ANH: first day unease.

today [0] i taught two classes, each of which were 75 minutes long. in each i felt like i was saying obvious things [1] and wondered if i was boring the students into a desperation of some kind, that i just stop talking and dismiss class early.

i don't know why, but i hate being boring ... well actually, i do know:

first, i don't want to enforce the stereotype that maths is hard and boring;
on the other hand, it takes a while for me to get anywhere interesting.

a lot of times i struggle with writing lectures because i can't convince myself of really, is this it? come on! there has to be something interesting in this whole topic..!

last night, after deliberating on and off, i finally put some lecture notes down on paper .. at about 1am. then i promptly fell asleep, glad to be rid of the task.

i don't think i succeeded. i don't think i "get" the students yet, and i don't think they "get" me.


[0] that is, on friday: i finished this post later that night.

[1] which, of course, they were .. to me. that's not a statement of arrogance; any seasoned calculus instructor would probably tell you the same.

ANH*: life comes first, then blogging.

.. yes, it's been pretty quiet on this blog. i guess you could say that i've been busy:

i'm teaching two courses this fall,
with plenty of preparations to do;

i'm starting a new job at a new department,
which involves figuring out how things work here;

i'm still trying to find an apartment, which is incredibly frustrating ..!

*sighs*
maybe things will calm down soon.


^* this is the pre-amble that will probably lead to a year-long series, where i'll comment (read: complain) about starting a new academic life in a new department .. this time, as an assistant professor on the tenure track.

as longtime readers of this blog may recall: i get weirded out by the term "professor." in fact, during my postdoc i told my student not to call me that, due to inaccuracies. now that it's part of the job title, i suppose i can't really escape it anymore ..

.. anyway, expect this to read like a "how-NOT-to" type of guide ..!

lastly, for star wars fans it's probably clear that ANH is short for a new hope; for the longest time i thought i'd fade out from academia like obi-wan kenobi, but apparently that didn't work out .. or rather, that did work out.

not quite ARR! numbers and symbols, as viewed by a man of letters.

these are some excerpts from the pleasures and sorrows of work by a. de botton, one of my favorite authors. he has a way of revealing the sublime aspects about everyday life.

anyway, this is his take on science ..

"Gone were the days of geniuses in their observatories and workshops, single-handedly rerouting scientific history. We had entered the sober era of the collaborative laboratory, where astrophysicists and aeronautical engineers banded themselves together for decade-long assaults on minor mysteries, resisting the media's attempts to raise any one of their number into a contemporary Galileo. A company might limit itself to perfecting the performance of silver-zinc batteries in zero-gravity conditions, rightly sensing the foolishness of expanding to address further puzzles in satellite electrics. A scientist might spend a lifetime examining the properties of titanium at high temperatures or the behaviour of hydrogen at the moment of ignition. The sum total of one's contributions to mankind might end up in an issue of the Journal of Advanced Propulsion Methods."

.. and this is his take on maths, viewed from the non-technical viewpoint:

"Noting my puzzlement, Ian told me that he was calculating the force of gravity at work on the cable, and that in his equation $l$ stood for the length of the span, $w$ for the effective weight per unit of length, and $T_H$ for the constant along the line. He explained that transmission engineers were unusually blessed in having at their fingertips a highly precise, efficient and universal vocabulary with which to convey even the most labyrinthine electrical scenarios, so that from Iran to Chile, $\psi$ referred to electric flux, $\mu$ to permeability, $\mathcal{P}$ to pereance, and $\alpha$ to the temperature coefficient of resistance.

I was struck by how impoverished ordinary language can be by contrast, requiring its user to arrange inordinate numbers of words in tottering and unstable piles in order to communicate meanings infinitely more basic than anything related to an electrical network. I found myself wishing that the rest of mankind would follow the engineers' example and agree on a series of symbols which could point incontrovertibly to certain elusive, vaporous, ad often painful psychological states -- a code which might help us to feel less tongue-tied and less lonely, and enable us to resolve arguments with swift and silent exchanges of equations.

