a trip down math memory lane.

my motivation for research is very low today. since i woke up belatedly, at 11am [1]) i don't think i've had any thoughts towards anything mathematically new or substantial.

part of me wonders ..

1. if this is unconscious disappointment in myself: i wanted to wake up early and get a good start,
   
   (despite a late night of friends, beer, revelry, and a propane bbq grill)
   
   but it's already a dismal start and from this, what follows will only be salvage.
   
   
2. if my mood reflects the weather outside. the last two days were warm and sunny, one of them humid. today is cool and cloudy grey: fine weather, despite a little rain drizzle.
   
   it reminds me of pittsburgh weather and those happy times.
   
   but it is hard to hold to absolutes, and not compare relatively: yesterday made such a fine evening, and today there is a comfortable yet uneasy chill in the air.
   

it's hard to say.

---

over the past few weeks i've written notes on many pages, and having no inclination or mind for mathematics today, it was a conveniently productive task to do.

it's startling. it's like reading my old journals from younger years, but these are instead of the mathematical sort.

there are all sorts of digressions into separate questions related or unrelated to my work, at hand. some of these read like ambitions and others like delusions of grandeur.

i didn't realise how much i filter, when talking research with the advisor. with him i stick to the successes and the instructive failures: the notions that make worthwhile meeting time.

some notes from a day, two weeks ago, contained a flaw in reasoning. it took a few days of further, confusing notes and scratch-work before i deduced there was an error.

revising my ideas, this destroyed the fleeting, hopeful fancies of what this flaw would have implied. so i forgot them, and with them i forgot a few good ideas which became 'guilty by association.'



[1] yes, i missed the defense that i mentioned, yesterday. so much for honor. \:

"we are men of honor. lies do not become us."

i never thought i'd encounter this sort of dilemma:

    i've been invited at least twice [1] to the same party, which will celebrate the thesis defense of a friendly acquaintance of mine.

    so i suppose that, as a personal matter of honor, i must attend the defense. usually, this wouldn't be so tricky of a decision, but there are two difficulties:

1. the defense begins at 9 am.
   
   yes: am, meaning ante meridiem or morning .. or as i prefer to call it: pre-noon. informally, it means either sleep-time or breakfast-time, or if i wake early, math-time.
   
   
2. the thesis concerns algebraic number theory and uses 'scary' words like p-adic cohomology.
   
   me, i'm just a poor analyst [2] who only knows a definition or two from basic (co)homology and can barely remember how to do any diagram-chasing!
   

[sighs]

it is a difficult thing, to struggle with matters of honor.


[1] once was through email, but that was a mass email to invite .. well, each and every math grad student. the two other times were the ph.d. candidate (who will defend) and one of the hosts/tenants of the house/party site.

[2] to this day, i don't quite know what to call myself, other than the broad umbrella term that is 'analyst.' i've met quasi-conformal geometers, geometric analysts, and geometric function theorists, but i don't know if i would call myself one of them.

let me prove a few more theorems and write a few more papers; then i will stake a claim! (:

on a lighter note .. monkeyBALL.

i may be weird when i called the boundary of an n-cube an n-square ..

.. but from the way one author describes the Monkeysphere, i think he means a Monkeyball. q:

beware your soul ..

i have been LaTeXing lately, and perhaps it will become a paper ..

.. after a long, arduous process, of course,
with many edits and rewrites,

but perhaps .. \:

---

i might have mentioned this before, but when left unchecked, LaTeX will consume your soul. computer programming of any sort, even excessive typing, will do this.

the problem, i think, occurs when one forgets that the code is not the important thing. for instance, it's easy to obsess over making LaTeX look even prettier than it already is ..

.. or when writing a program in C,
where you may write a function procedure
to handle the exception caused by the previous function,
which you had written to replace the earlier function,
which didn't have an exception,
but didn't handle cases in the object type that you wanted,
because your previous class construct didn't compile,
and you never figured out why,
and at the time you thought it easier to start from new.

one thing leads to another, then another, and at some point you realise that you're obsessing over something that has nothing to do with good mathematics or good programming or good anything.

i've heard that graphics in LaTeX are such quagmires.

so suggests the advice i've been told, when once i studied computer science:

unless you are really good at coding, NEVER improvise in front of the computer screen. always code with a plan in mind, and if the plan changes, sort it out first and then go back to coding.

