Tuesday, February 28, 2006

ramble of this and that.

I'm currently on holiday: this week is Spring Break @ U of M, which still strikes me strangely. Why call it Spring Break when it occurs during Winter Term?

I suppose it would be too disheartening to call it Winter Break, or perhaps too ambiguous. Or maybe it is honesty, because perhaps the administration knows that all the undergrads will travel to warmer climes, and have a proper spring there.



As usual I brought my work with me on the plane and here in California, which suits the occasion: my friend still works during the day, and I try in vain or pretend to work on my maths while he's away. Then on evenings, we set the work aside in favor of some diversion.

It works very well, until I ask him how his day went, and then he asks me mine. Sometimes I wonder whether my friends out in the "real world" think that I live some sort of "double life." One acquaintance whom I met in Manchester, NH, put it this way:

we'll never get to know that side of you, will we? that side of you which is pronounced for a large fraction of every working day of the week. you can understand me because what i do is easy to say, but we mightn't be able to reciprocate to you.

I didn't have a good response to this, but I suppose it's part of the territory of academics and whackos. q: There's a reason why we call it "theory," becuase what we study is not immediately natural to daily life. We suppose and ponder the possible or deem what is impossible. Asking someone to practice this rigorous and technical mode of imagination is a hard thing to ask.



As for the actual work, I've recently been browsing some of the literature on traces of Sobolev functions (cf. Adams's book). There's a fair bit of abstract machinery at work, which makes sense, I suppose.

For those who don't know, a trace operator on the class of Sobolev functions (on a bounded, regular domain) is some method of examining function data on the boundary of the domain of definition. This is inherently troublesome, because the boundary of a regular domain is (n-1)-dimensional and hence a null set w.r.t. Lebesgue n-measure. Since Sobolev functions are defined up to null sets, it's amazing enough that you can say anything on the boundary.

The price to pay is this abstract machinery. In Adams there are norms and Banach space constructions all over the place, and if I recognise it rightly, the context is given as some Bochner integration. It's not as abstract as other fields, say modern algebraic geometry, but I can't envision very easily the right pictures for Banace space theory.

Even trained as I've been, some modes of abstraction still don't sit well with me. Vector bundles are cool, but I never said I know how to use them. Banach spaces in functional analysis should not be taken lightly; in that very realm one encounters the Hahn-Banach theorem, which is a mystical hammer in its own right.

Maybe I've just been thinking too geometrically, as of late. It's easy for me to forget that thinking mathematically needn't always mean thinking geometrically; at the very least, my number theory friends have convinced me of that.

Perhaps later I will write about geometry and logic, and with these, address these recent questions of mathematical proof and verification (cf. talks by Devlin and Hales).

Friday, February 24, 2006

getting older; new work to do.

As of last week, I think we've settled this extension business in the smooth category. Now all that's left is to give a talk about it. You guessed it: there's always a catch.

Always. [sighs]

Oh well. I've got three weeks to pare it down into something manageable for two hours on a Thursday, and 'twill a friendly, forgiving crowd at that ..



Lately I don't feel like a newcomer anymore, but I still feel like a mathematical tyro. It's not an uncomfortable feeling, if only because of practice. I've now been accustomed to feeling foolish, unaware, and slow-witted about half of the moments of the day. It's a feeling like getting older, where you don't improve with age and instead, resign yourself to those faults and vices that will never go away.

I'm not certain what causes this sentiment. Perhaps it's the mental response to the end of this research investigation, and I'm assessing what, if anything, I''ve accomplished in my graduate career, or even during the brief years of my life.

Perhaps it's because we're nearing prospective student weekend, that time of year where we current grads meet and chat with our potential successors: our versions 2.0. I could swear that either the incoming grads are better prepared every year, or that I was woefully unprepared when first I came to Michigan.

Perhaps it is simply the timing of my third year, second semester at UM, and I realise that if all goes according to plan, then the game is half-over and the thesis nowhere near half-complete.

Most likely this is nothing new. I've been feeling old all my life, and maybe my body's biology has finally caught up.



Maybe I should say more positive things. It's too easy to resort to pessimism and disillusionment. So I guess I'll say a little about work.

It's more extension stuff, but I get to work once more with Sobolev spaces and revisit something slightly classic: the Dirichlet Problem, the Poisson extension operator, and its topological and regularity properties. In low dimensions, say 2 or 3, it might be a more viable tool than the Gehring-Schoenflies extension because of concreteness. Then again, harmonic homeomorphisms are a tricky hope: sometimes minimizing energy doesn't mean preserving mass, and in general, topological embeddings of space may be much to hope for.

