so, if anyone asks ..

like all other working mathematicians out there, i am often asked ..

.."so what do you do all day?"

lately i've felt like answering:

"i've been trying to build impossible geometric objects, in order to show that certain mass distributions [1] cannot possibly exist."

as you may guess, my recent obsession has to do with finishing a proof by contradiction .. which i suspect is a means of inference that most people are uncomfortable with.

if my experience in teaching basis analysis has any weight here, i think that even mathematics majors at university have trouble with this method of proof.
..
..
.. thinking about it, that kind of answer isn't terribly helpful .. that is, to me. more often than not, conversations of this kind only on go downhill after that question; either (a) i say something too complicated and the other person, not understanding, feels dumb, or (b) i make it sound too easy and the other person wonders why i bother working on that kind of problem.

more likely, the other person would probably be wondering what would drive a person to think all day about things that might not exist ..?

often i just can't win with this kind of thing.



[1] this is my colloquial expression for what's known as a measure; i think i borrowed (read: plagiarised) the term from falcοner's book, in fact. before settling on this, i tried to use the term "prοbaβility distributιon" but this usually misled my audience that my work is related to statistics of some kind ..

so it begins .. again, yet not again.

odd:
after posting a few days ago, i suddenly feel like posting again.

---

as i mentioned last time, yesterday was my first day of teaching for the semester. i can only say that i felt very boring.

come on: everyone knows what vector addition is!
do i really have to go over the first section of the first chapter?

surely there is a more efficient way .. maybe i should have built a worksheet, have everyone work it out in a few minutes, go over the answers, and then go on to something more interesting ..

.. thinking about it, maybe i should have done exactly that! [1]

---

apart from that, there's little else to say. today was the first department meeting for faculty and i learned exactly how out of the loop i am about departmental and administrative affairs.

..
it's a strange thing, being a tenure-track faculty member. it's like being suddenly thrown into the real world and realising that you have to grow up.

[1] that said, if you're reading this and will actually try this approach in your own first day of class, then let me know how it works ..!

what i didn't.

i don't know what happened.

tomorrow is my first day of teaching for the spring semester;
today i spent 8 hours in the office, i was busy all day ..

.. and yet i still haven't written out my lecture notes!

oh well: my classes meet in the afternoon, so i guess i'll write them tomorrow morning.

---

it's been a while since i've last posted in this blog. i thought i'd spend the winter break making sense of my life and all that's happened, this past fall ..

.. what with this new position at a new university and all ..

.. but things still don't make sense. most days of the week i'm making it up as i go along, just trying .. trying my best to get it all done and stay sane at the same time.

there never seems enough time to do it all: teaching, research, faculty meetings and advising and so on. more precisely, there is never enough time in the sizes and shapes that i want them [1].

if i could identify a change in my life, then i'd say that time now comes in fractured form.

---

so i shouldn't talk about what i did during winter break [2]. it would be more appropriate to say what i didn't do.

i didn't go to the office,
i didn't answer any student emails that i didn't have to answer.

i didn't make sense of my life,
i didn't travel out of town,
i didn't make any new goals.

for the most part, i didn't want to do anything.

i wanted to, i tried to write up notes for a research idea. there ended up being a flaw in the argument and so i thought, off-&-on about the problem ..

.. but not so deeply as to make it too much like work;
i think i got somewhere with it.

this past week i realised that, starting tomorrow, i will have to start doing things for a while: commitments, duties, promises ..

it's starting again. whether it makes sense or not, this new job and life, there are things to do, again.

[1] if you spend enough time staring at weekly schedules, such as the default format for gοogle calendar, then time stops feeling 1-dimensional and linear. instead, it becomes more and more like a very weird tetris game in 2-D, fitting commitments into rapidly dwindling empty spaces.

[2] today was also the first day of spring classes, so quite a few colleagues asked me that question anyway.

ARR!.. imagine computers as research collaborators?

sometimes i wonder if i should be a mathematician at all .. lately i've been reminiscing about the dreams my younger self had: among them was to be a successful novelist, perhaps in science fiction.

so when i read news articles like this one ..

