Sunday, July 30, 2006

a good (but incomplete) work session, and a hilarious excerpt.

personally, i think i know why g.h. hardy was famous for having a 4-hour workday: after four hours of (intense) mathematical thinking, one quickly runs out of ideas and determination.

today's work session felt productive. i identified a few issues about some ideas i had (concerning the geometry behind a particular computation) but i've yet to carry out those ideas.

for some reason, i keep losing my resolve in working those gory details out. they need to be done, and they might clear (or tip over) the first of two hurdles at this stage of the problem. but they remain undone.



maybe tomorrow i will immerse myself in that gore. tonight, i'll read a book of m. kapovich about hyperbolic geometry and discrete groups.

so here is the promised excerpt. (p 32)

3.2 CAT(λ)-spaces

The term "CAT(κ)-space" was (I believe) introduced by M. Gromov in his essay [Gro87] and has nothing to do with cats: CAT stands for Cartan-Alexandrov-Toponogov. I think that the historically correct term should be RAT(λ): Rauch-Alexandrov-Toponogov. However the name CAT(.) is already widely used and, besides, who likes rats anyway ....

Saturday, July 29, 2006

ho, ho! my own library catalogue!

apparently it's not as hard to make my own library catalog as i had thought (in my last post). indeed, it's now available on LibraryThing.com, at the following URL:

http://www.librarything.com/catalog.php?view=grey_ghost
(or just click on the post title, above)

it's not a complete list, because i'm in the office, and half of my maths books remain at home, waiting to be catalogued.

but i'm up to 56 volumes, which is pretty cool. i didn't realise that i had so many, and many of them i still read or reference!

libraries, public and personal.

thesis work improves .. there is still much to do, but i finally made it to the library, found a book on group actions,

(the Geometry of Discrete Groups by Alan Beardon is working very nicely)

and discovered a few facts, which fit in either of two categories.
  1. "the truth is out there" (a la x-files): those that i weren't certain were true but wanted to be true,

  2. pleasant surprises: those that i didn't know existed, but do make my life simpler.
i suppose it pays to visit the mathematics library, every so often. there's more of a story, but i'll save it for another day.



despite how small it is, i'm considering cataloguing my collection of mathematical texts and using the Library of Congress arrangement system.

it's for no other reasons than sheer whim and a nagging concern that i'm missing a few books here and there. sure, i could resolve the second reason simply by refusing to lend books to anyone ..

.. but that's just mean.

Wednesday, July 26, 2006

about that miscalculation ..

as mentioned in a recent post edit, yesterday i miscalculated an integral [1] and spent an entire weekend in self-delusion. other than puzzled confusion as to why that special case wouldn't generalise, it was a rather pleasant weekend.
i suppose it's another shred of evidence for the case of pessimism.

after all, moments of happiness and other positive synchronicities lie in an exceptional set of measure zero, and almost surely, things never work out as you'd like.

so at last count, i have .. let's see: zero (0) research results to report, which isn't very many at all. [2]

as an interesting contrast, this is also the same number of results that a non-mathematician, working a regular job and earning a median-level income, would have.

having arrived at this conclusion, i decided that since i was being just as unproductive as if i weren't doing any mathematics, i might as well not work at all and have a bit of fun instead.

so last night i watched two episodes of a medical drama on the FOX channel called "House" with my flatmate, then met with more friends at a local bar (leopold bros, for those who know), and after the bar closed down, we walked over to a friend's house, ordered pizza, and played many marvelous games of foozball.

so today i sat down and thought about the problem and its parts:

  • what i thought was true (but isn't),
  • what, if true, i could use to prove something,
  • what i didn't know, but would narrow down this ever-increasing list of things to know and not to know,

and in the end, the only thing left was a simple observation i made, when working out that ill-fated miscalculation. it's my last lead, a small lead ..

.. but it's still a lead.

the mystery endures, and it's geometric in nature: i don't know enough about group actions and high-dimensional hyperbolic geometry, but i'll need to learn enough and to know enough, otherwise the trail will turn cold and i''ll have to face that dark abyss of mathematician's block -- terrible cousin to writer's block, and more terrible, i'd say.

it's going to be a long few months, i fear.


[1] to be entirely accurate, it was more like an infinite number of integrals .. countably infinite, in fact.

also, i didn't actually calculate them, either; i was doing the analyst thing and trying to bound them.


[2] depressingly enough, this isn't the lowest possible number, either.

a few months ago in a discussion with the advisor, we realised that my "proof" wasn't actually a proof, making it a day where i had minus one (-1) research results to report.

Saturday, July 22, 2006

a work session, & titles/abstracts from the arXiv. [EDITED]

edit [25 july 2006]: the reason why the computation went so well is because i was stupid and made an integration error. sorry, folks.

maybe i'm a "one good work session a day" sort of guy.

during this morning's session at the coffeehouse, i ran a computation with a few simplifying assumptions and everything worked out well ..