Well, it's work. If I've learned anything as a grad student, it's that steady work is something to be appreciated and sought out, because good ideas are rare in their happenstance.

Saturday, February 18, 2006

article post.

i found these off the blogs of a few online acquaintances i've beFriended on livejournal. one of them concerns computer science, and both are on "science" news archives.


This article can be found at phyorg.com and this excerpt discusses a prejudice about cs that i've heard before.

(To clarify, the article seems to be in a Q & A format, but there are no Q's or A's anywhere. The responses are from Bernard Chazelle, professor of computer science at Princeton University.)

comment if you like; i still haven't formed an opinion about this one.

Princeton professor foresees computer science revolution

.. Computer science is not just about gaming, not just about the Internet. Computer science theory offers a profound window through which to view the world. Computing promises to be the most disruptive scientific paradigm since quantum mechanics. It will transform society in profound way.

Isn't computer science really just a stepchild of mathematics?

As the recent breakthroughs on Fermat's Last Theorem indicate, the field of mathematics has never been more fertile with new ideas. Mathematics is original and deep, but it does not force you to think differently. If a math giant from the past –- someone like Gauss – were to come back to Earth, he would have a lot of catching up to do but he would find that math is done much the same way that it was done during his life.


i expect some outrage from a few mathematicians about that line. under a particular interpretation, it almost sounds insulting.

Computer science, by contrast, is a new way of thinking, a new way of looking at things. For example, mathematics can't come near to describing the complexity of human endeavors in the way that computer science can. To make a literary analogy, mathematics produces the equivalent of one-liners – equations that are pithy, insightful, brilliant. Computer science is more like a novel by Tolstoy: it is messy and infuriatingly complex. But that is exactly what makes it unique and appealing -- computer algorithms are infinitely more capable of capturing nuances of complex reality in a way that pure mathematics cannot.

.. An algorithm is not a simple mathematical formula. It is a set of rules that govern a complex operation. You can look at Google as a giant algorithm. Or you can think of an economy or an ecological system as an algorithm in action.


i'm curious what prof. chazelle thinks "mathematics" is.



the title of the next article isn't what you think it means, but i admit, it spurred me to read it. you can find it on the new scientist website.

Hand waving boosts mathematics learning

Gestures that complement rather than simply illustrate verbal instructions can boost children's ability to complete problems in mathematics, researchers report.

.. Goldin-Meadow and her colleagues gave 160 children between the ages of eight and 10 a set of mathematical problems to solve. The students were randomly assigned to receive either verbal instructions alone or also with gestures. Those in the latter group either received gestures that copied or complemented the spoken guidance.

.. Children who saw the complementary gestures did best, solving three of the four addition problems correctly, on average. By comparison, those children who witnessed simple illustrative gestures typically solved fewer than two of the problems correctly. And students who received only verbal instructions solved only one of the four problems correctly, on average.


thinking naïvely about this, i suppose this method could help the learning process by incorporating a more visual method of understanding arithmetic. then again, teaching is subjective: can you really incorporate gestures and cues without changing other things in the lesson? were the examples done in class the same?

murky waters. such is the brain, and it amazes me how much we can learn.

Tuesday, February 14, 2006

lull in the storm.

i should say first that i have no well-defined measurement for what constitutes a 'good maths day.' despite the sappy malevolence of the holiday, today was nonetheless a good maths day.

my ideas seemed to have good flow, and i wrote well; it might be a little strange to say the latter bit, but anyone out there who's had to write up several proofs for a problem set will likely understand that usual conflict of ideas and written words. there seemed little or none of that, today.

progress was going so well that i completed a full thought, which in this case meant three pages of condensed work. i thought it wise to leave the office and the coffeehouses, not to do any more thesis work or try doing any of my side research. instead, i left to lift weights, heard a good seminar talk, and committed to some light reading at Borders Books and Music for an hour or two before heading back home.

there must have been other things in between, but i forget them, and they are not worth remembering.

i think i'm finally close to finishing this step in my research.

if my work is correct (which i'll spend tomorrow night checking, inevitably) then there are two last things to solve, and one of them should be sufficiently general that i can likely look it up.

it might be over, soon enough; then i can jump back into that frustration called the unknown, and scramble to find new results. i like to think of it as fighting the good fight. q:

Monday, February 13, 2006

a few accomplishments, two of them mathematical.

i felt quite proud of myself this weekend. among other things ..
  1. I chose not to attend my usual Student Geometry Topology Seminar but instead dared the AIM Seminar (Applied & Industrial Mathematics) because I thought I would see a talk on wavelets; the abstract did say something about image segmentation, and it seems one of the usual suspects, after all.

