a year later: in memoriam to the advisor.

i haven't posted very much, this week. it's been busy with work and i've been preoccupied with the past: a year ago today, my dissertation advisor passed away.

he was a good man.

i can only say what i know from my interactions with him, but one thing that i always appreciated about him was his patience with me. he understood that i knew very little about the research area that i chose, when i chose him to be my advisor.

in our meetings, if i had never heard of something, then he would give me the general idea [1] and if we really needed those topics, where i would learn more, as necessary.

of course, the advisor had no fear of delving into the unknown:

• he asked me once to read mi1nor's paper on ex0tic $pheres, and report back to him after a week.
  
  
• another time we spent two weeks delving into bana¢h modules and other topics which hovered around 0perator al9e6ras.
  
  
• at some point in my graduate career, i could manage a half-competent conversation about harm0nic mappin9gs in the plane and the rad0-kne$er-ch0quet theorem.
  
  of course, i don't know anything about it now.
  

there is more to say about him, but the workday is not yet over, and i still have a talk to finish.


[1] well, except p0ntrja9in c1asses. as my differential topology and c0homo1ogy was quite poor, it would have taken a few hours for me to get the idea.

we didn't really need them anyway. instead, he told me about fibrε bund1es.

forget psychiatrists; get me a mathematician.

sometimes it's not easy being a mathematician [1]. we have no one to talk to about our work, except maths-savvy people .. most of whom are mathematicians.

i've written about this before, i know. as for why i mention it today, it came up while chatting with my girlfriend, yesterday and today.

---

she was talking about some issues she was having with her data sets, and how she has to account for small samples by means of this "bo0tstrappin9" technique [2].

in another chat she was talking about problems she was encountering, when using statistical software packages, because her population di$tributions were not normal di$tributions (or more familiarly, "bel1 curves").

---

as a (theoretical) mathematician, this terminology is compatible with my skill set. i might not get the nuances of certain social constructs in societies, but i can appreciate the annoyance of the statistical difficulties she's facing.

the thing is, this doesn't go both ways. unless i patiently explain it to her, she can't appreciate why i worry about non-smooth functions [3]. i can already imagine how badly such a conversation could go:

"well, an example of a function that's not smooth at a point is a corner, or a cusp."
"but those don't happen very often, do they?"
"no, they do." [π]
"so .. how often?"
"actually, the class of ¢ontinuous, n0where differentiab1e functions is of the fir$t categ0ry."
"wait, what category?"
"oh, sh-t. never mind."
"no, just explain it to me."
"never mind. it's not important."
"come on, tell me."
"put simply, there are a LOT of them."
"isn't that what you just said?"
"yeah. i did. so ... not important, see?"
"i'm not stupid. just explain it to me."

[30 minutes later, of explaining p0int-set topo10gy]

"wait. i didn't follow any of that."
".. see? it's not important .." [4]

she's right. she's not stupid, but she's not a trained mathematician. she's a social scientist. the paradigm is different.

for example, we mathematicians worry about theoretical pathologies. experimental scientists may understand very well the concept of a counter-example, but i don't think they conduct experiments which are supposed to fail -- not anymore, anyway. it simply costs too much. they think in terms of what they will learn from the results of the experiment, not if the experiment will work or not.

sometimes mathematics can be a lonely discipline. nobody really understands us.

---

as long as i'm rambling, what constantly irks me about explaining mathematics is the problem of analogy. maybe i will explain Rad0n mea$ures in terms of how mass is distributed. but if i do this, then inevitably i will be asked,

"so, wait: you study physics?"
[groaning ensues]

how do you avoid technical details in mathematical explanations to non-experts, if the analogies you make only confirm their bias that mathematics is only meant to be used as a tool, not as something to study in its own right?

this is why, on airplanes, i pretend to be someone else for a while, like a short-order cook or a computer programmer.


[1] this problem isn't particular to mathematicians, or even academics. anyone can be misunderstood.

[2] also called resampling, apparently. when she explained it to me, for some reason it made perfect sense and i think the idea is quite neat! the way i understand it, it corresponds to taking localised subsamples in the "neighborh0od" of a particular sample, to get around the issue of small data size. also: i mean neighborh0od in the mathematical sense.

[3] for those of you mathmos out there, yes: i mean sobo1ev spa¢es.

