old friends, mathematical needs.

in high school i had this friend who was (and remains) a leftist. he loved political science and history, and his goal was to become a labor union organizer.

so he went off to university and enrolled specifically into a school of industrial labor and relations. when it came time to graduate, the labor unions found him too bookish and lacking in experience.

what irony:
according to these unions, i may have more "experience" than him,
though he would be worth a dozen of me, in effort and heart.

however, today i'm writing about him and not me.
perhaps i'll tell what stories i know, another day.

---

i say that he is bookish, because he likes books and loves knowledge,
he became a student of library science, and following the trend,
he has learned about information science.

he has become a techie.
he now knows how to write code;
i think he can out-code me, most days of the week.

he has a few weaknesses, though:
he likes math but doesn't believe he can do much of it.
odd, then, that he asks me about my work, wants to understand it,
and does .. to some heuristic extent.

---

in this story, here is where i come in:
you see, he asked me to look through a few papers for him,
about some applied mathematics and algo.

earlier today i browsed through a paper by dre2ner from 1985, about an optimization problem in the plane. mostly it's linear algebra stuff, and i think it was written for people who need maths but have specialised in other things.

the next paper is by m@rc0u1ides and dre2ner, about compressing space -- collections of N dimensional points -- into a planar representation. i suspect this will be similar, except error terms now become important, but i'm curious if there's any interesting geometry.

---

strange, when i think about it:

ten years ago, my friend and i would have talked about the communist revolution and whether it made any sense at all. we would have talked about voltaire -- rather, he would talk and i would listen and nod -- and perhaps we'd talk about ancient philosophies.

i think that when we talk next, it will be about symmetric, positive semi-definite matrices, norms and homogeneity, and the geometry of linear algebra.

weekend indulgences.

lately i've been splitting up my mathematical pursuits into two kinds:

1. mathematics that i promised to do;
   
2. mathematics that i want to do, right away.
   

this can't possibly be healthy. i blame this on my habit of making promises too easily, and having no sense of how long it takes to do anything mathematical.

this weekend i am indulging myself.

these past weekdays i've attempted -- sometimes succeeded -- in writing a first draft of a paper, as derived from the results in my frustratingly long thesis. admittedly, most of it involved cutting and pasting, second-guessing, rewording, summarizing, and other under-handed tricks.

---

so what is the indulgence?

i get to think about new math, sort out a new idea of which i am still suspicious. the last time i met my math sibs, i sketched this same idea for them and persisted that it was only an idea, not a proof.

they thought i was being modest;
they thought it was a sketch of proof.

well, i think i found a gap -- something subtle, nothing serious -- but it requires careful thought. i tell you: nobody ever takes me seriously. i don't say that i'm worried, for nothing!

---

anyways, a gap is a gap. i tried an idea and it's not perfect, which means a new idea is in order.

this reminds me. my late advisor had this manner about him:

when he was alive and well, a year or more ago, i'd walk in for our weekly meeting. he ask how things were going. i'd often say i thought about this, i think it's close to a proof, but something doesn't quite work.

he would want to know. he would want to know how close i was. he would get excited about it. i think little gaps in proof were like riddles to him: among other things, it's probably why he was a good mathematician.

to me, this gap is a weird little riddle, and the ultimate endgame doesn't worry me so much. i have to say, it's just a weekend indulgence, but i'm excited about working it out.

pondering: what am i, again?

wow. i'm part of the mathematics genealogy project now. it appears when i am google-sought.

does that mean i'm no longer a graduate student?

i don't feel much like a postdoc, but that's fine: i'm not a postdoc yet. the job begins in august/september.

on the other hand, i don't feel much like a student either, but i suppose i am.

i can still check out maths books from the library;
i can still log into university computers;
i still have an office.

then again, i'm unemployed.
maybe i am a quasi-student!

on a lighter note: romantic neurosis!

i have been fretting over moving in the fall, about making these sorts of plans;

i've considered the remote possibility that if i meet a woman now and if we started dating, then in 1-2 months time, this will lead to heartache, complications, and the like;

i've even put myself in situations where i will not meet very many strangers, while in ann arbor ..