.. but a little too well.

looking closely, the computation doesn't depend at all on the group structure, the underlying geometry, or even key properties of the function data. as a result, it can't possibly be right.

the principle at hand: never blindly compute. most nontrivial computations require some underlying reason or "leverage" for why they should work, whether geometric or function theoretic.

i can only assume that my simplifying assumptions were too strong. i should work without assumptions first, and work geometrically for this type of leverage .. that is, if there is any.



of course, the word to emphasize is "should."

the internet is too addictive, the office is too distractingly quiet, and i'm not sure whether i can accomplish good mathematics tonight; i don't work as well in the evenings as i used to .. \:

so before i leave the office and try a second session, i found a few interesting preprint titles and abstracts from the arXiv and the SNS @ Pisa website.
Harmonic Univalent Mappings and Linearly Connected Domains by M. Chuaqui and R. Hernández (5 pages)

apparently there is a way to detect the univalence of a harmonic mapping, by studying how its complex dilatation relates with the linear connectivity constant of its image set.

Graphs of W1,1-Maps with Values into S1: Relaxed Energies, Minimal Connections, and Lifting by M. Giaquinta and D. Mucci.

i wonder: why into the 1-circle?

The sharp quantitative Sobolev inequality for functions of bounded variation by N. Fusco, F. Maggi, and A. Pratelli.

abstract: The classical Sobolev embedding theorem of the space of functions of bounded variation BV(\Rn) into Ln¢(\Rn) is proved in a sharp quantitative form.

A Generalization of Reifenberg's Theorem in R3 by G. David, T. De Pauw, and T. Toro.

the last time i heard about a reifenberg condition was at a GFT talk this past fall, as an alternative possibility to whitney flat forms in testing for lipschitz parametrizations. the last citation i saw about it was dated from 1995, in a paper by t. toro.

perhaps the lipschitz condition is too much to ask for: in the abstract, the result of the authors is formulated with bi-hölder conditions, rather than bi-lipschitz conditions.

An Isoperimetric Inequality on the lp Balls by. S. Sodin

i can't resist hearing about isoperimetric inequalities. apparently the isoperimetric profile involves a logarithm, for when 1 < p < 2.

the nature of computation.

this would have been more relevant had i written it on thursday or friday, but life is imperfect.

anyways, my meeting with the advisor went reasonably well, despite the fact that i had no new research results to share after a month's time. in that discussion, we identified a necessary step or two in order to push the problem further.

in fact, they are computations; if they work out, then the problem proceeds, and if they don't, the thesis problem is killed.

"so it's like a two-hurdle race, then?" i asked him.
the advisor nodded.


wow. saying it in this way makes it sound like high stakes, but then again, we were in the same circumstance a year ago: we needed a particular extension theorem, and it condensed to a computation and eventually, to an unnaturally looking critical exponent of integrability.

i just hadn't thought of it as risk, last year. put another way, it's strange that i think of the current problem in terms of risk now. i suppose it's the human tendency of rationalising the past, and hoping that our efforts have not been in vain.



computations don't mean the same thing to me as they used to mean. i take the firm belief that if what you study has enough structure to make sense of computations, then you're already quite fortunate.

think, for a moment: for those mathmos out there, think about the structure of the objects that you regularly study, as well as other objects or concepts that you've seen, at some point in your studies.

how often are these structures "nice enough" that you can jot down relations between the relevant objects in a simple way? it must mean that there are enough deep but understood notions in the theory, which account for otherwise intractable difficulties and mysteries.

as an example, i like to think of the gauss-bonnet formula(e) for (Riemannian) surfaces. one can relate notions of geometry and topology quantitatively with a handful of symbols:



[courtesy of MathWorld]

it's something that i've told my non-mathmo friends on occasion, but it's hard to say whether they can appreciate what this means. for example, i might tell them that i study functions between manifolds (which is not a total lie) and they might ask:

them: "so what do the formulas look like for these functions?"
me: "i don't know, but i can draw a picture for you."
them: "wait. a picture?"
me: "it's one of a few ways i can understand what's going on."


at a naive level, i think it suggests to the non-mathematician how "abstract" the work of mathematicians can be.



anyways, back to work. it might be a computation, but it's not necessarily an easy one!

Thursday, July 20, 2006

i am a contrarian.

the night before meeting with the advisor, i feel as if i must accomplish something in mathematics, yet i really don't want to.

the night after meeting with the advisor, i feel like there's no need to rush and accomplish anything, yet i feel like doing mathematics anyway.

if you ever want me to do something, try reverse psychology. it might actually work.

Monday, July 17, 2006

at the library.

at the moment:
communing with the mathematical journal ann. acad. sci. fenn. math. in hard copy, and with the scanner.

as to why:
they're papers written in 1977 and 1981. if you know the nature of papers and journals and online access, then you'll probably know that online versions of research papers stop around the time of the internet boom (ca. 1990 or '91).

current daydreams and wishes:
  • i wish i had my own scanner.
  • i wish this library scanner were faster.
  • i wish that i was as smart as d. sullivan, or p. tukia, or j. väisälä.
lesson learned:
  • behind every great idea is a lot of details and justification.
  • i should have read these papers, far earlier.

    thanks to L. for reminding me of this deep concept called "references."