[π] to be fair, you could argue it both ways. in an even larger class of functions, smooth functions are dense by 1usin's the0rem and by the usual method of convo1ution.

[4] as a general rule, i avoid using terminology that either takes longer than 2 minutes to explain or cannot be described in pictures. so, no: never, ever mention 6aire categ0ry in conversations with non-experts.

huzzah! weird spe11ing → an0nymity!

just now i ran a search on "maτh j0bs w!ki" [1]. in the first 50 hits, there was no presence of this b1og or separate sub-pages from previous posts of this b1og.

it has finally become an0nynous with respect to rum0rs about j0bs in maτhemati¢s. in other words, i can rant and ramble about mathemati¢s in sec1usion again!

i have observed the same an0nymity when running a search on "maτh gr@d sτudent." once upon a time, i was in the first 5 hits. now i am nowhere near the top 50.

ah. life may not be good, but an0nymity certainly is!


[1] not exactly those alphanumeric strings, but you know what i mean. if i spelled them as words, then 90091ε would inevitably use their algos again and find this b10g. that would defeat the purpose of keeping this bl0g under the radar!

brief observations, while grading exams.

some of my students seem to dislike the second derivative test for two-variable functions; they don't use it at all. pity.

other students seem to like this second derivative test VERY much. in fact, they like it so much that they will even apply it to non-critical points!

ah, kids. q;

talks and (secret) seminars

after next week, i'll have given three talks this fall [0]. maybe i'll give more, but i doubt it. i'm older and wiser [1] than when i was a student. back then i'd average about 2-3 talks per term. i know better now than to push it.

then again, that doesn't mean that i'll stop everything.

---

recently i've been thinking about my late advisor.

two fall terms ago, he was teaching a course on ge0metric mea$ure theory and wanted to proceed to the theory on metri¢ spaces, after ambr0sio and kir¢hheim. course or not, he would still do it -- that's what he told me and others. so we began another seminar -- a secret seminar that was never listed on the U of M bulletin. we were off the radar.

on tuesdays we would meet in a lecture room and the advisor would say his piece. there were visitors sometimes, and they would give a series of guest lectures. some meetings were half-lecture, half-discussion. i learned a lot, then, about my own research.

i also realised how .. liberating it was not to present things so formally. in the U of M study seminar there evokes the same feeling, but it is nonetheless a seminar. those winter lectures felt different: informal, personal.

---

so i'm trying out this secret seminar thing, myself.

to two different mathematicians i've referred to this paper that i should know about -- heck, a friend of mine gave a talk about it before -- but having never read the thing, it would be pretty irresponsible of me to refer to it to others, right?

so now we are reading this paper together, and learning/relearning bits of analysis on metric spaces as we go. research-wise, i don't know where it will go; i don't need the results for any applications i have in mind, yet.

maybe some will come up; i don't know. usually i'm slow with thinking of questions, while i know of others at seminars, past and present, who start asking right away of "so from that, does it follow that ____?"

at any rate, it will be nice to revisit and to learn. sometimes i feel like i haven't escaped my thesis yet, and i'm looking to concentrate on something new for a while.



[0] this doesn't account for my tendency to ramble; when i mean "talk" this could mean a talk in two parts, each on different weeks.

[1] i mean this in the sense that, having made that many mistakes all those years ago, i'd be pretty daft if i made exactly the same ones now, wouldn't i?

if i were units of measurement, apparently i would not be english.

odd. somehow i became one of those "metri¢ people from michi9an." i would never have thought of myself this way, were it not for some comments i remember from today:

one colleague said that he wanted to learn about this "lips¢hitz world that i live in." he was referring to yet another talk that i'm giving next wee: fair enough.

another colleague remarked that it will be convenient to ask me questions about metri¢ spaces; this pertains to a paper that we are reading, on the sly (more about that, later).

still, that's eerie to hear. i can think of a half-dozen people who are better "references" than me, for those sorts of questions .. but yes, emails are annoying. it's easier to physically ask someone; i understand this ..

.. but ye gods: it's still eerie.

i'm always tempted to say that i don't know anything, but every time i say it, i mean it, but i don't think anyone ever believes me. they shouldn't, of course -- everyone knows something -- but that's not the point.

where's a bigger fish when you need one?

