long Island for over a year.. I'm not certain how I feel about that.

The paper is officially done; it's just a matter of waiting for copies to be printed out, and then I can mail the article to the journal editor. After that, it's out of our hands: done, and done.

Someone told me today that the Winter Holidays for UM are only eight days long, and after he says this we both shudder. That's too short a time, I think to myself. There's so much to do in so little time. Too often I promised myself that as soon as classes were done, I could think about topics of research and all the reading that I meant to do, if only to enrich my understanding of this term's coursework or the odd topic that looks interesting.

Eight days? I can't do it all in that time.

As a result, I've done my best to select a few goals:

• to rewrite my Quasiconformal Mapping notes. I don't know this topic nearly as well as I should, or more aptly, as well as an analysis student at University of Michigan should.
  
  As yet, QC Maps seems to me only a pretty theory, but there must be a reason why the UM analysts are so interested in these objects. What is the motivation, and how are QC mappings used in "applications" to further other theories?


• To work on extremal problems on CC spaces. This is some remaining work from last week's conversations; at the very least, this gives me a good excuse to learn the Calculus of Variations, and how potential theory comes into play in an actual setting.

Let's hope that times works out.

Not the Worst-Case Scenario ..

.. but sufficiently bad, and I can't rightly say much about it. The prof for one of my classes has e-mailed me my grade, and the only certainty which I can say is that I'm glad that I checked my e-mail after I had hosted a small dinner and get-together tonight.

Had the order been otherwise, I would have been a rather poor host and the dinner not very entertaining. It's a small matter, yes, but important: I'm a firm believer that commitments are, in their own way, serious promises and despite convenience or adversity, one must always be consistent and steadfast to them.

If that is so, then what about my commitments to my coursework? As a graduate student, haven't I made an implicit promise to perform well as I learn the subject matter of courses and lectures? Perhaps I have, but a few marks and letters on paper would say that my commitment has faltered.

This semester was red marks and low scores on homeworks, all throughout my classes. The grades give little impact on my future, near or far, but they confirm my long-standing suspicion, that something is amiss in my graduate education and career. I can't point out with confidence what is wrong, or what is the true cause for this performance, but I can make a few guesses:

• It could have been a post-Quals slump. Having passed my exams this past fall, I fell into the trap of relaxing a little: flipping back to the problems and research topics that had interested me before, and spending time to examine them when I could (or should) have been studying for classes and doing problem sets. I fled campus every so often to attend conferences and to pursue these matters, leaving those student responsibilities behind.
  
  
• It could have been my anti-social attitude once again, or rather my perfectionist attitude gone awry. Having gotten behind in my courses (for whichever reasons) something in me refused to speak freely with my classmates and inquire about missing entries in my class notes or hints and suggestions for a late problem set or two. My responsibility this was, and mine to bear; honorable principle would tell me not to seek undeserved help.
  

Poor judgment, some apathy, and foolhardy choices: I don't know.

Thinking about it now, I'm not sure what I'm doing, or if the road I'm heading has a definite and secure end in sight. I do get the feeling that I'm not "playing nice with others" and have become something of a deviant, even amongst the analysis students in the department. I don't feel like I fit in anymore, which may already be an easy thing for a math grad student to say ..

.. but watching the general mood and attitude of my peers and fellow students, I begin to wonder why my goals seem suddenly divergent with theirs. Am I looking in the right places and choosing the right decisions? Is this really the correct path when I receive warning signs in the form of poor grades? What is wrong with this picture, this portrait of Michigan and me?

Aw, hell. I can't answer these questions right now. So it's nobody's fault but possibly my own and now I'm debating whether it is indeed my own; this is fodder for thoughts of another time, and I need some sleep before tomorrow's bout of grading the Calc I Final.

Addendum (3:56 am): There is some good news. I've said so before, but my collaborative PDE paper is finally done; one small change/addition to the manuscript and it will be ready to be sent off to the journal. I suppose one small good thing had to come out of all of this.

One hurdle to go ..

My last Functional Analysis problem set is done, or rather, I decree it to be done. If I ever see it again it will likely be returned with dozens of red marks per page, despite the fact that I took great pains to check the details. Certain things are inevitable, it seems .. and people wonder at the origins of my paranoia and pessimism.

