an annoying splinter of an idea.

i had this research idea from last week, but sometimes i wish i hadn't thought of it.

you see, it doesn't work.

on the way to the airport last thursday, i worked out a simple example of a "measure-induced metric" on the unit interval, and that didn't work.

then again, i didn't expect it to work because it shouldn't; there are underlying reasons, and that later idea was meant to exploit those reasons.

so during the random moments i could muster over these holidays, i tried a more complicated example and hence: the idea that i mentioned earlier.

it not only fails to work, but it makes no sense; if i can finish one particular argument, then it will lead to a contradiction, and that's no good at all. it would mean that there is some undetected error in my work -- an error i can't quite understand, and that irritates me to no end.

i don't mind making errors; they're inevitable, and i've made my peace with that. but i can't stand making an error whose consequences are evident but whose source is secret.

it's like having a friend who knows a good riddle, knows that you don't know the answer, and won't tell you no matter how many times you ask ..

.. but perhaps i should focus on the positive. at least, everyone seems to tell me that.

---

at any rate, my best guess is that by modifying the example, i've turned a purely metric condition into something .. sobolev, or more aptly, function-theoretic.

the end result looks like a variational problem and i don't quite know how to formulate it. it's almost as if the functional is "in the wrong place," but i can't make that precise ..

moreover, is it even worth formulating? the example was meant to generalise into a setting which hasn't enough structure for variational problems!

on the other hand, i'm curious to see if other perspectives will illuminate the situation.

if only by analogy, the modified example reminds me of .. something like elasticity in light of finite energy, and i'm tempted to ask the folks at syracuse if they've seen anything like it.

then again, it could be something simple or crackpot. i'd hate to waste their time .. that, and look like an idiot.

---

usually such conundrums are worth even a little something, if only for the trouble they cause. however, i remain of the opinion that i might be better off without this idea.

at least, i would be less disturbed if the idea came in early january, when i'd be back in ann arbor and more inclined to work.

unhappily, it is still very late december, i am visiting family, and the holiday season hasn't yet passed.

my family are already of the opinion that i am a little crazy and "not quite right," and i'd rather not convince them any more of that.

inspiration .. or rather, motivated confusion .. strikes at such inopportune times!

it's a conspiracy.

it's been said that the last mathematical universalist was J. Henri Poincaré, and he died sometime early last century. since then, mathematicians have demonstrated the tendency to specialise in their own fields and then subfields and then (sub)²fields ..

but occasionally, mathematics appears as a conspiracy. there are too many coincidences ..

.. and i don't mean this upcoming film called "the number 23" with jim carrey (see the trailer @ apple.com), which looks really atrocious.

more precisely, i mean how concepts are related.

for example, earlier this summer i was browsing through c. villani's book, topics on optimal transportation because i had heard in a lecture, years ago @ Carnegie Mellon University, of a way to perceive the Wasser$tein space of measures on a metric (length?) space as having some sort of Riem@nnian structure.

the analogy is initially formal, though in the case of euclidean space there is a way of discussing this rigorously, through a complicated construction of gradient flows.

so i decided that this is really a formal thing, and not much can be done. of course, i was wrong.

lott and villani have recently worked on optimal transport and Ricci curvature (in the aleksandrov sense), and lott has even made sense of a Riemannian connection and curvature of the Wasserstein space, in the case of a compact smooth manifold.

as i may have said once, i'm perfectly willing to be both happy and wrong. the days when i am right are the pessimistic days, when the worst-case scenario actually happens.

---

the other case at hand involves the notion of 'concentration of measure' which i learned from hearing lectures a year or two ago in a class @ um. the ideas were wholly nonintuitive yet intriguing, and it was really quite something!

but i thought that it would simply be an idle pursuit, that i'd never actually see it used in my field of interest. again, i was wrong.

more, lott-villani have just recently related notions from the 'concentration of measure' phenomenon to local/global Poincaré inequalities on length spaces -- the latter being the 'bread and butter' to metric analysts like myself.

amazing. if i didn't have to research and write a thesis, i'd love to explore these notions and see what can be said. but alas .. if i want to finish in five years .. \:

grading day.

6:30am

the second alarm goes off. i groan and stumble out of bed and due to sleepiness, it takes me twice as long to brush my teeth.

7:45am

i make it to the exam room, ready to proctor. we begin late, as always.

10:15am

deliberations begin on grading rubrics, and then actual grading. by happy fortune, someone has brought a communal supply of coffee, as well as a batch of donuts and pastries.

4:30pm

the grading is finished, and we enter exam scores on the computer. we'll worry about the curve another day.

5:30pm

alcohol and darts ensue @ ABC. my dart throwing is idiot savant; in one turn i hit the bullseye, another turn i miss the dartboard.

between a long day and alcohol-diminished inhibition, i make wisecracks to the surprise of my company. apparently i'm not often humorous.

9:00pm

i tabulate tentative course grades; they will be finalised after tomorrow's office hours.

12:00 midnight

my body crashes from sheer exhaustion. (looking forward to it, actually.)

wow.

EDIT (AS OF 14 DEC '06): below, the first link may only work for subscribers. the second link may be more accessible.

holy cr@p. i'm listed on mathsci.net ..

.. or rather, 'we.' it's a co-authored article, after all.

i mean, i realise it was accepted .. but i still don't believe it. it hasn't sunk in. after looking at a host of research problems over several years and never solving any of them, this is the one which makes it.

this, a fun one that my co-author/friend and i worked out over our common travels, and played email tag with latest drafts for months.

maybe the universe has a sense of humor, after all.

at any rate, for those who care: here's the title/abstract and a PDF.

'conjectures' and writing.

i feel like i spent today reading raven's bones, or shaking a bag of runes.

apart from a little writing, most of today's thinking time went to making guesses pertaining my research, and on a whim, i called some of them 'conjectures.'

i don't really know what makes a conjecture, in the same way that i don't know when a fact is a proposition or a theorem. i suppose i've never asked anyone before.

a conjecture sounds to me something deep, a really good guess based on good evidence, or one of a few possibilities which exhaust the known methods and tricks but which still eludes proof.

random guesses should not be conjectures -- not without a great deal of contemplation first.

so today, i made guesses: guesses which 'feel right,' but i cannot determine how prove them, or even how to begin to prove them.

it makes me feel like a crackpot, or less unseemly, i feel like i'm playing 'grown-up' and making 'conjectures,' pretending to do what research faculty actually do.