.. but i have not looked at a girl, run through long-term scenarios, and deeming them too dangerous, jumped out of a window .. even if they have said something cute or quoted a movie i know.

heck: that stuff's for comic strips,
not neurotics in real life! (;

maths in collaboration.

slowly but surely, the writing proceeds.

today was also productive in another way .. or at least it felt productive. in the afternoon i met with my younger math sibs and talked about math and about research and about open questions in gmt.

i don't know if it will become a full-fledged collaboration, but our backgrounds have a nontrivial intersection and we can understand each other well enough, mathematically.

even if it works out, the timing is difficult: both of them are applying for jobs this fall, and i'm already busy with plenty of things, like converting my thesis into a paper and working on another collaboration with another research group ..

.. which reminds me: i should get back to them.

---

i like making plans, and i prefer collaborations when there is a set plan in mind. it's not enough to have a goal in mind, or a fixed research problem: there should be a plan.

if i prove this and if you prove that,
and if our strategy works,
then done: everybody's happy.

of course, nothing is ever so neatly organised as that, and plans often have a way of going awry. in fact, answering questions is hard enough.

i don't know that many mathematicians, but often i hear this sort of dialogue:

Q: say, i have this question. do you know if ____?
pA [1]: well, if you assume this, then _____.

we don't actually answer questions in full generality, unless we are so lucky that our proof techniques fit the situation perfectly. often they don't, and we can only say what we are familiar with.

we draw pictures,
we ask questions,
we whet the problem down by reductions,
we examine the worst-case scenarios,
and we hope.

i guess i'm saying nothing important here and nothing too interesting, either. i guess i worry a lot, when in collaboration:

everyone is talking or no one is talking;

i wonder if this will really work because i've forgotten how not to doubt;

i worry if my ideas are wrong or unfruitful and that they lead me -- worse, a team of us -- down a wild goose chase, with no goose in sight.

it's one thing if i waste my own time, in the mornings, thinking to myself.

for others who make a point to show up, at my office and at this or that time and day -- how frustrating it would be, for them, if only after hours do we see that my idea is silly!

so i worry.

if it's just my idea and if it doesn't work, oh well. but if it is to become a group idea, then yikes:

now i must have a good idea, a plan;

make sure it's good,
have space in it so that others can work and contribute,

and not force them to sit down, listen to me drone on,
until i show them what might not work.

[1] that is, a partial Answer.

in which biases reveal themselves.

at the moment i am not terribly proud of myself, but perhaps mildly proud.

not very proud: a friend of mine gave me his thesis to read, some months ago. it weighs at least 80 pages and i've barely made a dent in it.

mildly proud: on the other hand, just now i read through an entire paper of another friend, which weighs 15 pages.



this confirms my original fears and inclinations:

when i do cut out a paper from my thesis, it cannot be too long,
otherwise weak-minded people like me will never read it.

chalkboards and ladders.

i'm in the office now. there's nothing like writing on a chalkboard. it has the right sort of unfinished quality to it, but unlike waste paper, it captures some sort of care and attention.

i'm checking some details, now. i've had this idea for a little while -- two months, perhaps -- and it's about time that i check it and study its weaknesses.

an idea can make you powerful, because it can give you hope, and direction too. nonetheless, an idea is a fixed thing in time and memory: the point is to form new ideas from old. an idea should only be a step on a ladder. one should learn whether or not the ladder points somewhere, and if so, whether or not the ladder is tall enough.

arrived, but unready.

maybe it was different a few years ago. now i am back in this university town, ann arbor, but i'm not of the right mind to do mathematics yet.

i don't think i am serious enough, yet. maybe it's the lack of sleep. hours ago, i boarded an overnight airplane flight to return to michigan today.

if i start working on an idea now, then i will not commit to it. it will be mathematical wanderlust and it will be a waste of paper. more than anything, it will be jogging my memory, remembering what problems i was thinking about before and if i remember any good, juicy leads .. which i won't.

it is good to remember, but it would still be a waste of paper.

i don't know if i'm well-rested enough to write, either. specifically, i mean: write something that i won't immediately delete tomorrow.