One take-home exam for Differentiable Manifolds is left, then I can start reading the papers that my collaborator sent or suggested to me. More and more I find classes strange .. strange in the sense that the contents of lectures are fascinating and fine things; I'm learning a lot .. but too often I find that at any given moment there are other mathematics, specifically other topics or problems that I'd rather be doing or thinking about ..

On a somewhat related note, my collaborator is coming into town, which in an ideal setting means that we'll have time to catch up and work on the paper oft mentioned before. In the most ideal of settings, we can finally put that paper to rest and to start a new, more geometric-minded one.

Now that is motivating mathematics!

Thoughts on Geometric Analysis, and on PDE.

[After a brief experiment with LiveJournal, I've decided that from now on, my mathematics posts will be put here. See comment above for personal rants.]

Certain things in life are inevitable. For instance, just when you think that the job is done, instead it resurfaces and you must tend to it once more. So it goes with the manuscript.

My collaborator has written back and informed me of a paper, where similar results (that I took some pains to prove) had been proven already. I suppose in a way this is actually a good thing which has happened; at the very least, I was very likely proving the right things, and now there is a reference to see if my method of proof could be right.

On a different note, a friend of mine (and fellow graduate student in analysis: see here) has made progress on a Uniformization Theorem in the metric measure space context.

(I'll omit the technical details; you can bug him about them instead)

This is a fine thing, indeed. It's an enviable result in alarming generality, much like other work in that area of study: "Geometric Analysis" and in particular, "Analysis on Metric Spaces."

I prefer to think of this as topology-driven analysis, which strikes me as quite hard: hard in the sense that the methods of proof can be at times quite axiomatic and definition-driven. The geometry becomes rather intangible and there is less visual intuition at your disposal; I can't seem to make heads or tails of it.

By contrast, potential theory seems to make perfect sense. PDEs make sense. Variational problems make sense. Occasionally I wonder if I'm in the right sub-field. I may have found an academic niche, but it may also be dangerous to become too comfortable in a single thing. I may lose my powers of creativity and flexibility in the process.

Already so much mathematics doesn't make any sense to me, and what little I know is enough to confuse me. It's frightening.

I really don't know if there is a point to all of these things that I write. That's the nature and the purpose of a blog, I suppose: to settle the malaise and uncertainty in thought, by releasing them into the world. But perhaps there's more to it than that.

Seeing what goes on in my department and what my peers have accomplished, it's quite easy to feel that my own efforts are made smaller and less in worth. It still seems like I haven't done anything of consequence and that it will remain so for a long time, and meanwhile I see great things happening, all around me ...

Good News, Bad News ..

Some good news: My part of the draft is done, and sent back again to my collaborator friend for inspection. The ball's in his court for the second time around, so I'm not expecting any big changes on his part.

It finally feels like honest work. If anything, it's a reminder for me: if or when I ever become a mathematician and a proper researcher ..

ALWAYS WRITE OUT THE DETAILS!

.. or at least, give clues to the reader, as how to reconstruct the argument. It saves so much time and energy. That begs the question: must I have to feel worn out from proving little details to assure honest work from my research? Is there anything wrong with a quick and easy answer?

I begin to wonder if I have an exceedingly acute sense of guilt, or just a penchant for academic sado-masochism.

.. and now, some bad news: it's already 2 in the morning, and I have to teach in 6 hours and 30 minutes. ARGH ......

With that in mind, it feels like this entire term (up to a set of measure zero) has been bad news. More and more I believe it to be poor planning on my part, whether it is allocating time for teaching/preparing lessons or classes or homework or research or leaving for a conference.

But that 8:30 teaching assignment .. that is particularly frustrating. Remind me to plan my days for academic comfort, next year.

Thoughts as the Term Ends ..

The semester soon ends. This should be a good thing, but I'm still advisor-less, or as I like to put it, an "academic orphan." I've thought about possible candidates but still haven't talked to either of them. It either means that I'm lazier than I had thought, or more gutless than I had thought.

A month ago my undergraduate advisor had jokingly assured me that I could always come back and work with him .. at least, I thought he was joking. It was nice of him to say, at any rate.

On the other side of things, I think I'm making the most of "orphan" life.

At the expense of the chaos that is this semester's coursework and teaching, I've managed to attend a few conferences and think about a few research problems. A joint paper might be in the works, and another in the future: they will concern partial differential equations on a particular type of control manifold.

Partial differential equations. It seems like a dirty word amongst the pure mathematicians in my department .. well, except the PDE people. More and more it makes me wonder if I'm in the right place, or if I'm interested in the right things.