---

this writing feels like 'pretend' as well. it began some days ago, when i felt the need to codeify the progress of my research in some final, clear form.

in the past few months i've been following several threads which are loosely related; they wind in some common directions and fray in other directions. my mind isn't very good and i cannot keep all of it straight in my head.

week after week, i make claims and form arguments of proof between mental doldrums, and they change depending on new insights or discovered errors.

the 'big picture' has never been easy for me, and complexity is not my strength. so i write and i clarify in hopes that some things become obvious,

.. much like how i keep a personal journal in order to avoid seeking a therapist .. well, among other reasons.

if i were clever enough, or if my memory were better, i wouldn't write so much and my notes wouldn't be so neat. maybe it's a crutch and does me both harm and good, or maybe it's like a prosthetic leg and lets me keep up with the herd.

i don't know. but it makes my research easier.

i have four pages of "progress" and part of a page of these guesses. i haven't done any problem-solving today, proven anything new, or exhausted a line of argument. my guesses aren't even conjectures yet.

if it weren't a weekend, it would be a waste of a workday. then again, progress is rare and most research days are days wasted.

on book request spam.

november turns into december: it's time to unclutter the in-box again.

so today i deleted 318 emails that i've collected over (less than) one month's time, and most of them are spam: the majority are dead giveaways, by the nonsensical title and unknown sender.

then's there's unintentional spam from the math department. today being 2 December 2006, in the last week four emails were forwarded to the entire department in the form of book requests. three were from grad students, and one from a full professor.

these all take a common shape: someone wasn't able to find a book from the library, then emails the entire department. the idea is that the book in question is so specialized that someone else in the department must have checked it out.

the request is very reasonable: the requester wants to borrow the book for an hour, in order to check a fact or two. at worst, they want to make a copy of one chapter and return the book to the borrower.

of course, this situation is absurd. as grad students and faculty, our library privileges permit us to keep renewing our borrowed items indefinitely. when you are the borrower, then this is very convenient .. provided that you don't abuse your privileges.

for instance, i'm guilty of this vice. i once had a book checked out from the library for 1 1/2 years, possibly longer.

this is exactly the purpose behind loan recalls for university library resources. no one member of the university should be able to hoard library resources which are for the good of the many.

so to those who make book requests: if you need the book, then recall it. you might need it only for an hour or so, but chances are, the borrower also needed it for a little while. if they truly needed the book, then they would buy his/her own copy.

thoughts about the Rιesz Representatiοn Theorem.

i thought of this some months ago, but hadn't bothered to write it until now. so let it be said: never underestimate the likelihood of procrastination. q:

---

in calculus we teach students how to compute derivatives and possibly about what it means, geometrically, for a function to be differentiable at a point. but we give only a cursory nod to the notion of continuity; it's usually brushed aside in the standard repetoire, at least.

in such a case, calculus comes first. but any mathematician worth his salt will realise that differentiability is a rare property. in a first course in analysis or topology, continuity is the name of the game; calculus then gets the cursory nod.

so let us ponder continuity of functions.

---

in a first course in linear algebra, most examples of linear spaces are copies of euclidean space Rⁿ of varying dimension. the norm is always the standard 2-norm. if one is taught abstract linear spaces, then polynomials of degree n serve the same purpose as Rⁿ; the same norm of yore will do.

but if you have a daring lecturer, then (s)he will give you the example of the class of continuous functions on the closed interval [0,1]; it's a linear space, and i'll call it C^o for short. apply the max-norm, and you have a normed linear space.

in linear algebra, there is also a notion of duality: what linear functionals act on vectors in a continuous manner). in finite dimensions, dual spaces aren't that fascinating and the notion of "weak convergence" is unnecessary, but many strange novelties occur if one moves to infinite dimensions.

---

here is what i find to be the real kicker, though. thus far, our discussion about C^o has been topological and linear algebraic in nature. other than a norm, there's not much analysis going on.

now invoke the Rιesz Representatiοn Theorem (or some version of it): the dual space of C^o is precisely the space of measures μ on [0,1], where the action is by integration:

f → ∫_[0,1] f dμ

in other words, applying the notion of linearity and "length" to continuous functions will produce calculus, in the form of measure and integration.

this still astonishes me, to this day. analysis seems to come out of nowhere, and somehow as a natural outcome.

good memory == smart?

EDIT (AS OF TUESDAY @ 9 PM): apparently i don't know euler's formula, either. i suppose it's all the more reason why i should be impressed by those who do.

---

today's calc ii class was a review/refresher on how to manipulate taylor series, and i informed my students that certain series are good to remember.

then one student asks about the homework. to start that particular problem, i begin writing down the Taylor series for sin x about x = 0 .. or maybe it was e^x; i forget.

what i do remember is turning around, ready to start an explanation, and there is already a hand up. preparing for the worst, i call on the owner of the raised hand.

"you knew the taylor series of that function, off the top of your head?"

"er, yeah."

"wow."

"well, write it down enough times, and it tends to stick."

then someone else asked if we need to know that series for the exam. i think i caused a minor tumult by saying "yes, and these three others."

---

what surprised me was .. well, the surprise. i mean, it's just a formula, and it's a well-known series. it's like knowing that the harmonic series diverges.

i dare say that any mathematics major or grad student worth his or her salt would know the taylor series for e^x off the top of their head.

would you be impressed if you met someone who remembers:

the quadratic formula?
or the pythagorean theorem?
or that sin(π/6) = 1/2?

i might be a little delighted if someone did know euler's formula:

e^iπ - 1 = 0.
e^iπ + 1 = 0.

for instance, the guy from xkcd knows it. great comic, that one.

to be 'educated'

i actually wrote this as a footnote on another blog post, but i think it stands on its own:

to some, being 'cultured' or 'educated' means being well-read in literature, being able to talk philosophy and politik and human nature, being able to appreciate classical music and fine art.

my demands on being 'educated' include those and more. why ignore our techie side? for example, an 'educated' person should be able

1. to appreciate why the harmonic series Σ_n n^-1 diverges,
   
   
2. to appreciate the classical proof of why there are infinitely many prime numbers.
   

that doesn't mean i ask such persons to prove these results. i just ask that they be listeners without glazed eyes. if we should see beauty in symphonies and in baroque art, we should also see beauty in geometry and in number.

less demandingly, i would ask that an 'educated' person be able to build a simple webpage.

it would also be fair to ask that such persons know a little science, but then i risk not being an educated person! q:

post grading blues.

an hour ago (~1:40am) i was still grading calc ii exams. between that and the bike ride home through the rain, i think i've lost a large dose of humanity.

i feel cold and cold-blooded, cynical, and worn out, too. i feel like impassive stone.

all that frustration at not having any time for research has somehow vanished; maybe it lies hidden under too many hours of waking and too many instances of misapplied convergence tests to infinite series. i don't feel the past few nights of fitful sleep and insomnia .. not yet, at least, and i fear they lie in wait for the morning.

argh. tomorrow morning is in a few hours!

my hair is wet and my eyes are wide open; in an hour i will be dead to the world and in a few more hours, jolted awake by necessity and past promises.

i hate having to be responsible. more responsibility means less sleep. \:

time mismanagement.

i should learn to plan more efficiently.

for instance, my friday nights are almost always free. if the week ends well, then i partake in beer with fellow math grads until early evening, and then i go my separate way to be alone for a while.

i never get much work done then, anyways. i should be grading calc ii homeworks then, so that i wouldn't have to grade them now.

you see, i really, really don't want to grade them now.

instead, i would rather be reading from a text by Dupré and Gillette about Banach modules, or lounging in my recliner and reading the latest issue of Bon Appétit magazine.

argh. work comes before .. er, more enjoyable work, and of course, play.

sometimes i really, really hate mathematics.

i'm not sure how it happened, but during my meeting with the advisor i screwed up the same proof two weeks in a row. it's not a new theorem, and not even the proof of a theorem, but a single line in a proof.

one. single. line.

i hate being inept, i hate wasting the advisor's time, and for a little while today, i hated mathematics.