---

the thesis casts a long shadow. now i hesitate with my ideas, because i'm afraid they are not grand enough, expansive enough.

today i think i'll make good on my promise of a list of open questions.

questions can be innocent, and there is little risk if one asks a silly question. you don't have to solve anything, either: just think about what you don't know, realise what you don't know, and then formulate it. make a guess, if you feel bold about it.

thoughts about mathematics, without actually doing any mathematics.

not much maths since i skipped the conference early on friday.

i was able to write a few paragraphs of a proof in LaTeX on an airplane .. until my battery went out. then it was just me and skymall magazine and what i would purchase if i had a spare $1000 that was good for no other purpose.

i've been thinking of jotting down my own list of questions and open problems that i've heard about, in mathematics.

sometimes one can say a lot, simply by asking a question.

for example, asking a question tells the interrogated what you find interesting and what makes you curious. asking tells a little of what you know and are willing to try or have tried.

besides, my mind is untrustworthy; if i don't write these bits down, i might never remember them again.

along with this, there have been a few notebooks that i've accrued, over the years: their contents are nothing but scratchwork and random questions that have come to mind. perhaps i'll organise these up, in written form, when i clean out my office at U of M.

the conference, so far.

admittedly, i have this minor type of distrust when it comes to close mathematical colleagues. if they tell me that they liked a talk i gave, then i would like to believe them ..

.. but i cannot.
i still consider the possibility that they are just being supportive or nice.

perhaps they know how neurotic i am and unconsciously they want to placate me -- hence they do not lie, but they may exaggerate. this is possible.

however, this suspicious reasoning fails if a complete stranger walks up to me and says the same things. this happened today -- mine was even the last talk, too, which is always the most painful one to sit through -- so i think i must accept that it was a good talk.

i received a few questions, some which took a little while to answer. nonetheless it was fun to answer them.

maybe all that time writing a thesis has colored my opinions. i don't expect others to like the objects that i study, because they are no longer as exciting as i once thought they were. i'll still study them for a while, but having written and (re-)edited dozens of pages about derιvatiοns, i've had my fill of them.

---

this has been an interesting conference already. i've met all sorts of people, including friends of friends. in particular, i keep meeting students of one particular mathematician, but i've still not met that mathematician himself!

(actually, that goes for a few mathematicians.)

i'm being exposed to new ideas. just today i finally learned what a baιre-1 function is and what a Darbοux function is. in a talk i heard about a Banach space construction motivated by seeking mathematical foundations for the feyηman paτh integral.

strangely enough, i also spoke with one prof about a geometric problem with some application to Lιe grοup representatιοns. it just sort of happened. perhaps he misheard me: i said that i was interested in geοmetric measurε theοry, and maybe he heard ge0metry instead. nice guy, though, and his ideas are interesting.

it's nice to absorb truly new things, for a little while. it's nice not to be completely myself for a while.

on travels, and writing.

i'll be traveling for the next two weeks (more or less) [0]:

ann arbor → chicago → san francisco → las vegas → ann arbor;

some of it is business, some of it pleasure.
actually, most of it pleasure. (:

yesterday i debated whether to bring my laptop with me.

i tried to think objectively about how heavily it actually is, including the power adapter.

i thought about how much work i would actually do, as opposed to surfing the net, writing emails that can wait ..

.. and of course, blogging. q:

i made up my mind today: it's coming with me.

to save space, i won't bring any fancy shoes, and i'll wear running shoes for the whole time. we mathematicians aren't the sort to dress up, anyway.