    Instead, I was pleasantly surprised to see in action the spaces:

    • W1,10 (the vanishing Sobolev space)
    • BMO (functions of bounded mean oscillation - they even cited work by J. Garnett and P.W. Jones, on dyadic stuff!)
    • BV (functions of bounded variation)
    • and their dual spaces
    • and order-1 distributions (i.e. the distributional divergences of functions from such spaces)
    It was like seeing old friends again, and more so than I mean. Whenever I see old friends, there is little time and it runs short: so it was with these nice function spaces. The speaker discussed them only in the last twenty minutes or so, and the talk ended late because of an excess of details. In the meanwhile, I wonder how many in the audience understood the context of these functions .. \:
  2. I fled the country with friends and had a pleasant time in Windsor (the city in Canada across the river from Detroit), although I felt a little old, since the drinking age over there is 18 and the bars and clubs were full of young whippersnappers and other creatures of the night ..

    .. and despite that tempting environment, I did adhere to my axioms.

  3. Most of all, last night I thought about a problem of my own, and I was able to solve it with a covering theorem!

    Granted, it wasn't the 5B-Covering Theorem (I don't work in spaces of that generality, usually) but Vitali's Covering Lemma in good old Rn. I felt like something I learned actually stuck and was made useful .. as if my brain's cortex has been worth something, after all!

    Let's hope it remains a proof tonight, and wasn't the outcome of an overexcited mind. \:

Friday, February 10, 2006

i will make a solemn, cynical vow.

i will not write recommendation letters for calculus students.

what is there to say? calculus is too basic to form any assessment on a student.

if the student studies science or engineering, then (s)he should have completed her/his calculus coursework and should instead ask their differential equations or linear algebra instructor for her/his coveted letter.

the content of those later courses might be advanced enough to form an opinion of whether the student is any good.

you may ask: what if the student has just learned calculus I or II, and hasn't yet taken an advanced course?

should they really be applying to these research programs, if they are not yet prepared?

or you could ask: what if the student's program doesn't require more advanced mathematics?

if the program's content isn't mathematically heavy, then should you really be asking for a letter of recommendation from your mathematics instructor?

in fact, half the days of the week i wonder whether the concept of an american "undergraduate education" is really education, or if it is pre-job training (read: hoop-jumping and busywork) and a fiscal investment (read: high tuition).

don't mind me, i guess. bitter thoughts prevail quite easily in me.

Thursday, February 09, 2006

thursday night routine.

as i feared, the study of the smooth category continues. this is the sharpness result that will never go away; i fear now that it will haunt me until past my fifth year!

but it's thursday, after (1) my research meeting with the advisor, (2) the Analysis Study Seminar, and (3) a few hours at the Brown Jug. if my motivation levels were low last night, then now they have reached a singularity towards -∞. it doesn't help that i should draft and finalise a recommendation letter for a former student of mine.

i haven't attended the post-AnSS Brown Jug excursions for a while .. not since my prelim, now that i think about it. maybe it took that long for me to convince myself that my committee didn't take my poor performance personally. it's funny how irrational fears can dissuade and discourage us.

also, since last fall, some visiting profs had returned to their home universities and some of the postdocs left this term for professional opportunities. the table feels emptier, which is risky for me ..

.. now i have to be witty and entertaining! q:

Wednesday, February 08, 2006

mental inertia.

one of these days i'll determine some dependable way of convincing myself to work. naively it would seem easy, since i have no other life. paradoxically enough, mine is the life of a lazy workaholic.

as of now, i've hit a work lull and in danger of lulling the rest of the evening away .. which is bad: if anything, i have to check whether a mapping i've constructed is continuously differentiable, and if so, then it might be the final nail in this gluing manifolds chapter of my thesis work.

then again, i've said that before with some tone of hope, only to have it dashed upon jagged and piercing rocks .. so let's not get too excited here.

the best solution on the table, thus far, hearkens to old habits: i still work well in coffeehouses, and have done so since my undergrad days.

to think: when i was a wee undergrad, all i really wanted was an office. in retrospect, this is kin to a child's wish to be like the bigger kids (read: grad students)

i suppose the price of asking for something you want is getting just that. it almost makes you want to cede away your desire and wants.

but anyways, i've spent too long in the office and the thought of going home is not often a happy one. how that came to be is unknown to me.

i should get to work, and so demanding, i should resolve these petty matters. tomorrow is another meeting with the advisor, and the glimmer of hope (that the end is near) is too bright for me to sink into lazy oblivion .. at least not tonight.

all right. time to verify the smoothness of a mapping.

looks to be a busy summer ..