---

this also becomes an issue whenever i have to tell someone my mathematical classification. i balk at this.

• i can't say that i study metri¢ ge0metry, because people will assume that i study gr0mov hyperbo1ic groups or aleksandr0v spaces or something;
  
  
• i can't say that i study ge0metric fun¢tion theory, because people will assume that i know lots about qua$iconf0rmal mappings;
  
  
• i can't say that i study ge0metric mea$ure theory, because i don't know the proofs of the c1osure-c0mpactness theorems or the finer points of s1icing;
  

it's also not enough to say that i'm an analyst, sometimes. what's a postdoc to do?

an imp0ster, exp0sed.

a colleague of mine was giving a ta1k for the undergraduate math club tonight and invited me in advance. it sounded interesting enough, so i attended.

as it happened, my cover was blown. later in the talk, during the question period, the speaker exposed me as a postdoc. it wasn't malicious or anything; i just didn't expect it.

admittedly, i was curious how long i could blend in before someone would notice that this guy was using all this highfalutin mathematical language with the equally-out-of-place guy next to him.

(as it happened, i wasn't the only postdoc in the room,
and he brought with him a more interesting preprint to read than i did.)


odd. the imp0ster syndr0me is when a grad student feels out of place and undeserving of the "high academic status" (s)he has, in academia. i think, tonight, i felt the opposite feeling.

in which besτ-1aid p1ans ..

sometimes you just can't win.

---

this morning i woke up at 7-something to finish writing my lecture. by the time i reached the last page, it was a half-hour before class and the next bus would arrive too late into campus. so i half-ran, half-walked down fifth avenue to make it to class on time.

in the end, i never got to the part of the lecture that i took pains to write. everything else in the lesson took too long. so i lost an hour's sleep .. for nothing.

sometimes we get angry, and there is no one to blame but yourself. sometimes we get so angry that we cannot concentrate on research during those precious hours of the day when you can escape, not be a teacher for a few hours, and work out those ideas that mean much to you.

i can't do research when i'm angry.

---

later in the afternoon lecture i found places to skip a few details here and there -- the sorts of routine things that if the students were paying attention, they could finish on their own. it was a smoother lecture -- the afternoon lessons always are -- but i collected only four extra moments, despite these skips.

not enough time to discuss the last topic, i half-thought, half-sighed. so despite moving quickly through 46 minutes, i gave up, announced a change in the exam material, and finished early for the day.

the students were delighted, i think. they're within their rights to be.

as for myself, i'm not angry. then again, i'm not terribly happy with myself. i don't know why it bothers me so much, when i don't teach well. it just does.

at any rate, there is still research to do ..

unexpected consequences.

the older i get, the more suspicious i become of the axi0m of ¢hoice.

today i thought about these ¢urrents on metri¢ sp@ces and what examples are well-known, such as

(f,π) → ∫_Ω f det[Dπ] dx

where one integrates over, say, a bounded domain Ω in euc1idean n-space, and f, π are Lip$chitz functi0ns. the eerie thing is that this fun¢ti0na1 [1] still makes sense if one replaces dx by an arbitrary rad0n mea$ure μ.

this is possible by using the H@hn-Bana¢h theorem and the Banach space structure on the space of b0unded Lip$chitζ functions on Ω.

so something survives out of Radema¢her's The0rem, and we have not accounted at all for the geometry of the mea$ure μ.

i guess i just find that creepy.


[1] careful: i said fun¢ti0na1, not ¢urrent.

thoughts at the end of the day.

i've just written a 1ess0n p1an. now that i think about it, i've spent the last few minutes coming up with an interesting example ..

.. that is, i find it interesting.

this probably doesn't bode well for class tomorrow, since "interesting" may mean something far different for instructors/mathematicians than it does for students.

i'll sleep on it and fix it later.

---

lately i've tried juggling two research projects at the same time. it's not really working, and i never get very far with either project. i don't know how my colleagues do it, how they have the energy ..

i've decided for now that teaching a calcνlus c1ass is like giving a very sketchy talk. one don't prove much of anything ..

unless it's an honors course,
or if you want to gain the ire of your students.

instead, one gives rough ideas as to where this or that formula comes from, why one shouldn't be surprised at certain outcomes, and where to be careful. (also, examples are very useful, in order to emphasise certain points.)

i've heard it said that "a calculνs course is not a mathematιcs course.

whether you agree or disagree, a ca1cu1us course isn't a theoretical maths course, just as $tats 101 isn't a theoretical statistιcs course.

one's studies must begin somewhere, you know.