---

as for why i'm bringing prospero [1] along, the answer is simple. the writing isn't going too badly. i've hit a good spot and i'd rather not stop.

looking at my thesis now, it's gory and too long and there are too many details. also, i'm too wordy.

now, i get a second chance to fix all of that.
i can write something shorter and better and to the point.

appropriate or not, i think i see how everything fits together now, how it all happens. i think i see the whole story.

before, my proofs were piecemeal and i wrote what i could, much like how a friend will get off the train and readily tell you his impressions of the trip he just took. sometimes it is too much, other times he almost skips entire days or weeks.

i think i can tell the story well, now.

i have no illusions about my travels and my work time. there will be very little.

i'm content with working on a 5-hour train ride,
between flights on the west coast,
in the mornings before my family -- after long-winded deliberations -- makes plans for the day.

believe it or not, i feel like writing. perhaps not right now and not every minute of every single day, but yesterday and today i wrote, and tomorrow i want to write some more.

[0] so, caveat emptor: i mightn't be updating this blog as often as in recent time.

[1] i must have mentioned this before: prospero is the name of my laptop. my last laptop was named arielle, and my last computer -- a desktop that kept crashing -- was named caliban.

goals and dares.

i think i have been lazy for too long. since last week tuesday i have not been working diligently -- perhaps i spend some few hours working, each day, but those are negligible.

i might as well be recalling what i did the day before, plan a little more, but not get my hands dirty at all. that's no way to work.

---

the weekend nears, and after that there is a conference. after that, i see my family for 1.5 weeks. so there are only a few days left in ann arbor, where i have complete freedom to work as i like.

so i've squandered a week and some days.
i guess that's not much, but it's led to this.

what can i do in a weekend? i want to write a paper ..no, two papers.. and what could i accomplish in two days? i want to think about a joint project, ponder the subtleties, and how can i get into a groove, in two short days?

---

but there's the fallacy, you see: the illusion that your goal is too big to fit in such a small window of time. sure, it looks hopeless ..

.. until you realise that goals are never atomic.

goals are always a sequence of sub-goals,
which are further sub-sequences of tasks
..and iterating..
they are large but finite sequences of trivial errands.

a lot of errands can be done in one day, and i have two days. sure, i promised to meet friends tomorrow and have a bit of fun, but this is not to say that i have no time for goals.

if these goals are worth doing at all, then there will be time for them.


i think i am too used to being worried and frustrated and pessimistic. everyone tells me to be optimistic, and occasionally i try it out. when i do and when it works, i realise again how empowering it is ..

.. until it lasts, anyway.

---

on an unrelated note, i met with my mathematical sibs today. they have ambition, they have less fear than i do. from their suggestion we thought about a conjecture together.

it's something i'd have never suggested, myself. during other summers, i might have suggested we read a paper together, learn a little something. then from our lessons, see if we can go just a little further and prove something new.


goals are one thing; dares are another.
i don't dare conjectures. call it residual pessimism, but

having tried on my own,
thinking about this open problem or that,
and going absolutely nowhere,

i don't have much stomach to dare anymore.

next week: stranger in a strange land?

not a very productive day, so far. at best i may re-write a talk today in preparation for chicago next week.

this should be interesting.

the conference is recurring, year after year,
the participants are a regular crowd, colleagues and friends
-- and even after attending the last one --
i will still feel like the "new guy."

oh well.

i should make new friends and meet new people, anyway. if i were a parent, i suppose i'd say that "it builds character."

this will require a more active participation.
i won't be able to assume very much familiarity.

i'll not only have to explain the odd bits in my thesis -- if anyone even asks about it -- but to explain what my research area is.

not everyone cares about metric spaces, or suspects that you can do any good analysis on spaces other than manifolds.

i don't think this is a crowd which uses the word "quasιconfοrmal" very often, or "sοbolev space" or "rectifiabιlity."