There seemed to be no shortage of feelings to which the engineers' brevity might be profitably applied. If only a letter could have been identified, for example, with which elegantly to allude the strange desire one occasionally has to elicit love from people one does not even particularly like ($\beta$, say); or the irritation evoked when acquaintances seem to be more worried about one's illnesses than one is oneself ($\omega$); or the still vaguer sense one can sometimes have that different periods of one's life are in coexistence, so that one would have only to return to one's childhood home to find everything the same as it once was, with no one having died and nothing having changed ($\xi$). Possessed of such a notational system, one would be able to compress the free-floating nostalgia and anxiety fo a typical Sunday afternoon into a single pellucid and unambiguous sequence ($\beta + \omega | \xi \times 2$)" and attract sympathy and compassion from the friends around whom one would otherwise have grunted unhelpfully."

ARR! in which maths could bring about the end of the world (unless algebraιc geοmetry saves us)?

predictions are always hard, especially when it comes to the future. this one, however, concerns actual maths for once:

"Our conclusion is there is a small but definite chance that RSA and classic Diffie-Hellman will not be usable for encryption purposes in four to five years,” said Stamos, referring to the two most commonly used encryption methods.
..
RSA and Diffie-Hellman encryption are both underpinned by a mathematical challenge known as the discrete logarithm problem. That problem is computationally difficult to solve, ensuring that encrypted data can only be decoded quickly with knowledge of the secret key used to encode it in the first place. Breaking RSA or Diffie-Hellman encryption today requires using vast computing resources for significant periods of time.

However, it is possible that algorithms able to solve the discrete logarithm problem quickly could exist. “We rely on that efficient algorithm not being found,” said Jarved Samuel, a cryptographer who works for security consultancy ISEC Partners and presented alongside Stamos. “If it is found the cryptosystem is broken."

~ from "Math Advances Raise the Prospect of an Internet Security Crisis" @mit:techreview

related to this, algebraic geometry might actually be useful for something .. soon, which means that i'll never heard the end of it from a few of my colleagues!

anyway, another excerpt from the article reads:

Stamos called on the security industry to think about how to move away from Diffie-Hellman and RSA, and specifically to use an alternative known as elliptic curve cryptography (ECC), which is significantly younger but relies on more intractable mathematical challenges to secure encrypted data.

The U.S. National Security Agency has for years recommended ECC as the most reliable cryptographic protection available. In 2005 the agency released a toolkit called SuiteB featuring encryption algorithms to be used to protect government information. SuiteB makes use of ECC and eschews RSA and Diffie-Hellman. A classified encryption toolkit, SuiteA, is used internally by the NSA and is also believed to be based on ECC.

ARR! statistics about .. well, mathematics.

i like the way this guy thinks:

"... but I think we're starting to see a new kind of metamathematics, where people use statistical methods to study the structure of mathematics itself.  This is mathematics as actually done by people, so it involves issues of taste and style.  These are subjective things.  But I suspect there are some features of math that are fairly independent of who is doing it.  Maybe some theorems are 'important' in a fairly objective sense - important crossroads that most travelers tend to stop at.  And someday we may understand why."

~ from "The network of mathematics" @johnbaez:g+ (via mathbabe)

reading this reminds me also of the flysρeck project, regarding the use of formal proof:as expounded on their wiki:

" How does a formal proof differ from a traditional mathematical proof?

Traditional mathematical proofs are written in a way to make them easily understood by mathematicians. Routine logical steps are omitted. An enormous amount of context is assumed on the part of the reader. Proofs, especially in topology and geometry, rely on intuitive arguments in situations where a trained mathematician would be capable of translating those intuitive arguments into a more rigorous argument.

In a formal proof, all the intermediate logical steps are supplied. No appeal is made to intuition, even if the translation from intuition to logic is routine. Thus, a formal proof is less intuitive, and yet less susceptible to logical errors."

the way i understand it, traditional proofs are like pseudocode whereas formal proofs would be real computer programs that you can "compile" against the standard axioms .. although it would probably resemble machine code.

i think this comparison also highlights their comparative advantages nicely:

the formal proof mightn't be readable, but at least you know it runs and witness how it does ..