---

after that debacle of a meeting and two seminars later, i sat at my office desk and thought. in other words, i stared at the chalkboard and it stared right back.

then i decided to make a cup of coffee, and as i was pouring the water into the back chamber of giulia [1] the idea suddenly hit me. by the time the coffee was ready, i came up with a correct proof ..

.. well, it looks correct, but so did the last two 'proofs' of mine. but looking at these steps, it's obvious. f*cking obvious.

so i sipped the coffee angrily, hardly enjoying it at all.

---

[1] yes, the coffee maker is called giulia, and no, i didn't name it; my officemate kevin tu¢ker did.

when we were trying to come up with names, i suggested we could name it after a beautiful woman and immediately, he thought of giulia (pronounced: julia, but trochaically).

a grad population, over the years.

a first-year grad student, my desk was in 1061 east hall with a dozen others. just earlier i remembered those former officemates of mine:

• i'm still officemates with ryan, after all these years. mike, whose desk was across from me, is now a little down the hall from my current office. marie and jared had corner desks, that year; he's now down the hall and she's now on the fifth floor.
  
  
• john is still a um student, but his advisor is at yale, so he is there for the year (or longer).
  
  
• i only see kyung yong when i am walking to the library or moving up- and downstairs. young kwon is still somewhere in ann arbor; few ever see him.
  
  
• eliana transferred to chicago, to be closer to her husband.
  
  
• alison returned to her home state of florida.
  
  
• dave decided to be an ecologist/biologist. he's now on the other end of the diag, and northwest of east hall.
  
  
• ken is no longer with us.
  
  
• ray, yuan, and another chinese student finished their actuarial programs, and are now working people.
  

much has happened in three years. the world turns, the years pass, and we grow older and have more stories to tell. the new kids still look fresh-faced, even the second-year students.

they haven't seen their peers leave .. yet.

in which i read the funnypages.

last night i (re)discovered a webcomic called xkcd. i don't know what the title means, but every so often they have these hilarious mathematical/nerd comics.

for example, [1]

full-sized image
link to this comic

and these also crack me up, even if physics are involved:

link to this comic


(can't find the link for this one, but it's on the website)

[1] this artwork is NOT mine, but the work of one Randall Munroe. You can read about him here.

this is turning into week three of trying to understand a proof that is only two paragraphs long (and maybe an estimate, but not much more). i've been working day after day and overall there's a little progress, but the big picture isn't fully clear to me yet.

yes: it is something important and worth knowing, but i can't quite detect whether i'll be able to extract anything new from the argument or prove anything new or make a useful observation.

so even 'victory' won't really be victory. we will gain nothing, but merely regain the certainty in something already known.

there will be something, maybe another lead after this proof is settled. between the advisor and me, something will turn up; i believe it. there are always questions, even if i cannot think of them as quickly or easily as others can.

life goes on, wherever it's going.

---

october turns into november, and if i'm still on the five-year plan, there's about a year's time left on the clock. a year before i'm supposed to be ready to look for jobs.

i'm still nowhere. i spent between 12 and 14 months on the first thesis problem, which is sunk, and after 2 months i've no real progress on this second one. nothing, except a souped-up example which lends nothing to a good theory.

two months is nothing, and yes, i'm impatient. but i don't know what i'm looking for or where this project will go and honestly, i don't know if i'll succeed. this is a dark time in my life ..

.. yet one day, somehow i'm supposed to look back to now and think about how much free, unhindered time i have and why i was so worried over nothing. if time travel were possible, i think i'd travel to that future and punch my future self for being so nostalgically saccharine and dismissive. smug bastard.

---

i don't feel tired, but it sounds like i am.

in which i dare the darkness.

tonight i remained in the office longer than i'd have liked. it was necessary, though, because i was dabbling in the dark arts known as algebra ..

.. and i couldn't very well permit that stuff follow me home now, could i?

it would taint the very air, lengthen shadows to sinister darkness, spoil the soy milk and make stale the coffee ..

[shudders]

it's not that i detest all things algebra; for instance, i like lie groups and lie algebras. kleinian groups form very interesting illustrations via tessellation. i also like hilbert spaces and linear spaces in general; the weak-star topology would be nothing without the notion of duality, which is a linear notion.

but outright algebra -- modules over rings and left- vs right-actions and (argh) universal properties (bleah) -- they just makes me uneasy ..

.. in the same way that superman hates magic, or how batman hates moving into action without a foolproof plan [1]. let's face it:

give the villains half a chance, and they will lay a trap for our heroes. the heroes will escape, thwart the villains' nefarious plans, and save the day, but there's always that annoying trap towards a dénouement.

anyway, moving along ..

i also kept muttering to myself this mantra: it's not really algebra, and tried to convince myself that this wasn't evil.

"look's there's a norm! it can't be all bad."

"C^*-algebra is just a convenient name. the elements are functions, and remember: functions are our friends .."

"you're in the commutative case. think: L^∞, measures .. happy things .."

it worked, at least for a little while. but eventually i could bear it no longer, jotted down quickly some remarks on my notes, and left to find .. an algebra book from the science library.

but all's not lost: the identifier "Banach" appears twice on the title, and "module" only once.

still, i would never have thought that i'd need this algebra in order to get my work done.

[1] by this i refer to the grim, gritty batman, and not the pansy batman of the '70s tv show or the joel schumacher-directed debacles. give me a christopher nolan-type batman, or even the gothic batman of tim burton.

in my book, batman should be a borderline personality, whose paranoia and meticulous planning keeps the crime of gotham at bay.

fitful day, little done.

i couldn't concentrate today. at most i reread some class notes from 2 1/2 years ago that i don't remember taking. all the better to reread them, i suppose, and better yet, they concern results from a paper of J. Cheeger that i was going to read anyway.

then, absurdly enough, i took notes.

yes, you read that correctly: i took reading notes from my class notes. they weren't hard to follow, but somehow the process of notetaking sharpens my understanding, much like taking notes during research seminars.

indeed, there's a 5% chance that you might actually look at seminar notes again, but the point is to keep busy and listen more attentively; you can't slack off if you're recording the nuances of the talk.

still .. absurd. yes, absurd.
but i couldn't help myself.

---

there's an off-and-on feeling i get, and though i can predict it coming i can never shake it off when it does. it's a feeling of being overwhelmed, but it's not so much stress as being impressed.

take this cheeger paper that i mentioned. borrowing generalisations from the quasi-world theory, he reproves a form of rademacher's theorem which is genuinely new in a modern, more general context ..

.. but that's not all.

impressive as it is, that theorem is used as an application to deduce a structure theorem: generalised cotangent bundles exist for a class of metric (measure) spaces.

amazing. absolutely amazing.

it's not exactly what i study, but an elephant of a question arises: how in the world do i add to that?!?

under the cool light of calm and reason, one can always think of something, perhaps by identifying weak points and asking questions.

but for me, that takes a while. i have to withdraw and let it sink in me, much like how i am quiet after a good film or concert. sometimes i see or hear or understand and in a single moment it is too much, and i can say or do nothing.

who'd have thought there would be a downside in being able to appreciate something?

teaching and science and other thoughts.

i learned several things after teaching today.