conversely, i think they will say words like "baιre categοry" and "pean&omircon; derιvative" and i might give dumb looks in return.

so if i want to be understood and to understand others, then i'd better be convincing and have a good memory!

interesting ..!

i haven't checked the arxiv in a while. i must be getting sloppy!

anyways, two recent preprints have come up on my radar. the links are below, but to avoid this blog arising from google searches, i'll only list the arxiv numbers and not the authors' names.

i have also obscured certain keywords with typographical symbols. as non-robots, those of you in the know will ..er, know what i mean.

i am curious, though: if somehow this blog comes up on a google search for these two preprints, do warn me. it means that i have to become even more careful!

arXiv:0806.0021 (submitted 30 may '08)

Abstract: Using isοperimetry and symmetrizatiοn we provide a unified framework to study the classical and lοgarithmic sοbolev inequalities. In particular, we obtain new Gaussιan symmetrizatiοn inequalities and connect them with lοgarithmic sοbolev inequalities. Our methods are very general and can be easily adapted to more general contexts.

arXiv:0806:0052 (submitted 31 May '08)

Abstract: We derive sharp sοbolev inequalities for sοboleν spaces on metrιc spaces. In particular, we obtain new sharp sοbolev emβeddings and Fabεr-Krahη estimates for Hörmaηder vect0r fields.

i find this second preprint particularly interesting. i wonder if it relates to another well-known problem in the area of sub-Riemanniaη geοmetry.

i had given up on thinking about Steιner Sυmmetrizatiοn, but it hadn't occurred to me that symmetrizatιon techniques on the function side of things would work.

these preprints are short; maybe i should print them out ..

in which i discuss.. chemistry?!?

earlier i was tutoring a high school student in chemistry, and we were on the subject of isomers. interestingly enough, the section in her textbook read:

"an introduction to functional groups"

and i blinked.

of course, it had nothing to do with functional analysis or with groups from abstract algebra. odd, though, how these scientists use similar-sounding jargon.

---

i think maths have made me too paranoid for basic science. for one of her homework problems, it read:

"find all isomers of alkanes with 4 carbon atoms and with 5 carbon atoms."

really, this is a combinatorics problem, even though the student doesn't know what combinatorics means.

i asked her if she was sure that she addressed all the possibilities, and i think i made her paranoid about it. we went through some of the fine print -- e.g. double bonds between carbons are not allowed -- and then it came down to some innocent sounding questions, such as:

"how do you know that these two structures are really different?"

as an example, think of pentane and 2-methylbutane, pictured here. they are intuitively different, of course, but how do you explain the difference?

more to the point,
how do you explain this without anything hard,
like topology?

maybe it wasn't fair of me to ask her this. her teacher would probably not ask this. but i asked her anyway, and now i have to think of a simple answer.

so i tried to explain it in terms of cut points.

look: if you remove this atom from this first molecule, then it splits into three pieces. in this second molecule, if you remove any atom, there are always two pieces, not three. so they can't be the same, because we get different numbers of pieces.

so i reduced to the absurd. then again, it wasn't mathematics and it didn't require rigorous proof ..

.. still, the answer irks me. maybe i shouldn't have asked the question(s) in the first place. the student could have gone the rest of her life less worried, not having to worry about whether she got all the isomers or not.

maybe i should leave science to the scientists and science tutors.

the summer will be short.

i've gotten back to my old day-to-day routine:

i wake up late, have breakfast and a coffee, stay in my apartment until the afternoon. while there, i think about a particular problem, work out some thoughts with paper and pen.