---

then again, meandering can have its benefits. while updating a preliminary bibliography, i remembered a theorem from one reference.

i think i can generalise the result, now.
so the LaTeX ceases, and the mathematics continues.

concerning some controversy of late ..

on a more serious note, i learned today that there is a banff protocol concerning increases in mathematical journal prices and (nonviolent) protest:

THE BANFF PROTOCOL

We agree neither to submit to, referee for, nor participate in the operation of any journal that charges an excessively high per page subscription fee, as compared to the average of the 25 highest impact journals in pure mathematics**

as for my stand .. let's say that i'll wait until tenure before i make a stand. \:

in which i acquit myself of murder, and debug a proof.

this may be understatement, but mathematics can be confusing. i'd even say that it can be quite hard.

today over coffee i worked for a few hours, until my observations gradually suggested statements of an absurd inclination. i don't know how much logic everyone knows ..

(though i know that many of you do know enough logic, say, to be dangerous)

.. but i am referring to proofs by contradiction; colloquially speaking, they refer to verifying the validity of logical propositions to the absurdity of their negations.

---

to give a silly example, one might ask whether it is true or not that this morning, i killed a man in beijing with my bare hands.

well, suppose that i did; that would have meant that i was physically in beijing this morning, but that can't be true because i have an alibi: the barista at caribou coffee knows that i (or someone like me) had ordered a coffee and a cookie, and whittled away a morning, staring at pages of strange symbols while looking confused and frustrated.

so i couldn't have been in beijing, and so i couldn't have committed that murder. it's not rigorous proof, but then again, real life doesn't make a very good logical framework ..

.. so you'll just have to trust me: it wasn't me. (:

---

i still can't determine what, exactly, is wrong with a recent argument of mine. most of it seems right, if only because i've attempted to write a transparent, lazy argument which borrows most of its strength from the good results of others ..

.. but something remains amiss. i thought of an example which is inconveniently close to a counter-example to the conclusion of that argument.

in the last hour i worked, i resorted to troubleshooting and what computer programmers might recognise as 'debugging.' in other words, i narrowed down the errors to a few places in my argument-code, and now it's a matter of testing those places.

i'll input test data and look for valid output, but the data isn't numerical. instead i'll input a known metric measure space and the output will be a collection of geometric objects.

(the objects are metric currents, for those who know.)

so it's down to this: getting my hands dirty with the details of an example .. but isn't that what mathematical analysis is supposed to be? (:

contemplating hardy.

i remember reading that g.h. hardy would work four hours a day, from 9am to 1pm, and then spend the rest of the day to other pursuits, such as cricket or tennis.

whether this was actually true, i can see how it works. lately when i do mathematics, i feel like i do problem solving by trying to "think my way out" of a problem. it's easier to make progress when i've been thinking in that area for a longer period of time ..

.. but after four hours, if you can't "think your way out," then .. well, you're "trapped." maybe it's better to leave it alone, let it lie, and do something else for a while.

who knows? maybe it's best for the unconscious mind to process the situation, without the interruption of conscious, demanding thoughts.


as for what to do when you're not doing math, xkcd comics are perfect for the techie. today's was particularly soothing, for the mathematician:

fluency and a little hope.

life .. isn't bad, lately:
research-wise, i mean.

some ideas are working, and others are not, leaving unutterable mysteries for other days to ponder .. and 'unutterable' in that i still cannot find the right formulation.

---

i forgot my notebooks and folders yesterday, when i arrived at a coffeehouse after playing basketball and drinking large amounts of water. but i had some paper, and realised that i could work straightaway.

the questions and context came that easily to mind. perhaps it was the caffeine stimulus, but i thought in a few new directions and now there's plenty to investigate.


it's an interesting feeling: i feel productive, and i feel fluent.

i've often told my students that mathematics is a language, and it takes time and practice to do it well. but sometimes to do really well, one does the same as when learning a spoken language: immerse in it.

i've been thinking about this problem and the relevant literature since last fall, a nine-month immersion, and i think it's finally sinking in. it's not just simple phrases, like "hello" and "at what time does the train arrive?" more than that, i feel like i can have an actual conversation .. albeit halting and unsophisticated, but i feel the fluency.


i also feel hope .. or as much hope as i can feel, at the moment.

having no more brainpower for research tonight, i'm collecting my thoughts and endeavors into LaTeX format, and realising that it adds up to a sparse number of pages. never mind omitting some background and writing mere outlines of proof ..