1. i really, really hate British units of measure. units of mass aren't so common, and instead pounds refer to a unit of gravitational force. fine: if that's the physics definition, then i can't and shouldn't argue it.
   
   but it doesn't make any sense. how do you discuss density of solids, then?
   
   you'd figure that the Brits wouldn't mind Newtons (N) as a unit of force. it's named after one of them, after all. everyone remembers what Newton did, but nobody recalls the deeds of, say, Alfred the Great .. who, oddly enough, is the only English king to be named 'Great.'
   
   frustrating.
   
   
2. basic mechanics isn't as simple-minded as i thought. moreover, my students seem much more inclined to physics than, say, volumes of revolution or polar coordinates. they actually seemed lively .. though that could be more my volley of errors that people kept noticing.
   
   as i was fumbling with certain concepts, such as force induced by pressure in a fluid and gravitational force, a few of my more vocal students jumped in with plenty of good pointers. admittedly, i didn't know they had it in them.

again: works and days.

if it were summertime i'd wait a few days and then undergo a post-conference slump. but the fall term remains in full swing and i spent most of today "taking care of business" ..

prep and teaching,
then attending a grad union committee meeting,
then attending Geometric Measure Theory lecture,

then i was nudged out of my office, because my officemate's calc ii students flooded into the floor and chairs of our already crowded hovel in east hall.

retreating to the psychology atrium (which is far quieter than the mathematics atrium) i tried to focus and to think.

but in the end my thoughts were sporadic and ineffectual. coming home after fetching supplies from trader joe's, the mood's been the same.

i can't get any research done today .. the issue isn't even research; the confusion lies in understanding an existing proof of an 8 year-old theorem. it leads to nothing new and possibly something instructive, but at the core it is prerequisite before anything else is done.

this seems to echo conversations from this past weekend with friends and peers:

i've said before that the content of my talk would make a tolerable paper, but i would be the wrong person to write it. it would be an anthology of topics, and most of the results would be previously known and the only novelty would be to see them in one place.

others have said that there is something valuable in making connections, even if the results are known. it would solidify and further the body of knowledge in the field. i don't disagree with them, or more accurately they haven't disagreed with me.

but they weren't listening.

i didn't dismiss the role of such a paper. my point was that i'm the wrong person to write it. the most important quality of a student or a young researcher is creativity and innovation.

i'm not agreeing with g.h. hardy and i'm not saying that 'mathematics is a young man's game,' but in the sociology that is mathematical academe, new researchers have different priorities than established, tenured researchers. my priority is to create or to discover something useful.

i haven't had very many good ideas. in recent history, i have had two good ideas which have worked, but one isn't really worth anything and the other is a computation that any analyst could do. countless bad ideas make the rest.

but i digress.

in sum, i was distracted today by obligations, and i did very little thinking. the object of my thoughts isn't new, but it must precede any future innovation. i'm not clever enough to be impatient, and i'm not as patient as i need to be.

---

maybe tomorrow will be better. the obligations and errands will meet me again tomorrow and greedily snatch away my time.

october soon ends. in a year's time i might have to look for jobs, and i don't know if i'll be ready by then.

how the talk went.

preamble: i have to stop giving talks for the first time at conferences. indeed, i sound like a nervous twit when i do ..


minus signs.

1. i went over my allotted time: a conference sin!
   
   
2. i misread the audience. it was an AMS Special Session, but the same crowd lingered for all the talks, so i wasted two slides giving definitions that everyone already knew.
   
   
3. i should have drawn diagrams on the transparencies themselves, instead of drawing them on-the-fly, on the whiteboard behind the lectern. this happened four times.
   
   
4. i think i misrepresented myself and may have intimidated some of the audience by mentioning milnor's exotic differential structures on the 7-sphere.
   
   
5. i never reached my last two slides, which would have said something about harmonic functions. a great tragedy ..
   

plus signs.

1. at the least, i think i spoke clearly and that everyone followed me.
   
   
2. i did get a few curious questions and many informative comments and suggestions. analysts are such nice people!
   

now it's a matter of writing a research article based on what i talked about. oh, joy of joys ..

disasters happen and are waiting to happen.

EDIT (FROM 22 OCT 2006): as it happened, on friday night, i was informed that I wouldn't be talking last on sunday morning, but on saturday morning instead.

life again throws curve balls.

---

it's a day or two to go before the AMS Conference @ Cincinnati, OH. by fortune of fate, i'm talking last on sunday, which is fine; someone has to be last. i've already written the talk and need only print out the slides ..

.. but i have a bad feeling about this.

i'll be talking about the Thesis Problem that Was (read: Died) and haven't gotten to writing up the preprint yet. i simply have no time. how does anyone ever have time to work on new research, write up results, teach a section of 31 students, and attend classes and seminars?!?

small favors: i've never been so happy as to have a "Preliminary Report" category before.

---

you'd think that with two days off this week (UM has a 'Fall Break'), i'd be living it up and coasting and enjoying life .. but i'm not.

monday and tuesday off meant an obsessive few days of research: equivalently, i ended up over-thinking the details that don't matter and overlooking the details that do.

today what i thought was a proof .. wasn't.

i realised this only as i was explaining it to the advisor, who was the one to point it out. so i looked foolish again. you'd think that i'd be used to the feeling by now .. but i'm not.

---

i'm losing an obscene amount of time to teaching .. or at least, it saps my mental energy. what i thought was a good worksheet confused my class more than it educated, and tomorrow is a lesson on

"yes, this is how you do the worksheet;
no, you probably won't see something this complicated on an exam .. i hope."

i hate hedging my bets. it doesn't mean that i'd rather write my own exams, but .. i hate having to keep marching on pace because i can't let my students learn at their own pace.

uniformity might be great for convergence of functions, but it's terrible for students and learning.

on a side note, i think i inadverently made one of my students cry. this is exactly why i should never make judgment calls, because nobody should EVER trust my judgment.

argh.
conference talk. teaching. planning. packing.

i must get to it, because sometimes we make promises that we must keep, despite time and circumstance.

something from computer science.

i like reading essays from a particular author/programmer/"startupper" by the name of paul graham.

irrationally, i distrust capitalists and the rich, but any hacker that can use the word "isomorphic" colloquially and correctly is all right in my book; in his essays, paul graham has, so he is all right.

it could also be that he writes essays that i want to believe, like "why nerds are unpopular" (an uplifting piece) and "what you'll wish you'd known" (something like a speech).

anyways, here are two excerpts from his essay, "the hundred-year language" :

Any programming language can be divided into two parts: some set of fundamental operators that play the role of axioms, and the rest of the language, which could in principle be written in terms of these fundamental operators.

I think the fundamental operators are the most important factor in a language's long term survival. The rest you can change. It's like the rule that in buying a house you should consider location first of all. Everything else you can fix later, but you can't fix the location.

I think it's important not just that the axioms be well chosen, but that there be few of them. Mathematicians have always felt this way about axioms-- the fewer, the better-- and I think they're onto something.

so paul knows axioms! dandy! but this second bit seems to me a little insightful.