.. full of conferences!
April 7-9 @ Minneapolis, Minnesota

The Ninth Rivière-Fabes Symposium on Analysis and PDE

Invited speakers: Char1es Fefferm@n, Wilhelm $chlag, Isabe11e Ga11agher, Alexandru I0nescu, Dieg0 Maldonad0

May 8-12 @ Ann Arbor, MI

Modern Developments in Geometric Function Theory:
Workshop for Graduate Students and Postdocs


Confirmed speakers: Mari0 B0nk, Alexandre Eremenk0, Juha Heinonen, Tadeu$z Iwanie¢, Steffen R0hde

June 19-23 @ Będlewo Research and Conference Center (Poland)

Analysis and Partial Differential Equations:
In honor of Professor Bo9dan B0jarski


Topics: Quasiconf0rmal Mappings and Related PDEs, Function Spaces, Distributions and Measures, Pseud0differential Operators and Index Theory, Riemann-Hi1bert Type Transmission Problems, Fredh01m Pairs, Complex Methods in PDEs, Calculus of Variati0ns and Null La9rangians, Ana1ysis on Metri¢ Spaces

July 12-15 @ Urbana-Champaign, Illinois

Conference on Geometric Analysis and Applications

Confirmed speakers: Zoltan M. Ba1ogh, Anthony B1och, Mario B0nk, Roger Br0ckett, Der-Chen Chan9, Monique ¢hyba, Giovanna ¢itti, Michael Cow1ing, Selim Esed0glu, Bruce K1einer, Adam K0ranyi, Pertti Matti1a, Jean Petit0t, Alessandro Sarτi

Thursday, February 02, 2006

student seminar title/abstact

(cross-posted @ the QC Mappings on Fractals blog)

to all other interested parties;

I've submitted an abstract for UM Student Analysis Seminar this week, and the format will be a little different from previous times and traditional seminars.

This is a call for open problems: they don't need to be high-powered and result in a publication, but an opening to stimulate thought. The goal is to raise interest in varying subfields in analysis, and possibly lead to collaborations amongst us students.

So, anyone know a good problem? q:



Tues, 7 Feb 2006:
Title: Open Problem Session
Speaker: none
Abstract: Interested parties will bring unsolved problems (related to mathematical analysis; possibly research level) and we will spend the hour discussing them, and hopefully make progress in solving them.

When choosing problems, we would prefer that the background not be too lengthy; problems of a general background (e.g. complex analysis, measure theory) are encouraged.

after the dust settles ..

.. a technical issue remains in the latest edition of my thesis work. i don't see how to reconcile one bit.

it's not the collars or the tubular neighborhoods, the isotopies or the vector bundles. none of that abstract machinery is bothering me at the moment, which is surprising enough in itself. the issue actually arises from ..

.. the existence and uniqueness theorem(s) for systems of ODE.

yes, ODE. argh .. to reach this far and get stuck on ODE? [1] well, not explicit ODE; i mean the theory, but still ..



to clarify, the standard Picard Iteration theorem uses a contraction fixed-point theorem in the setting of an integral operator. boundedness of the source function for the ODE isn't enough; the function must also be Lipschitz in the "dependent variable" to make the right estimate for a contraction.

it would mean that the flow arising from the source function be a differentiable mapping with Lipschitz derivatives (C1,1-regularity). i don't want that kind of regularity, and neither does my advisor.

despite my usual inclination, i honestly hope that i've made a mistake and overlooked something .. perhaps the source function arises from something which permits the Lipschitz property so that we needn't assume it ..

.. ye gods. i just want to finish this so that i can get back to forming new results, play around with my beloved sobolev spaces. is that so much to ask?



[1] strangely enough, my other research will likely and soon encounter ODE troubles of its own .. the ugly nonlinear and non-closed form type, i fear.

this, of course, is provided that i get around to settling details about selective Steiner symmetrizations.

a boy's work is never done, is it?