.. there's not much.
then again, i'm not done, but what can i say?
my hope is a fragile sort.

open .. sesame? allspice? caraway?

it's a strange thing. whether classes run or not, my mornings have been as they always are ..

.. so why is it so hard now, to work diligently and accomplish something for my week's mathematical endeavors?

---

today i was distracted by too many things: the recent humidity (and a relative warm day in an non-air-conditioned apartment), a promise to play basketball in the afternoon, and another commitment in the evening.

i entertained an idea today, but it doesn't seem right. for one thing, it's not quite well-defined, but it's temptingly close ..

.. and it's the only thing that i can think of, which seems not unreasonable to prove. a month or two ago i had formulated a rough conjecture

(rather, i'd call it a claim;
it's not grandiose enough to be called a conjecture.)

now i'm close to something, but i can't quite formulate what the new form of the claim/conjecture should be.

perhaps i should explain that.

---

over these past months, we've encountered plenty of instructive examples and cautionary non-examples which show that, as i would have stated, the old conjecture is false.

however, interestingly and frustratingly enough, all the non-examples the advisor and i have considered are not that far off from that initial claim.

it's almost as if we need a magic word or a password, and then the gates will open. if we had the right claim -- if we could formulate what seems true and fits the phenomena we see in examples -- then i could set out and try to prove it.

i cam mumble or scribble something, but it wouldn't be precise .. yet. but i'm working on it.

premonitions of the job search.

i forgot to write that on wednesday, the department hosted a "be prepared for the job market" workshop.

it lasted 3 hours, with a few short breaks in between, and discussed the importance of teaching statements and teaching recommendations, curricula vitae, and letters of recommendation.

it was informative, especially the last part where postdocs talked about their own experiences with applying for jobs.

on the whole, i became nervous and a little worried about the immediate future, but i wouldn't say that i am scared out of my wits. there's enough time to make plans and contigency plans ..


.. although a friend of mine did inform me that cemetaries often use bulldozers to do their digging, these days.

so my contigency plan of becoming a professional grave-digger has just been made kaput. \:

oh well.
time for a new Plan B.

Federεr: 1, me: 0.

thinking about it, that title looks more like a score from a pro tennis match.

but instead of rοger federεer, i mean herbert federεr, one of the grand old men of geοmetric measurε theοry. i learned today that a classically-stated version of an argument of mine was already known as Federer's Support Theorem.

so today, he beat me to a theorem .. by a dozen years or more. \:

---

it never fails:
every time i think i have a good idea, it turns out to be someone else's!

it's too bad that a ph.d. thesis can't be a really cool exposition; then this wouldn't be an issue anymore. but alas: we must pursue originality in our ideas .. |:

back to work, i guess.

spring fever, work and distractions.

lately it's been difficult to concentrate on work. i blame it on spring fever:

the warm, feel-good weather,
the hurried exodus of the undergrads,

the end of winter term and classes and seminars,
or any of our "regularly-scheduled programming."

i have always found it difficult to maintain a schedule, especially when the constraints are within your control.

if i must be in class at 2pm, if that is truly an immutable commitment, then i can plan appropriately: i will wake up at 8:30am and start work straightaway at 9. it will be diligent work, because i know i'd rather not deal with the afternoon lassitudes of office life.

i will even remember to eat lunch, because i know i do not want to be hungry in midst of class; my longhand notes would suffer, otherwise.

as the saying goes, the constraints make the problem. if anything, it's what i tell my calc i students when they have to solve basic optimization problems. q:

but being too flexible is problematic, because no single decision matters so much more than another.

if i wake at 10am but am free to work the same number of hours and arrange my dayspan until 2am, then why should i wake at 8am?

if i have nothing planned all afternoon and if there is only work to do, then when should i focus my mind and work hard? it ends up that while working, i might write a little, then look past the window to empty sidewalks, and then think of whether this is the best use of my time.

i think of whether i enjoy doing something else, whether i should have taken a holiday so that i could settle this insidious inclination of being distracted; then, having enjoyed myself, i can settle contentedly to work.

like many other mathematicians, i am terrible at making decisions which don't matter.

---

earlier today i received some spam email from southwest airlines, and their featured destination today is virginia beach.

i sat and reread the email a few times, speculating.
i imagined warm sun and hot sand.
then i imagined small groups of others: a few families, groups of friends,

and i imagined myself alone,
not knowing what to do with myself, despite being on holiday.

then i sighed, i logged out of my email account ..

.. but i haven't deleted that email yet:
the futility of hope, you could say.