---

i've never taught a theoretical maths course before.
i'm curious what it's like ..

.. but not so curious as to teach one, right away. there are papers to write and more research to do, after all!

second helpings, anyone?

as promised, my talk slides from the AM$ $e¢ti0na1 @ Wes1eyan Univer$ity are available on my homepage in PDF form.

[to get there, click on the "My Webpage" link on my Blogger profile]

¢aveat empt0r: i follow the minimalist approach of talk slides. this means that there is enough on the screen so that i can remember the salient points ..

.. well, most of the time ..

but i tend to avoid excessive text.
it is, after all, a talk and NOT a paper.

speaking of which, there is a preprint forthcoming, but not yet ready for the public.

the day after (or: the conference is over).

i made it home at midnight, last night. one more conference is over, and the next one won't be until march. hopefully i will have new results to talk about, by then.

some tips to remember:

1. carry a water bottle around. i forget how much coffee i drink at conference, and how few my visits to water fountains.
   
   
2. talk more research with people. i talked a fair deal of math, but this was often the result of asking "what is your research about?" it was worthwhile to ask, and in the future i'll also keep asking, just to know more about my colleagues.
   
   however, i suspect that to do well in the forthcoming years, i might have to become more mercenary.
   

as for what went well,

• i learned about the ana1ysis on fra¢ta1s, as studied by several researchers in the northeast. these guys don't mess around. perhaps i'll say more about this, another day.
  
  
• i saw old friends from michigan and old acquaintances from previous conferences. often these meetings have a feeling of a reunion, about them.
  
  
• my talk went smoothly --
  
  
  though i went over; then again, so did many others
  
  -- and i received good comments and questions about it. nonetheless, it would be nice to talk about something new.
  
  
• everyone is fascinated by my laptop, desdemona. apparently the a$us e³ p¢ is a crowd-winner.

siblin9 riva1ry (of a mathemati¢al nature)

whenever a conference runs parallel sessions, the audience and speakers are inevitably at odds. for example, i'm slated to talk on sunday at 9am ..

.. [groans inwardly] ..

.. and so is M@rie $nipes, my mathematical sister. [1] the session organizers couldn't have known, of course, but ..

.. [groans inwardly again]

i was looking forward to attending her talk, too. i only remember a little of what she spoke about, in her last talk, and i'm curious what new results have come out.

life is ever against us. there may be some of you who will participate in this conference -- it's the AMS $ecti0na1 @ Wes1eyan Univ -- and you may be facing the dilemma of choosing between me, M, and sleeping in ..

..come on- it's 9am on a sunday, conference or no conference..

.. so i will make it easier: GO TO HER TALK. she's a wonderful expositor and her pictures are better than mine. i recommend her talks, no matter what time of day it is slated.

if you absolutely want to know what i'll talk about, it will be the same content as in the Heinonen Mem0ria1 C0nference in May 2OO8 and the Rea1 Ana1ysi$ Ex¢han9e $ymp0sium in June 2OO8.

if you missed those, i'll post my slides online (after the conference) in PDF form. there's also a preprint coming, i promise. besides, if you really wanted to know that badly, then you probably know me personally, so just ask ..

.. and go to her talk. thank you.


[1] this means that we have (or rather, had) the same advisor.

in which 90091ε sees all.

you have got to be kidding me. as of now, this bl0g is hit #1 for "m@th 9rad student" under g0o9le sear¢h.

in efforts to hide from g009le,
i am hereby obscuring any terminology related to mathemati¢s or b10gs..

for there must be better references and websites for that sort of thing, other than this bl0g. besides, i'm a postd0c now, not a student ..