(the problem is almost never resolved.)

when i get tired of this, i go and wander outside, confused and uncertain;

this usually involves lunch and another coffee, and usually a lot of thinking about what to do next. half the time i decide to delay the decision and walk to the office;

i'll figure out what to do once i get there,
and sometimes, i actually do.

it's not a bad life.
i wouldn't mind waking up earlier, but there's no immediate gain to this:

i'd only quit my first work session earlier, have lunch earlier,
and confront the "what do i do now?" question sooner.

i've been avoiding making long term plans, which is unwise.

goals are one thing. i have many goals, but goals are abstract entities. if you never plan out your goals -- never say when you will do this task or that -- then apart from sheer luck, will you really fulfill them?

time is short.

i move in two months to a new apartment, a four-hour drive away. to make things more complicated, i haven't found that new apartment yet.

in those two months, i have a paper or two to write and a joint research project (or two) to work on.

within those months, two weeks have been set aside already: one week (or so) for a conference in chicago, and one week to see family on the west coast. likely there will be more weeks to remove, for other purposes.

so i have six weeks to write [1] and to collaborate. that's not much time. maybe i should cut short all this time off.

maybe i should wake up earlier after all. \:

[1] i knew it: the thesis was just the first stage.
there is always more writing to do.

books change (when you re-read them).

this past thursday i told a colleague that i'm taking the rest of the week off from mathematics. strictly speaking, i lied.

what i meant was: i'm not going to worry about turning the thesis into a paper, or meeting my current research commitments until next week. it was just easier to say that i was taking time off.

---

so these last few days i've done a little thinking. mostly, i've been idle and i've been reading recreationally. among other things,

snow crash by neal stephenson,
dark continent: europe's twentieth century by mark mazower,
invisible cities by italo calvino.

.. all good stuff.

calvino reads like jorge luis borges,
or perhaps borges reads like calvino,
or perhaps they have a literary ancestor, like whom they read.

no matter, that;
this is supposedly a blog about mathematics, right?

---

recently i've been thinking about BV functions [1] and the calculus of variations. a long time ago i borrowed from the library two books by e. giu$ti.

back then i had just begun graduate school and wanted to learn all sorts of things. i remember reading (rather: browsing) those books, and many others, with wonder and curiosity.

i had no motives or motivations;
i just wanted to learn.
i was a student, then.

these days, i am more like a researcher,
a luckless mercenary of ideas.

some days ago i borrowed the giu$ti books again, browsed them again. so far i can't seem to adapt those techniques to the problem at hand; it's just a different setting.

the ideas remain fascinating, and i noted something that i never noted before. there are morals in these books -- at least, early on, in the preface or first chapter of each -- so that the detailed, technical follow-up makes some sort of sense.

if you haven't seen the calculus of variations in book or talk form, then let me tell you: the estimates can get very gory. almost always, i dwell on one thing for too long, and then i am lost with what comes next.

perhaps i should mention one passage from direct method$ in the c@lculu$ of v@riati0n$: [pp. 4]

Actually, in order to apply the Weierstrass theorem, it is necessary that the functional F be lower semicontinuous, and that the set V in which one looks for the minimum be compact. These two properties are in some sense in competition; in order to have the semicontinuity it is preferable to endow V with a relatively strong topology; the fewer convergent sequences exist, the easier the functional is semicontinuous. On the contrary, for the compactness it is better to have the opposite: the weaker the topology, the easier for a sequence to converge.

maybe upon a little thought, this tension becomes clear. often i make much of things, but this is like a moral.

however, if you are new to these sorts of things -- functional analysis, theory of distributions -- you mightn't have considered the value of choosing just the right topology and narrow your focus to the right kind of function.

maybe you wouldn't have worried about topology at all. you'd have left it for the knot theorists and to those who play with homology theories. some functional analysis courses are taught in a "standard way" and one forgets that there is a lot of topology in the background.

at any rate, i like these books.

giu$ti also gives some attention to history and progress. this is valuable for someone like me, who moonlights in this area and doesn't know what is already known and in what direction the research area has gone or will go.

at any rate, there is still maths to do. by tomorrow i should think about thesis-derived matters again. the problem awaits!

[1] rather, functions of bounded variation. in some sense, these are the functions that you may have encountered in a first course on measure theory. see the wiki.