Languages evolve slowly because they're not really technologies. Languages are notation. A program is a formal description of the problem you want a computer to solve for you. So the rate of evolution in programming languages is more like the rate of evolution in mathematical notation than, say, transportation or communications. Mathematical notation does evolve, but not with the giant leaps you see in technology.

i can't resist another excerpt. as you can imagine, i didn't finish the essay before posting this .. but this is the last, i promise.

it's worth reading because as mathematicians, numbers are numbers, and what paul describes is something like ordinals from set theory. but from a programmer's perspective .. mmmmrrrppgh.

There are more shocking prospects even than that. The Lisp that McCarthy described in 1960, for example, didn't have numbers. Logically, you don't need to have a separate notion of numbers, because you can represent them as lists: the integer n could be represented as a list of n elements. You can do math this way. It's just unbearably inefficient.

teaching, an exam, and the toll on research.

last night was exam grading until 1am or so.

that in itself mightn't have been so bad .. digging ditches would have been worse, for example .. but it was what came before:

• insomnia, then prepping at 8am, and teaching at 10am,
  
  
• urgently thinking about research for a few short hours in the noise and disturbance of east hall,
  
  
• lecture at 2pm, seminar(s) at 3pm, and then office hours until 5:30,
  

and then you know the rest: exam proctoring and grading. i acknowledge that plenty of people out there have long days, every day, and they have my respect.

it's just that i don't cope so well .. not anymore, at least. i don't play so well with others either.

---

purely empirically, an 'exam week' isn't that much more work: an additional office hour than usual, and a change in prep routine. it's the stress and concern that changes things.

students care about mathematics and class for the first time in weeks because they finally encounter something that affects their immediate existence (that being a grade). some are nervous, and some are close to panic.

i know i'm not worried about how my students do; they'll do as well as they've studied and prepared themselves. i know i'm not worried about how well i've taught because, apart from a stray lesson or two, it doesn't matter: a student will choose to learn or not to learn, to learn well or not to learn well.

but by some sort of osmosis, it infects me and i worry a little. i find myself less capable of concentrating and less effective at research.

it's like upping the pace on a road run, where you can feel your leg turnover change to something quicker. there's a slight twinge and it feels unnatural.

you know you can maintain it for as long as you need, but when you switch back to your usual pace, that twinge will change to a soreness that won't go away for a while.

so today i returned to my usual pace.

---

today was also my meeting with the advisor, and it was fine. but it wasn't .. optimal. despite the hours of thinking i tried to fit in ..

(amidst the bits of stress from exam routine and between writing shifts, last weekend)

.. i walked into the advisor's office, empty-handed: no results, no remarks, no descriptions of any notable obstructions. i ran out of time and had nothing to say .. or rather, nothing new.

i hate it when that happens. i hate wasting people's time: the advisor's time.

these are parts of papers that i should read, these are claims and exercises i should do, and i should be trying to come up with my own good ideas. but like last week and the week before that, i ran out of time and i couldn't ..

.. and about the ideas, well, often i'm just not that brilliant. the odds are bad, but i try.

i try because someday, that will be my only option. as the saying goes, we are all students and we will always be students, but someday i won't be a graduate student and i won't have an advisor and i'll have to stand 'on my own two feet.'

quite simply, i try to be a mathematician, a researcher .. but more often than not, my endeavors come to nought and i end up just a student again.

".. but some things must be done."

in the last 24 hours i've spent my waking hours writing. rather, i've been LaTeXing.

it's a talk for an upcoming conference in two weeks; the rationale is that if i can convince myself to write a talk, then i might be further motivated to write a research article on the same subject. there are a few slides left, and i think the final form of it will be tolerable and not too boring.

i've even coded a few figures on some slides, so thanks to the graphicsx standard package, the audience will see a picture or two.

more, it will be a friendly audience and i think i will enjoy giving the talk. it's been a while since i've felt that way.

as i've told a friend of mine yesterday, i'm not looking forward to writing this paper, but some things must be done.

it feels to me like writing a eulogy, because the content will be salvage from my first thesis problem. as you may recall, that problem is dead or left for dead.

never mind the difference; the effect is the same.

to me, writing this paper is reliving that inevitable end.

very little of the inquiry is my original contribution; of a year spent on this project, there were less than two months where i was working on original ideas and not investigating the past work of others.

i explored many areas and ideas and learned a few things, but that's not the point; the point was to do or accomplish something, and specifically, something new and preferably interesting.

i can count two original "theorems" from that inquiry, and one of them is a computation anyone could do. the other is more an observation than a theorem, and there is that same "everyman" feeling to it.

i can't seem to articulate what i mean. i know that this first problem is not one of my failures, or a failure at all.

but i admit: identifying the flaw in the method seems one of the rare times when my contribution actually mattered. i did that. inherently negative as it may be, it is an accomplishment ..

.. and look what it did. now there's no more problem. lovely, wonderful, and bloody marvelous.

but no worries. there's a second problem, with mysteries at every turn.

it's all frontier land. there are few blazed trails, and i can choose not to take them. it's terra incognita, where you may define "success" in any terms you want.

but that theorem i proved .. remember that one? loosely speaking, it asserts that a subclass of concrete examples are not worth studying with this theory.

some days i feel like i inquire, if only to wait for another disaster, and for another problem to fall dead.

why do i have a bad feeling about this?
all of this ..?

before the meeting and after; thoughts on salvage and writing.

i wrote this most of this, last night; new text from tonight begins at the symbol '#.' an epilogue follows the symbol '##.'

i suppose it could have been too much to ask for: that is, three consecutive weeks of good research ..

two weeks ago, i made a conjecture and it might still be true. however, it is certain that it remains a conjecture tonight.

at the time, i could only sketch an argument for a rather special case, but it was too simple-minded. i would be embarrassed even to state it as a lemma, because it is too transparent; upon reading the hypothesis in the lemma, it would be obvious how the proof would go. [1]

there's no art in that: no boldness. but when i tried to generalise, there was a gap.

one week ago, i filled the gap and managed a weak version of the conjecture; it's true for a wider class of examples, but remains unknown in full generality.

it's of little interest, and more a toy than anything else. i suppose that one of its few strengths lies in its concreteness. to some degree, the weakness of transparency is gone, but as usual this proof reduces to that first special case.

i dare say, though, that the reduction is not so transparent. you might even call it clever, though it is n. weaver's cleverness and not mine.

# this past week i've been unable to concentrate. i don't know why. an idea i had this past weekend now makes no sense and i can't remember the motivation. i think i fooled myself, and it wouldn't be the first time.

i hate it when this happens. tomorrow i'll stop by the advisor's office and tell him that i have no results. the most frustating part is that i have essentially nothing to say, because it's not even worth mentioning what didn't work, because i know why and it didn't teach me anything.

the meeting will be fine; i know that.

i now know the advisor well enough that that hour and half-hour will be worthwhile, indispensible time. what bothers me is that the advisor is disturbingly clever and wise, and if i don't stop him he'll prove results which should be my task to prove. he can't help it; he became faculty @ um for many good reasons.

advising and guiding are fine things, but there is no honesty in my standing in another's shadow, because one day i will move away and i must face the sun myself.

the way is difficult and we may choose our ways, but some choices are meant to be formal.

