Conservation of Paranoia / Planning Ahead.

So I met with my advisor yesterday for the first time in .. wait, six weeks?

I'm still inclined towards disbelief, but between my advisor's one month holiday with his family and the travelling-planning issues for a week before and after, I suppose it must be right. Wow.

At any rate, we met and I showed sufficient progress, but apparently I was working on the wrong metric space. Instead of Euclidean space Rⁿ, the results were meant for its 1-point compactification, or if one wishes concreteness, on the n-sphere Sⁿ in Rⁿ⁺¹.

Thus for more than a week, I was worried about a issue that is, in fact, a non-issue: the (local) Sobolev regularity of the inversion map x → |x|^-2 x is problematic near the origin in Rⁿ, but the equivalent mapping is a diffeomorphism on Sⁿ, so I was worried over nothing ..

.. well, sort of.
Something else is now suspect.

For those mathematically-inclined, I'm trying to prove an extension theorem for a particular type of homeomorphism on particular types of domains. In my efforts to resolve the (non)issue, I began a systematic examination in how "large" a sub-domain can the old and new homeomorphisms agree in value.

This in turn led to the study of the functions' behaviors for points approaching infinity, and rather interesting limits, at that! Apparently these same limits might cause trouble for another function in a different part of the proof. It would have been fine were we in Rⁿ, but it may go wrong in Sⁿ.

I suppose some difficulties never go away, do they? (;

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On a less technical note, I gave my advisor a preprint of an article of my co-author's and mine, and he didn't seem too displeased by it. Then again, he didn't get much time to read it during our meeting, so I suppose we'll wait and see ..

We also talked a little about the future, and my previous guess now seems true: after this term I'll have gotten all my course requirements completed (both cognate and distributional), and tentatively my oral Prelim Exam is set at the end of the Fall 2005 Term.

If I pass, it would mean that I'll be a Ph.D. candidate by January. Zounds! "Candidate" has such a nice ring to it, too .. q:

a month passes ..

.. and my advisor has returned from .. where he was, earlier.

  (I never asked where he went, and he never mentioned it)

Again I am accountable for my mathematical deeds. Now all I need to do is find him and allow myself to be held accountable.

I guess it hasn't been too unproductive of a month .. but there are few results to speak of. Earlier in June, co-author and I ran into some technical troubles in the final stage of our proof ..

  (for those keeping score at home, this is the second project, not the first)

.. so now the project has been put back on indefinite hold. It only adds to my deep-rooted suspicion that as much as I like geometric problems, I'm likely little good at them and should stick to more analytical topics, such as PDE.

I did have an idea to circumvent that technical difficulty, the other day, but I suppose it has to wait until the weekend. Thesis work comes first.

Update Alert: my new mathematical webpage!

With any luck, the Fox Trot people won't sue me for copyright infringement .. but I just couldn't resist using that particular comic strip for an opening .. q:

Main changes: I've tweaked around the page layout a bit and altered the color scheme. Also, I designed a new "logo" which you can find on the upper-right corner here: it's meant as the infinitesmal picture of quasiconformality, treated as colored spheres and ellipsoids under the light of a distant light source.

Other changes: The list of UM Mathematics Seminars is still there, but since I predict that I won't be doing much traveling in the next few years (I do have a problem to solve and a thesis to write, after all), I've removed the Events / Conferences page from the list of Links.

Anyways, back to work. Enjoy the site!

Putting my money where my mouth is.

No work done today, but looking at my precedent of work hours and hours in the office, one could make the argument that "the night's still young" and "the day ain't over yet."

I continued with playing host to two friends of mine from high school, and then I played soccer with friends from the Department. Now it feels like time to work, but I can't identify whether that is my conscience urging me to work my status quo routine of hours, or whether it is something larger in scope ..

.. such as accepting and meeting my current fate head-on: having been handed a research problem by my advisor, to review the context and what is known, then to plan a strategy and execute it, baby step by baby step. I'm sorry to say that I'm still somewhat overwhelmed, and it's unclear where to start first:

• Do I read the background material more in depth? That way I can be aware of the trouble spots early on, and when they occur then I am prepared to combat them.
  
  
• Do I close the gap, and get some preliminary results done? It's not like I don't understand the problem nor misunderstand the broad strokes of its shape. Doing something might suggest to me where I should start my study of the background material ..

It's probably a convex combination of the two, or likely a feedback loop of repeating one and then the other .. and I should get to it! Now if I can only get myself to accomplish that something ..

I should keep my damned mouth shut ..

.. and stop this proclivity of wasting time on the Internet, particularly blogging and commenting on blog posts of others. In my hierarchy of must-do's, I should be:

1. Start work on my advisor-given thesis problem. Oh yes: I have a problem now, and I'm in the state of affairs where I actually understand the problem at hand, but apart from making parallels from previous results in the field, I'm at a loss for how even to approach it.
   
   Then again, if I knew how to solve it, then it would no longer be a problem, would it? q: Still, the inherent and newfound uncertainty of the matter is still unnerving to me.
   
   
2. Continue reading my advisor's book; I still feel like I don't know enough to start work in the Analysis on Metric Spaces. [1] Then again, half the days of the week I wake up and wonder if I'll ever feel ready to work in this field.
   
   
3. LaTeX my results on previous research. There are still details to check, but if I don't do them now then I'll be swamped by more immediate work: for example, the earlier items on this list!

Instead, over the last few days I've been posting on my blog, doing work for the last item on the Must-Do List, and consistently replying to a blog post about matters of Linear Algebra.

Below is the link, if you're really that keen on reading what I have to say. My comments are under the username "grey_ghost."

[from a LiveJournal mathematics community]

[1] Lately I've prefered to refer to the subject as "metric analysis" rather than "analysis on metric spaces." Less words make a single, comprehensible mouthful.

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. \:

---

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.

---

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.