Saturday, July 15, 2006

the month ends ..

.. and no, i don't mean june or july (though the ann arbor summer is now 5/8's over, which itself is scary) but that this month of travelling is over:
  • 1+ weeks in Poland
    (Bedlewo, and a little of Warsaw and Poznan)

  • 1+ weeks in New York
    (Long Island, and visits to Brooklyn and Queens)

  • 1- weeks in Illinois
    (Champaign-Urbana, and stopovers in Chicago)
it was a fine thing to see friends, colleagues, and family, and not worry so much about mathematics. i've gone a little behind from the progress i wanted, but i suppose that was inevitable: everyone else seems to think that my goals are ambitious and slightly unrealistic.

i haven't decided what i think about it. i've been wrong before and inevitably i will be wrong again ..



i think that being away from ann arbor is good for the health. sometimes it's too much to be in the presence of such mathematical fervor and ambition: at the very least it is too much for me.

as for this past conference in champaign, it was a pleasant break and i met some new colleagues .. even some fellow math grads: thesis students of friends, and who will form the next generation of the C-C space crowd.

having done little/no work in C-C spaces, i'm consistently surprised that the crowd remains so friendly and invites me to these gatherings.

it feels a little like having a dual citizenship, to live in both the quasi-world and the C-C world, with a close-to-expired visa into p-harmonic land. q:

Sunday, July 09, 2006

admissions of difficulty.

these past few days i've been "reading" a paper of Sullivan's from proceedings in a geometric topology conference held in georgia, sometime before 1980 (i think). i've been having three recurring thoughts about it.
  1. in less than a dozen pages, he proves a result that took a one-semester course to present .. and the current plan is that i extend the result.

    yikes ...

    i wonder if i can actually do what i said i would do .. or thinking more positively, i wonder how far i'll get. [1]

  2. this result is hard .. or at least, it is very hard for me.

    over these few years, i think i've developed a notion of what i am capable (or incapable) of understanding a particular notion at a given time and place. moreover, i might even detect why i might possibly understand something.

    for example, following seminar talks at gft [2] isn't too bad, if only because i sat through so many of them by now. compare this with when i was a first year: i can say with honesty that i was lost most of the time.

    i can't say that i understand sullivan's proof .. yet. it's a dangerous and worrisome thing when i ask myself, "why did he include this part, and why is this necessary?" because it often means that i really don't know what's going on: strategy, details, or otherwise.

    for example, i think i'm missing something that should even be obvious: i'm still not entirely sure where the "Lipschitz" comes from in this Lipschitz Structure Theorem. admittedly, it still seems somewhat magical.

    so .. there is a LOT of work to do .. and the scary thing is, by now i'm supposed to have worked on this project for over a year.

  3. this result is pretty cool. i can't really explain why, but it feels like it was done right.

    i only have these impressions when i read certain authors and works, such as elias stein, lars ahlfors, john milnor, fred gehring, and others. conversely, i wish i could add m. gromov to that same list, but that would be a lie; i simply cannot understand him most of the time, and can't appreciate it.
[1] the theorem is: every topological n-manifold, n ≠ 4, admits a Lipschitz structure; that is, transition maps between charts can be improved to bi-Lipschitz homeomorphisms. moreover, the structure is unique up to such homeomorphisms of the manifold.

[2] geometric function theory (seminar), otherwise known as the wednesday seminar for the analytically-inclined @ um.

Tuesday, July 04, 2006

at home, for a holiday.

not much to say, at the moment. i haven't done any (relevant) mathematics in a while, and after all the travelling is done, it might add up to three weeks' time.
  1. i spent one week in poland, listening to other people talk either about math that i haven't thought about since my undergrad days, or about math which has and will always feel new and strange (and cool) to me.

  2. the subsequent week i spent entertaining ideas from the conference, and reading a little about topics in optimal (mass) transportation.

    nothing came out of it, of course, and the only tangible memory from those days are a few pages from a legal pad about

    • what doesn't work;
    • what gaps lie between two existing frameworks;
    • what difficulties may arise if one pursues this further.

    in other words, it was an instructive waste of time. perhaps one day i'll have sorted out enough "wrong" ideas, so that i can finally have some "right" ones and prove something really interesting.

    well, a boy can dream, right?

  3. this week i'm home, visiting family and friends, and it's a 50:50 toss-up as to whether i will get any work done at all. at this point, re-learing the argument of Sullivan's Lipschitz manifold theorem is optimistic thinking, and new results look like an impossibility.

    it wouldn't usually be a problem, but the advisor returns by late july and stage 2 of the thesis problem was thrown to me in mid-june. some progress before that next meeting could prove .. say, useful?

  4. and immediately before the return of the advisor, there is a conference in illinois .. which means more interesting but "not quite thesis relevant" mathematics for another stretch of days, and little to no time for any thesis progress.
anyways, back to a little work before 4th of July festivities go in full swing. here's hoping.