"Some might argue that computers will never be able to match human ingenuity but it is difficult these days to argue they can't at least mimic many of our skills.

Take the eDavid painting robot. The computer-controlled arm - adapted from a welding machine - chooses from five brushes and 24 colours to create impressive artworks on canvas.

It works by snapping a photo of its subject matter and then making the necessary calculations to turn the image into a drawing or painting in a wide variety of styles.

Its creators admit that it has no awareness of what it is doing. But it is able to make decisions about things like shading and brushstrokes as it goes, tweaking its moves based on how the picture is evolving, rather than just creating a pre-determined image."

~ from "The quest to turn computers into creative artists" @bbc_tech

.. then i immediately begin to imagine the possibilities:

if the strength of a computer lies in being very efficient with a finite, fixed set of tools, then imagine if we could get computers to prove simple lemmata for us, just by giving them suitable hypotheses and a fixed set of axioms and existing lemmata ..

yes, it is hard enough to build a robust proof-checking program .. and some researchers have spent years of their lives focusing on a single, specific verification .. but i'm not talking about a universal engine:

i liken it to writing programs that play go or chess well. it's not that the computer can think on its own, but rather that it can traverse through the decision tree of possible games very efficiently. in fact, a competitive program mightn't even run through all the possibilities, but simply the games that grand masters have played before.

so imagine coding in all the basic, rigorous proofs from mathematics (e.g. the proof of the triangle inequality in Euclιdean space) and adding shortcuts into the space of all proofs. i wonder what lemmas a computer could tell me ..?

being a mathematician, sometimes i have mathematical daydreams.

a shot in the dark (but no updates yet: stay tuned)

(yes, it's been a while .. and no, i don't have time yet to write about what's happened since the last two posts ..)


[sighs]

i'm almost convinced that there is no good way to teach a first course in proofs to undergraduates. sometimes i even wonder if it's something that can be "taught" .. in the sense that if the student really wants to learn and to understand, then (s)he has to commit to a minimum amount of time for self-study and development.

it's like teaching someone how to be paranoid: as a skill, it only develops with time, experience, and stimuli ..

*sighs*

ye gods, i hate asking for money.

---

it's clearer to me now that there is a "rat race" to academia in general and to the sciences in particular. more and more i envision a future where i'll never stop writing grants and there will always be another meeting to sit in, another memorandum that i should have read (but have skimmed over, at best).

for a while i've wondered if i was cut out to be a mathematician, but i've made my peace with it now. it's been long enough that i wasn't going to cut it, then i would probably be doing something else by now.

i'm starting to wonder, though, if i'm cut out to be a professional mathematician.

the research is fine and the teaching, though time-consuming, is also fine and often enough fulfilling (if not enjoyable). as for the grants .. and the applications .. and the meetings, and so on;

i can see why many faculty "give up" upon earning tenure.

these professional aspects of the job were never advertised to me, as a ph.d. student; maybe the advisor was deliberately putting it in the background, if only so that we could have a greater focus, when working together. as a postdoc there seemed more and more of it, when discussing the nature of work with my colleagues.

who knows? maybe i've just always been naive;

my colleagues, near and far, seem quite able to maintain research as their primary focus and if anything, shape their other duties to complement this one singular priority. more and more i find this admirable.

maybe i'm just too new to this position, that these are all just growing pains, and that these shall pass with time and enough patience and a little humor. i don't know and it's hard to say.

i'm not giving up. it's just that i can see why others do.

ANH: from end to start, for now.

so it feels like ages since i last thought about a blog post of any kind. it seems like there's so much to saybut at the same time, none of it is really worth mentioning. that's always the difficulty of beginning a story at the beginning ..

.. so, being lazy at the moment, i'll not. i'll begin at the ending instead, which is today.