---

## perhaps i shouldn't call it a conjecture. the word "conjecture" is too grand and suggests something that is inherently worth pursuing. maybe i should have said "claim."

the meeting did go well. the advisor and i philosophised, and we pondered the direction this inquiry goes. the once-conjecture/now-claim is fine, but ultimately it was meant to follow a path of counterexamples and derive a necessary condition for the geometric function theory that i study.

if i prove it, then it is a little something. but i haven't proven it and at the moment i have no more ideas, so what stands is not theory, but a souped-up class of counterexamples based on another's existing work. even if i prove it, the result won't fully resolve the issues at hand.

there are deeper issues and more relevant connections to the existing theory of the analysis on metric spaces. in particular,

• there is the theory of upper gradients through rich geometries of curves;
  
• there is p. hajlasz's theory of sobolev spaces via lipschitz-like moduli of continuity;
  
• and there another nascent, but existing notion. as yet, there is no rich theory.
  

to paraphrase aristotle, even taxonomy is no easy feat. as mathematicians, we go further and attempt an understanding between elements of this taxonomy, and assess its relevance.

this is among the goals at hand. what does it mean, a sobolev space? let us have a metric and a measure; can we have calculus? are there many types, and are they so different?

now i am too philosophical. let me prove things before i say more.

---

this post grows long, but let me say: i had a thought, a few days ago. it was an old realisation, made acute and sharp by the passage of time and an unexpected reminder. the future nears, a conference quickly nears, and the job hunt will soon come; a year's time.

that first thesis problem is dead; i've left it for dead, but there is something to salvage. i want it to mean something, that a year of my work mean something. in a little while i will give a talk about it (a so-called preliminary report), and the advisor still thinks it a good idea that i write a paper about it.

it is good to have publications, if only for a strong c.v. to date, i have only one and that is a joint work. another can only help.

still .. why is it so hard to get to writing? i've heard a saying once that a mathematical paper is never finished, but given up. to write this, i will have to give up the possibility of progress for a while; i would write what is, and not what could be or is likely to be.

i think i am too used to uncertainty, and cannot bear it. it's silly but true.

[1] and yes, when i say "obvious" i do mean obvious. however, i didn't say that it's trivial .. though i dare say it's close.

an "immortal solution?"

a few weeks ago i remember the title of a preprint from the arXiv concerning things called "immortal solutions." i didn't think much of the title at the time, and in all honestly, i thought it was a catchy name to make the paper more readable.

but now the term is called an "eternal solution" and has made news, at least on ars mathematica. this work of p. smith contributes to the analysis of the Navier-Stokes equations (see wikipedia entry) which is one of "the biggest games in town." the clay mathematics institute, for example, formulates a problem concerning these PDE.

as i've read secondhand, the theory uses viscosity solutions and c. sormani @ cuny has written a little about it. if she vouches for it, i'll listen.

this world is too small. i've bumped into prof. sormani once, years ago at a conference at CUNY Graduate Center, though i don't think we'd remember one another.

some days i wish i worked in PDE.

in which string theory is mentioned.

i feel like i jumped on the bandwagon, which is never a good feeling.

this week i found two more articles denouncing string theory with the usual complaints and talking points. the first can be found @ american scientist online and the other is in new yorker magazine book review.

after reading the first article, i decided to hop off the fence and write something about it. i wonder if the trolls will swarm by that blog post, this weekend.

[shrugs]

at any rate, i found this excerpt from the new yorker article an interesting one:

String theory came into existence by accident. In the late nineteen-sixties, a couple of young physicists thumbing through mathematics books came upon a centuries-old formula that, miraculously, seemed to fit the latest experimental data about elementary particles. At first, no one had a clue why this should be. Within a few years, however, the hidden meaning of the formula emerged: if elementary particles were thought of as tiny wriggling strings, it all made sense. What were these strings supposed to be made of? Nothing, really. As one physicist put it, they were to be thought of as “tiny one-dimensional rips in the smooth fabric of space.”

why strings?
then again, why atoms of protons and electrons?

i wonder what the mathematical formula was. centuries old?

delusions of grandeur.

this might be a happy error arising from late-night fatigue, but earlier tonight i think i proved a theorem. by this i mean i really proved something: an original theorem that nobody else has proven before, though it is a generalisation of a theorem of n. weaver.

the conjecture still stands but i'm one step closer to it, if only because the theorem serves as motivation on why the conjecture should make any sense at all.

maybe in 24 hours i'll find a flaw and become depressed again, but for now i am happy and tired and tipsy from a glass of white port, and in a moment i will retire happily to bed.

let the disappointment come tomorrow, if it comes. tonight i am in happy delusion.

mathematical celebrity sightings.

when i really think about it, the most famous people i've ever met are the physicist sir rοger penrοse and the fields medalist terencε taο.

both of these meetings were accidents and forgettable events; these were before i was a graduate student and i didn't realise at the time that these were very great men, indeed.

i suppose it also means that i've actually met a knight of the british empire. (;

while i'm at it, i might as well keep boasting.

i have heard lectures from the grand old man of analysis, elιas m. steιn, fraηk mοrgan of the double-bubble conjecture, and the mother wavelet herself, ingrιd daubechιes.

less known, i've heard talks from jeff cheegεr, dennis sullιvan, l. craιg eνans, and peter jοnes.

but the real kicker is: i've never heard a lecture from fred gehriηg before.

reflections on time and days, and frustrations.

if i had to do it all over again, i think i would teach later in the mornings or in the early afternoons. in the block scheduling for calculus classes @ um, 1 to 2:30 would be ideal, i think.

i could wake in the pre-noon and enjoy my coffee. i would think for a few hours or work things out on paper or read a few references.

ultimately i'd resolve or muddle up some thoughts from my previous endeavors (it's hard to expect progress, every day!) and by noontime or later i'd be out of ideas, which would make a perfect time to have a little lunch and prepare lessons to teach.

11:30 to 1 wouldn't be bad, either: it would be a tight fit for the morning's work, though.

i'd like to believe that i could still wake up fresh and early if it means getting a decent amount of work done. i think of how tired i am, but then i think of how i'd rather not move to a third thesis problem ..

.. and it makes an early riser out of me. (;

all in all, i think my old habits are dwindling: except in cases of sheer willpower or pure curiosity, i can't work late into the night anymore. 1 am and i cannot think deeply any longer ..

.. unless the problem has marked me and i can remember it without hesitation, but that in itself is a bad, obsessive sign .. \:

---

and as you may have guessed, a problem has indeed marked me, with acute ambivalence. i love mathematics research; i also hate it, and often at the same time.

it brings out the worst in me, because i grumble too often and to myself. i become a troll and no longer a person.

too many in east hall know me now, and i have no peace. people wave or say hello to me, and i only nod curtly at them or do nothing. i don't answer the telephone and delay emails which need no immediate response.

my officemates are not mean and unfriendly enough as teachers, because their students keep flooding into our office for homework help!

from paper recycling bins i eagerly fish out pages with blank back sides. later i return them dejectedly to the bins, with back sides all cluttered with scrawls and figures and nothing useful.

it's a sequential depression: i get an idea and try it out, see how it looks .. but it fails. then another idea comes, and i try it out .. and it looks like sh*t. why did i even think of this idea? f*cking idiotic.

more ideas rise .. then fall, and after a while i cannot bear it any longer. therein lies the hate: when i work in mathematics, i have to confront my own folly and ignorance, and to allow myself my inclination towards errors and misjudgments.