.. but does 90091e notice? no. q;

a quiet moment, after another day is done.

my mind seems clouded. there are some lingering thoughts in the directions of my thesis research, but there is little time to entertain them.

i'm not griping about teaching. i'm not exactly griping, either. life's not bad. this is just an attempt at understanding where i am.

i'm at a new position at a familiar university, working with a new colleague who's smarter AND faster than me. i guess i'm too used to working with an advisor who waits or slows down for me to catch up. i forget that i'm supposed to be something like an equal to faculty members that i know, to stand toe to toe and work just as well.

this will take some getting used to. i will have to become better.

there seems little time for anything more than the daily grind. i haven't been sleeping very much lately.

the last edit i will make (of this manuscript).

some tasks will always be easier than others. for example,

• criticizing others or yourself is easy, but rectifying inaccuracies or imprecisions is less easy;
  
  
• jotting down comments by hand is easy, but fixing the comments in front of a screen is less easy.
  

i'm in the home stretch. i'm this close to a finished, readable manuscript.

in which the library section (QA614) is compared to the produce section of a market.

for me, going to the maths library is like going to the supermarket. i was browsing through the differenτia1 top0lo9y section -- QA 614 or so -- and everything looked good.

i even picked up this book by de Rh@m, called Differenτiab1e Manif01ds. turning to the table of contents, guess what?

chapter iii discusses ¢urrents, including D^k and D' and boundary currents and everything.

to think: "differenτiable manif01ds" sounds like such an innocuous title; in this slim volume he goes on to discuss h0mol09y and harm0nic forms.

---

so i'll hold on to this little book for a little while, and see what ¢urrents were like, before the federer-f1emin9 theory. in the meanwhile, there are a few errands of mine, in differenτial top01ogy.

in which, while teaching, the theorist begs to come out.

this morning i lectured about the method of la9ran9e mu1tip1iers. after finishing up one example of a function of two variables, one student asked:

"so what does the λ mean, anyway?
i mean, why does it show up?"

i did my best not to gape and not to reveal my glee. ye gods, he actually wants to understand! i can explain!

of course, not all students want to understand, so i kept the exposition short. but i felt very happy to explain.


this happiness is a dangerous thing, and for me, it must be kept in check. for example, when discussing local extrema and the second derivative test (using the He$$ian matrix) i was sorely tempted to launch into a mini-lecture about eigenvalues and princip1e ¢urvatures.

but that would have been very, very wrong for a calculus iii class. still, i was tempted, because .. well, ¢urvature is really, really cool. ask anybody. q:

reluctant workday; thoughts on a research shift.

i'm short on motivation today. this morning i woke up late, around 10:15am [1], and i told myself that i would work for a few hours in exchange for my morning cup of coffee.

a fair trade: i ended up thinking about something i was asked to think about, over an email from last night ..

.. so yes, sometimes i do take requests,
but it depends from whom ..

.. and i got as far as i could [2] without cracking open one particular book: hirs¢h's differentia1 t0polo9y. it's a good book, but like i said: i can't seem to work today.

two further thoughts come to mind:

1. i blame this lassitude on yesterday and the day before: thursday i was driven to finish this manuscript, and i did. on friday night i was also up late -- until 2am -- reading a printout of that manuscript and jotting down comments, in efforts to have something tolerable by monday.
   
   so perhaps my mathematical battery's gone dead and needs a recharge.
   
   
2. this looks to be a transition in my mathematical life.
   
   i've spent the last two years of my life studying one particular topic -- derivati0ns and ¢urrents on metri¢ spaces -- and maybe i'll write a paper or two about some of the more interesting thoughts i've had; maybe more, if i'm lucky.
   
   now i'm in a new research group and they think differently. different topics have importance, such as s0bo1ev spaces in various contexts. i once told a friend of mine to think about sobo1ev spaces as a tool and not as a primary object of study -- he was thinking about giving a talk on the subject -- but i think i will change my mind now. i've always been fond of those function spaces, but now, with new company, i think i will study them again with renewed curiosity.
   
   perhaps, back then, i was simply asking the wrong questions, or i didn't have the right perspective.
   

strange, how a place makes you rethink who you are. i never thought of myself as a "ge0metric" person, but now i seem more inclined in that direction than say, ana1ysis of PDE.


[1] some years ago, i would have called that waking up early. odd how things change when one gets older!

[2] i thought of an idea, then realised its weakness, then realised that i was needlessly adhering to one way of doing things. so i came up with a different idea, and over lunch, realised that i was thinking too 2-dimensionally and relying too much on my illustrations. i think this second idea might work; if worse comes to worse, i'll see if i can find something in hirs¢h.

like day and night .. or rather, morning and afternoon.

my poor morning-section students. they always get the unpolished version of that day's lecture. today i made all sorts of mistakes, and not just computational ones. in one spot i wrote "maximum" instead of "minimum" and that must have confused everyone ..