---

so today i gave a lecture about metric spaces to my students. it's a first course in analysis and the textbook [1] happens to cover the topic, which to me sounds like a license to expound on it for 75 minutes.

so i showed them the discrete metric on any set, and how the unit circle would look if the set were the euclidean plane. i showed them the L-infinity norm, how the unit circle looks like the usual unit square, and how short the proof is for its triangle inequality. this is in contrast to how the proof of the triangle inequality goes for the usual L-2 distance, which uses Cauchy-Schwarz and in turn, a nod to Pythagoreas's theorem.

i thought it was cool. it would be the kind of lecture that would have inspired me as a student .. but i don't know. i'm getting to know the students in my class, but i'm still learning all the time.



[1] we're using baby Rudin.

ARR! more machine now, than man .. twisted and evil.

"On the one hand, today’s computers feature programming and writing tools more powerful than anything available in the twentieth century. But, in a different way, each of these tasks would be much harder: on a modern machine, each man would face a more challenging battle with distraction ... Kafka, Kerouac, and Wozniak had one advantage over us: they worked on machines that did not readily do more than one thing at a time, easily yielding to our conflicting desires. And, while distraction was surely available—say, by reading the newspaper, or chatting with friends—there was a crucial difference. Today’s machines don’t just allow distraction; they promote it. The Web calls us constantly, like a carnival barker, and the machines, instead of keeping us on task, make it easy to get drawn in—and even add their own distractions to the mix. In short: we have built a generation of “distraction machines” that make great feats of concentrated effort harder instead of easier."

~ from "HOW TODAY'S COMPUTERS WEAKEN OUR BRAIN @newyorker "

on how our choices can haunt us later.

if i sit and think about it, then it feels i have a lot to say about the last two weeks, of this new job, at this university.

i don't know where to begin, though;
if i start now, then it will all come out as chaos.

maybe i've been writing too many lectures lately, and habit urges me to put some order or narrative into it. after all, life is simply a sequence of events; any additional order or structure on it is an inherently human contribution.

my guess is that it will take months for me to make sense of it all: these experiences, mistakes, small joys, and frequent setbacks. (i don't know.)

---

as for something small to share ..

.. during the first lecture of multivariable calculus, on a whim i decided to pronounce the letter z as zed, just like how they seem to do in europe and the u.k.

as a result, now i feel compelled to be consistent and remember, from now on, to refer to the vertical axis (in 3 dimensions) as the 'zed-axis" .. or else risk being caught as a pretentious snob!

ARR!.. apparently i still have more trigοnometry to learn.

well, i learned something new today:

It sounds cumbersome now, but doing multiplication by hand requires a lot more operations than addition does. When each operation takes a nontrivial amount of time (and is prone to a nontrivial amount of error), a procedure that lets you convert multiplication into addition is a real time-saver, and it can help increase accuracy.

The secret trig functions, like logarithms, made computations easier. Versine and haversine [1] were used the most often. Near the angle $\theta = 0$, $\cos(\theta)$ is very close to $1$. If you were doing a computation that had $1-\cos(\theta)$ in it, your computation might be ruined if your cosine table didn’t have enough significant figures. To illustrate, the cosine of $5$ degrees is $0.996194698$, and the cosine of $1$ degree is $0.999847695$. The difference $\cos(1^o)-\cos(5^o)$ is $0.003652997$. If you had three significant figures in your cosine table, you would only get 1 significant figure of precision in your answer, due to the leading zeroes in the difference. And a table with only three significant figures of precision would not be able to distinguish between 0 degree and 1 degree angles. In many cases, this wouldn’t matter, but it could be a problem if the errors built up over the course of a computation.

~ from "10 Secret Trig Functions Your Math Teachers Never Taught You" @sciam

in other news: it's been more than two weeks into this new job, and i still feel disoriented. often i feel exhausted, too.

on the bright side: i finally found an expensive apartment and signed a lease .. after a month of searching (and simultaneously teaching, for the last 2 1/2 weeks).

[1] these are defined, respectively, as $\textrm{versin}(\theta) = 1-\cos(\theta)$ and $\textrm{haversin}(\theta) = \frac{1}{2}\textrm{versin}(\theta)$. suggestively, "ha" mean half.