i think artlessly, and brutally. i have no guiding principles, and i stumble a lot.

above all else, the most frustrating bit is that whatever works for the problem, whatever the solution .. is easy. natural. it flows .. and you curse yourself for not seeing it sooner.

i dare say that mathematics is difficult, because ultimately it is easy, but frustratingly so.

conjectures

last week the advisor made a conjecture which seems on-target, but i wasn't sure what to think of it.

i suppose it's a natural reaction. if i knew how to think of it, i'd have proven it, and it wouldn't have been a conjecture now, would it?

today i made a conjecture which encapsulates the advisor's previous conjecture, though i wasn't thinking of it at the time.

more accurately, i made it last night over a glass of white port wine. then i realised that i was indeed drinking port and thus could not be trusted with conjectures, so i labelled the page i wrote on:

crackpot conjecture

and then happily continued with my whimsical speculations.

doing so is like giving free license to be wrong. sometimes i write "scratch paper" on a recycled page and then start messing it up with scrawls and diagrams and cross-outs. when there are enough ideas, then comes the good paper and the process of clarifying the ideas.

even to this day, i feel like i can only do mathematics, draft by draft; some habits do die hard.

the weird thing is that the conjecture still sounds plausible when i'm sober, and it would relate these metric co-tangent bundle and exterior differential notions with geometric measure theory.

the cool thing is that the advisor also thinks it's plausible.

the worrisome thing is that Weaver's notion of exterior differential may be rather rigid, and it mightn't be as interesting an object of study.

then again, it could be my paranoia: after you've killed one thesis problem (for the worse), there's always the fear that you'll do it again. who'd have thought that it would be so worrying to be right, for once?

article post: my brush with someone famous.

seed magazine (a science/techie mag) has an article on one of this year's fields medalists, and for once it's not grigori perelman. instead, they chose to highlight terence tao of ucla.

the really weird thing is that i've actually met terence tao, but at the time i didn't know who he was:

it was a little more than four years ago, i had just finished my undergrad and as luck would have it, the park city mathematics institute had flexible funding and accepted my application into their three-week summer program in park city, utah.

i arrived there in the afternoon and when wandering the hallways of the conference center, i bumped into two mathematicians hard at work with a pile of pages with scribbles in front of them. one of them, a young asian man, seemed to be explaining something to the other, a young woman.

"hard at work already, eh?" i joked. to my infinite future relief, i didn't say what i thought i saw: two grad students already prepping for the lectures by distinguished faculty.

"um, yes," the young man replied, and the two returned to work.

the next day, i found out that the young man was terry tao and the young woman his student. how did i find out?

by attending tao's lectures. brilliant man.

good news, for a change.

with all the griping and ranting and moaning i do on this blog, i feel obliged to say that today wasn't one of those days, but that did have something to do with the past few days.

---

wednesday was my talk at student analysis seminar, and it lacked luster. in retrospect i should have realised that i wouldn't end up discussing things that i truly wanted to discuss .. not nearly that much at all.

i should have talked about something else .. less dear to my heart, perhaps, and maybe something lighthearted and fun, if analysis can be like that at all. i think i took it too seriously and spent too much time consolidating sources and facts and areas of study .. to the extent that half that time was a waste, and later i was irrationally upset by my own folly.

but promises are promises, and abstracts have their deadlines, and we cannot retract all our bad decisions.

if all goes well i won't be giving another talk for a while. sometime before december i trust i'll be drafted into another analysis study seminar talk, and there is a talk i promised friends for a mid-october conference. but apart from those, i'd rather be silent for a while, and wait until i have something worth saying before i speak.

at any rate: an obligation or two now met, and over.

---

yesterday afternoon i showed the advisor some colorful pictures i drew, and he pondered my idea for a construction, said that it might work, but not to dwell on it for too long. trust me: this isn't as strange as it sounds.

the idea also seems to work, which i suppose is good .. even though it took me a month, off and on, to sort out ..

.. and considering that it is one variant of someone else's very concrete example, and that doesn't prove anything deep. it's merely a motivating example.

the real work is ahead: what do two examples teach us? is there anything interesting happening, geometrically? as my friend john mackay would say, is this fit at all for man or beast?

it's too much thought for a friday night. a small job's done, but there are many more left, and i'll start another one tomorrow.

---

i may have time to write that paper, at last .. provided i don't waste it. the time for it had to come, sooner or later, but in the midst of errands and duties, sometimes i forget how things realistically follow one another, and my hope ebbs to dull expectations from dull, glassy eyes.

i hope things brighten for a spell. i could use the time to be productive.

teaching and emails ..

i've received three emails in as many days about exam time conflicts.

could it be that the students are reading the syllabi and course policy closely, because they have a quiz on those things, tomorrow?

i should have done this years ago. q:

---

not much else to say. still waking up at 8:30, even today.

(note last night was a party until 2am, hosted by none other than kevin and hannah. i wonder how late they were up ..)

so this morning i managed to do a little work, even if it wasn't research; at least the first part of my talk (student analysis seminar) is done, and the work session felt like a summer morning's, again. quite nice.

other than that and a little teaching prep, i've done depressingly little. i must find some way to reconfigure my brain into being more productive more often, otherwise i'll either finish in five years and prove little to nothing, or finish in six years with not as little to nothing.

[sighs]

start-of-semester frustrations.

the last time i felt productive was sometime last weekend, possibly monday. i thought i had overcome some obstacle of proof and was deducing corollaries and generalisations .. in short, doing a mental victory dance.

the last time i felt like i was doing something worthwhile was on tuesday morning, between 9 and 11:30 am. but it wasn't productive per se; i realised that my proof didn't work after all, and sought to find adjustments to sort it out.

this worthy time came to a halt, since i promised to meet my first-year mentoring student for lunch .. and it was a good lunch, after all.

---

since then, it's felt like damage control.

teaching prep is a headache, with the handouts for the first day and whatnot, and i've caught the bad turn of the rotation for our student analysis reading group; tomorrow it's my turn to talk as well as my meeting with the advisor, and guess what?

my proof still doesn't work: not enough peace-&-quiet time to ponder it properly.

the first floor of east hall may as well be the first floor of shapiro undergrad library, where no work is actually done and where students gab and gab and walk around and gab with other people .. i'm beginning to like working at home, more and more.

next week is another talk to give, the consistent obligation of teaching prep, and down the road, pursuing this little thing called research and a thesis ..

.. right? remember those?

---

i've been making a few vows, lately, and here is a new one:

from mid-september to mid-october, i'm committing to nothing but teaching and research and writing. i refuse to give any more talks or volunteer for anything superfluous until late october; i've been too busy for too long and it's time to take control of matters.

love/hate the office.

east hall, home of the math dept @ um, is becoming quite populated.

in fact, ann arbor is becoming quite populated, with returning undergrads and their parents arriving in droves.

sometimes it seems like the office is a paradox of intent: it is a place to do work, yet it is one of the hardest places to accomplish any work. sometimes, i can barely hear myself think, or worse, barely able to have coherent thoughts of any depth or sophistication.

this term, i think i'll try to work at home [1] in the pre-noon hours, and when i have an idea to implement, i'll do so at the office during the afternoon. it's similar to how i treat computer programming:

never sit down to code unless you have a plan in mind.

in this case, it's never go to the office without knowing what mathematics you have in mind.