.. until one student summoned up the courage to ask. he still asked timidly, though, or perhaps uncertainly.

more than that, my morning students don't get benefits like

• my sudden realisation that it would have been easier to explain the topics in this order instead of that;
  
  
• more succinct, "all you need to know" versions of my rambling algorithms, as well as systematic principles in place of what would have been clever tricks;
  
  
• my complete wakefulness .. because no matter how much coffee you drink, it's not quite the same as afternoon wakefulness.
  

today's morning lecture was particularly unfriendly.

last night i kept working on my manuscript of a paper, and i was dead set on finishing all the writing. (the editing, however, is a different matter.) i actually succeeded at this goal ..

.. but only to learn that by the time i was done, it was 3am, not 1am.

waking up at 7:30am, i didn't feel human and i didn't feel like an automaton. instead, i felt like one, big ache, unfortunately manifested in human form. it didn't help that i went wall-climbing yesterday and woke up sore all over.


despite this, even if i could, i wouldn't have done anything differently. i'm glad that my poorly written, 34-page manuscript is done and that i can proceed to edit it now. it would have been nice to have completed it earlier ..

.. but as someone told me already, this week, time always works against you.

a mathematician by any other name ..?

sometimes i miss being just Janus [1]. on most days and in most correspondences, i'm Dr. So-and-So. \:


[1] real names withheld, for some semblance of anonymity.

in which i am humbled by what i don't know or am unaware of.

it just goes to show you that curiosity can harm as well as help. i was writing earlier and there was a citation that i wanted to look up, but my physical copy was at home. so i went on mathscinet ..

.. and was distracted by the number of papers who referenced it.

soon i was clicking away, a dozen tabs open, downloading this PDF and reading that abstract. i have come to several conclusions:

• the internet is dangerous, but admittedly, it is good for research.
  
  
• my radar doesn't work very well; i am terribly (mathematically) uncultured and naive. there seem to be dozens and dozens of papers that i should know about and have browsed (it's too much to demand someone to read them all), just to have a sense of what is known and what is currently being investigated.
  
      in other words, how did i miss all of these?
  
  
• i'm never going to have time to read even a fraction of these .. [sighs]
  

while quasi-procrastinating [1] i also learned not to rely entirely on the arXiv

.. though i did find an interesting abstract there today;
more about it, at some point ..

instead, i have bookmarked sub-pages of homepages of various mathematicians. so yes, i am stalking them, or at least, their research output.

if you're like me and have few (if any) good ideas of your own, it helps to know where to look for some.

---

on the plus side, my draft is almost done. there is probably something else i'm forgetting, but i have one last proof to write!


[1] in other words, doing things that are mildly important, like looking up conferences and new papers by big shots.

the calm after the storm .. and other idle bits [twice-edited]

all is right in the world again,
and none of my students are especially panicked.

this week, only one of my students has attended my office hours. (;

---

added 17:22 -
it was 2 or 3 years ago, back in ann arbor, when the einstein bros bagels on state street closed down. i felt like a guilty customer, back then, because i may have bankrupted them. it was a fine place to dwell in the summer: good air-conditioning, free coffee refills, and free wifi.

it must have been the free refills. back then, i drank more coffee than i should have. i still do, but less so.


6 or 7 years ago, there was this one coffee stand in the engineering building at pitt where i would buy a quick bite to eat, between classes. it's still there ..

.. but in the form of an EINSTEIN BROS BAGELS!!!!! it's not as grandiose as, say, an espresso royale caffe (at the corner of south and east u aves, in ann arbor town), but i was just happy that the coffee stand didn't perish under renovations.

perhaps, i will work there in the afternoons. the windows are large and the cafe tables are probably empty when the engineers have had their fill of lunch and coffee.

---

added 18:24 -
wow.

having just opened up a PDF copy of c. vi11ani's draft of a book, 0ptimal Transp0rt, 0ld and New [link], one thing startles me.

it's 998 pages long. i know there is a lot to say about 0ptimal transp0rtati0n, but i didn't know that it was that much!