.. whereas getting the idea for the formal proof would probably require some basic principles for why it could conceivably work, in which case one would probably have a traditional proof in mind.

going back to the idea of a network of all mathematics, i suppose that a formal proof would be checking the existence of a continuum from a statement (a given node on that network) to the fundamental axioms (or "roots" of the network).

ye gods: that would be a really complicated network!

disruptions, distractions.

so i was having dinner with my sister [0] when she asked me what i was thinking about lately. (i took this to mean mathematics.)

at first i couldn't think of anything, which was really weird,
especially since i was working on something that very morning;

i guess i've been really distracted lately.

---

to put things into perspective, the last five posts ..

(namely this, that, also this and this, and that too)

.. were dead-man's switches .. at least in erdös's sense [1]. they were compiled about a week in advance. i was away on the spur-of-the-moment holiday because .. well, the girl i'm dating was on holiday and we wanted to spend more time together. [2]

at first i thought i'd not bring any maths with me, if only as a romantic gesture .. but then i looked up my flight and it was 3 hours long: a long time, and i didn't have anything in mind to read at the time [e];

well, a laptop's not that heavy, and neither is that manuscript ..

so on the flight to rome, i idly decided to edit my $\LaTeX$. i sorted out one lemma, then another, and then in the middle of another proof, something didn't make sense.

closing the laptop, i tried to rework the proof in my mind .. only to perceive a gap. there was only an hour left in the flight to think of a patch before she and i would meet up again in the airport baggage claim .. and i started panicking ..

because the manuscript is supposed to be ready already, dammit! argh!

thus the plane touched down, decelerated [3], and arrived to its gate while i struggled without any viable ideas. dejected, i re-activated my cell phone, shuffled out of the plane and into another unfamiliar airport, and wondered ..

what kind of mathematician makes that kind of mistake, misses a gap like that?

sighing, i looked for the right baggage carousel from my flight ..

this would prove to be a distracted trip; in the mornings before we got up, i'd fumble with details in my head and feign sleepiness when she asked what was wrong.

.. to be continued?

[0] i.e. a mathematical sibling.

[1] according to this article and other accounts i've read, erdös would consider a friend "dead" if he stopped working on mathematics, so the term "dead man's switch" isn't that far off. (i wonder, then, if erdös considered "death" to be a permanent condition!)

[2] i'm leaving finland in 2 weeks. she and i haven't spoken about the future yet .. at least in much detail. suffice it to say that it's a rather turbulent time in my life. i'm pleasantly surprised that i can focus on anything mathematical, at the moment!

[e] in case you were wondering, she and i were on separate flights. now that i think about it, i don't know if she's aware of how much and often i work. (personally i don't think i work all that much, but experience with family and previous girlfriends seems to suggest otherwise.)

on an epilogical note: the laptop did help after all. i had merely a dumbphone and her smartphone wasn't connecting properly to the hotel wifi, so it came in handy with looking up various cool locations to visit.

[3] "de-celerate" shouldn't be an actual word. acceleration, being a directional quantity, can take on both negative and positive values.

ARR! accounting for asymmetry .. socially?

i'm not completely convinced of this study, partly because there is no discussion of how the researchers made their observations and recorded their data:

"The brain, Cacioppo demonstrated, reacts more strongly to stimuli it deems negative. There is a greater surge in electrical activity. Thus, our attitudes are more heavily influenced by downbeat news than good news.
..
Here's the tricky part. Because of the disproportionate weight of the negative, balance does not mean a 50-50 equilibrium. Researchers have carefully charted the amount of time couples spend fighting vs. interacting positively. And they have found that a very specific ratio exists between the amount of positivity and negativity required to make married life satisfying to both partners.

That magic ratio is five to one. As long as there was five times as much positive feeling and interaction between husband and wife as there was negative, researchers found, the marriage was likely to be stable over time. In contrast, those couples who were heading for divorce were doing far too little on the positive side to compensate for the growing negativity between them."

~ from "Our Brain's Negative Bias" @psytoday

i mean, how does a researcher get access to a couple's daily life so that they can objectively measure how much time they spend fighting? (this is not to say that the study is bogus, but only points out how little i know about how to conduct social research.)