[1] note that "home" also means 'caribou coffee.'

oh great. fall term soon begins.

woefully unproductive: that's how the last few weeks have been. i've been stuck on this or that little lemma, and they are not terribly interesting. sometimes i feel like i merely play with numbers.

unsurprisingly and unfortunately, this unproductivity is my own fault; having promised too many people that i'd do too many things, there's little time (and peaceful time, at that) to do what i'm supposed to be doing:

mathematical research, and enough of it to make a thesis.

i'm also supposed to be teaching soon (after Labor Day), but that involves teaching calculus to children who look like fast-forwarded adults, and who don't care about calculus anyway. so that hardly counts.

too many seminars and talks to give, too many students already and more returning to town, and too much impatience on my part.

metric co-tangent bundles: a happy rant.

man, this co-tangent bundle stuff is so damned cool. right now i'd say that it's cooler than elvis.

to work in this theory, one has to use a little of everything:

the weak-* topology from functional analysis, and worrisome issues of metrizability and nets vs. sequences;

metric derivations which are reminiscient of differential geometry, but with Lipschitz and not smooth flavour;

and the bread and butter of the analysis on metric spaces: the boogeymen fractals, like Cantor sets, Sierpenski carpets, Laakso constructions, and other such things;

it is like seeing old friends, meeting faces that were always familiar but to whom you were never introduced, and then having friendly arm-wrestling contests and laughing after somebody wins.

i mean, it's hard, and sooner or later i'll post here again about how annoying and frustrating all of this is, but i like to think that it's interesting work.

in life, there are plenty of ways to practice drudgery and misery, but to do so in an interesting way .. that is rare, indeed!

we are creatures, driven by conflict.

this crackpottery [1] must end. if it doesn't, i might become obsessive and brooding and moody, and that serves no one.

i'm beginning to think that the tukia-sullivan tiling technique is simply incompatible with the notion of (weak) second derivatives, and to solve the problem, one needs an entirely new technique.

[see here for my reason(s)]

maybe i should pull the plug on this last thesis problem, and let it die. i've already been given this metric co-tangent theory on my plate, and maybe i should start thinking about what can be done there, and what might make a good thesis.

---

i hate giving up.

i've given up on too many things over these few years, whether they be research problems or not, and i don't see it ever reducing in number. what hope does it give, when all the things i've set out to accomplish have resulted in failure?

some people have told me that even good, even great mathematicians can't solve all problems they want, and that's true. i don't expect to be great or good or even mediocre, but what does that say about us?

that we settle for less, and dare not dream of mountains higher than those we've seen before and scaled? that we stop hoping for lofty heights and enslave ourselves in heavy chains, for our own protection against setback and failure?

what is the point, then, of ever attempting anything?

f*ck. i hate it when i argue my way into something intractable, mathematical or philosophical. this is exactly what i mean by becoming "brooding and moody."

education and knowledge have not made my life any better, but then again, was there any reason to expect them to do so?

---

[1] as you might guess, yes: 'crackpottery' is the state or quality of being a crackpot, and no: it's notin the dictionary. like the word 'carefreedom,' i think i made it up.

never trust groups; functional analysis and metric geometry, oh my!

my crackpot ideas (for reviving the dead thesis problem back into life) aren't working. that should come as no surprise, of course, but these days i can't help but take such things personally ..

.. which can't possibly be healthy. then again, how healthy can it be, to think about mathematics for most of the day? q:

so, after having thought a little about group actions in the contexts of the conformally-natural extension of circle homeomorphisms (cf Douady and Earle) and the hyperbolic tiling extension (cf Kirby, Siebenmann, Sullivan), i think i'm entitled to this opinion:

i hate group-equivariant mappings, at least when it comes to second derivatives.

you'd think that actions by isometries would be reasonably nice, but no. there is an inherent problem between equivariance via group conjugations F = g F g^-1 (which are, heuristically speaking, rescalings of space) and second derivatives (which are quantities describing curvature).

---

meanwhile, the metric co-tangent bundle theory does look interesting .. if that's what it's called.

between the work of N. Weaver and that of J. Cheeger, there are function algebras and abstract constructions running amok and amidst the analysis on metric spaces and measure-theoretic geometry. it feels like i'm walking around a space station and gawking at the various alien races, all the while asking myself,

"what am i supposed to be doing, here?"

and somehow, i suppose that when i answer that question, i'll know what my thesis problem is.

is it all fun and games?

upon the death of my thesis problem i devoted myself to being (mathematically) unproductive and avoided entangling alliances of all sorts.

it worked well until this afternoon.

as you may have guessed, i started entertaining crackpot theories, mostly about how to raise the problem from the dead. i'll mention something the moment that i see something rigorous in the works, but don't hold your breath; i won't.

---

as for the title, it seems like the p-harmonic folks are spending more time at games, especially tug-of-war!

• a friend of mine told me once about a game-theoretic formulation of mean curvature flow via pde,
  
  
• earlier this year, Peres, Schramm, Sheffield, and Wilson gave a formulation of the infinity-Laplacian in terms of "tug of war,"
  
  
• then earlier this summer, i heard a talk of Jan Malý about a game-theoretic version of the eikonal equation,
  

and now it seems that Peres and Sheffield have gone further and interpolated some of their earlier joint work:


Tug of war with noise: a game theoretic view of the p-Laplacian

Fix a bounded domain Ω in R^d, a continuous function F on the boundary of Ω, and constants ε > 0, p > 1, and q > 1 with p^-1 + q^-1 = 1. For each x in Ω, let u^ε(x) be the value for player I of the following two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed and the player who wins the toss chooses a vector v of length at most epsilon to add to the game position, after which a random "noise vector" with mean zero and variance (q/p)|v|² in each orthogonal direction is also added. The game ends when the game position reaches some y on the boundary of Ω, and player I's payoff is F(y).

We show that (for sufficiently regular Ω) as ε tends to zero the functions u^ε converge uniformly to the unique p-harmonic extension of F. Using a modified game (in which ε gets smaller as the game position approaches the boundary), we prove similar statements for general bounded domains Ω and resolutive functions F.

[stunned]

in my meeting with the advisor today, i think we killed the thesis problem ..

.. and no, 'killed' doesn't mean 'solved;'
it means 'unsolvable' or at least 'intractable.'
maybe it means 'impossible.'

i prefer the term 'insoluble,' myself.

i mean, i got a corollary out of the deal,
and if i write it up, maybe that will be a paper.

but for now, i think i'll crawl under a rock, and stay there for a while.

a personal library update.

today in the second floor atrium, there was a huge pile of mathematical books, many of them classics, and there was a sign which read:

free books

now i am eight books richer, though it's hard to say if i will ever seriously use them ..

.. but no matter. though there may be no such thing as a free lunch, there is such a thing as a free book!

---

i finally updated my LibraryThing mathematical book catalog, and again, you can view it here:

http://www.librarything.com/catalog.php?view=grey_ghost

i'm up to ninety-five (95) books, though not all of them are that "mathematical." judge them as you like.

anyways, back to work. the theorem won't prove itself, after all.

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 ....

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.

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.

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 W^1,1-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 l_p 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!

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.

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."
  

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:

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.