In case you thought I had given up on knot theory

August 4, 2007 in geekiness, links, math, thought

I guess it might actually be pretty cool. Professional mathematician and fellow wordpress.com blogger Terrance Tao has a really cool post summarizing a talk from the 2006 ICE about the marriage of knot theory and dynamical systems. Specifically, he talks about the use of limits of knots (whatever that means…) as a way of understanding the dynamics of the Lorenz attractor (pictured above), a quite pretty-looking “strange attractor” in . However, that analysis is based on interpreting the periodic orbits of the system as knots (when taken alone) or links (when taken together), and in general this doesn’t work. Some systems are positively riddled with periodic orbits, and some have practically none at all, so people started trying to interpret nonperiodic orbits as knots/links. This is accomplished by taking a sort of ergodic-flavored limit of the series of knots formed by picking a starting point in the space and a long time . The knots then come from tracing out the path of as it moves around for time and then connecting the final point to with a line segment to close the loop and form a knot.

The ergodic flavor comes from how the limit is taken and is better explained, I think, using the concrete example that Tao gives in his post. The goal is to understand the helicity of a 3-D dynamical system, which according to Wikipedia is “the extent to which corkscrew-like motion occurs.” The idea is to take two random starting points, and and a long time . Form the first knot as before, by tracing out the path through which moves in time and joining the final point to the starting point to form a closed loop. Do the same thing with to get another knot, .

Now we have two knots, and that are probably looped together in interesting ways. We’ll call link consisting of the two of them taken together (in math speak, ). In knot theory, the linking number of a link is a measure of how tangled up the separate loops of the link are, and is defined so that it’s the same no matter how you look at the link.

Taking a step back to look at the big picture, we can see that if a body of water is corkscrewing as it moves, the paths traced out by two particles in the water ( and here) will be wound together in a DNA-ish double spiral. Furthermore, the faster the water is corkscrewing, the more tightly coiled this spiral will be. Since the degree of spiralling is measured here by the linking number of , by dividing that linking number by the time we can get a measure of the average rate of corkscrewing over this time interval. Taking the limit of this normalized linking number as and “averaging” over all choices of starting points and gives us a measure of how much corkscrewing there is in the motion defined by .

Taking this sort of limit (of some quantity that cumulatively changes over time, averaged for that time and across the entire space) is the bread and butter of ergodic theory (the study of dynamical systems). The difficult thing about studying systems this way is making sure that these limits actually exist, and this is what most of the long proofs from the introductory stuff I studied were aimed at doing. Making sure a limit exists is usually done by hand-crafting the terms of the limit to make sure everything works, so it would be interesting if ideas from a totally different discipline (here, knot theory) could be used, essentially unchanged, as the basis for these finicky limits.

And, indeed, in this case the technique does work: the limit described above, also called the time-average of the linking number, is exactly equal to the helicity of the flow . To me (and, I imagine, to other mathematicians), this is interesting because it suggests that there is enough similarity between the structure of space as formalized by dynamical systems and as formalized by knot theory to make the one expressible in terms of the other. So, for all my frustration at the seeming arbitrariness and lack of rigor of knot theory, it is somehow a close cousin of ergodic theory, the subject of probably my favorite math class I’ve ever taken.

I’m endlessly fascinated by these higher-level structural isomorphisms, whether they’re found in mathematics, human knowledge representation, language, or the natural world. Probably unsurprisingly, I suspect that the awareness and embodiment of these sorts of structure is what makes thinking (or even living) things what they are. This doesn’t mean that I’m in the symbolic-representationalist camp of cognitive science, which holds that abstract structures and rules are represented explicitly in human representations of information. The fact that structural similarities (like the ergodic theory-knot theory one discussed here) often happen seemingly by accident is, I think, evidence against such explicitly representational systems. If you’re a long-time (and rather brave) reader, you might remember my rantings about the importance of emergence in science, and more generally ontology. I’m not sure right now how this relates, but I know that it does, so I’ll just leave you to mull over that until I get my hairbrained mind in order.

It’s LEARNIN’ TIME

April 18, 2007 in geekiness, links, math, science, thought, williams

It’s happening again. In spite of myself, in spite of being sleep deprived, hooked on caffeine, overworked, and thoroughly overwhelmed, I’m getting excited about learning. All of my senior friends are turning in their theses (the ones that have them, anyway) and I’m feeling really inspired. Even though I’m probably at least a year out of seriously thinking about my own thesis, the idea of doing a thesis is starting to take on a more immediate, corporeal quality, and that’s really exciting. I suppose that might be a good way to characterize my experience the last few weeks. I decided to turn down the opportunity to be a JA, opting instead to bum around here next semester and go to India in the spring, and after a bit of hand-wringing and “what have I done”-ing realize how absolutely amazing next year is going to be. I’ve been talking more with my research professor about projects for the summer, both things she wants me to do and things I want the chance to do. I’m halfway through the hellish two weeks I alluded to in my last post, and the math exam and tutorial paper went/are going better than I thought they would, and I feel like I’ve got a handle on some ideas for final projects. So, the immediate (and not-so-immediate) future is looking more and more corporeal, more immanent and more exciting every day.

For this, I’m very thankful. I’ve been a wreck most of the semester, worrying about getting through the rest of the summer/finding the right summer gig/figuring out next year, and to finally feel like things are a little more under control is a huge relief. I know this is sort of a rehashing of the themes from last week’s post, but I’ve been putting most of my thinking/writing muscle into writing paper, programs and problem sets so there’s not much left over for substantial stuff here.

Anyway, I’ve been reading Good Math, Bad Math, a neat blog about math, science, computers, and society (and Seed ScienceBlog). But I’ve got learnin’ to do , so that’s all for now.

Oh!  I made french bread again yesterday, and it was delicious!  Next time I’ll have to remember to take pictures, because baking bread sure is purty.

Dear god yes

February 17, 2007 in geekiness, math

Someone at WordPress.com apparently knows what’s what…you can now use to enter pretty math directly within posts. So, for example, here’s the definition of the entropy of a partition, that I had to on my own computer and then upload for an earlier post:

and here’s the code:

$latex \displaystyle H(\gamma) = -\sum_{i=1}^k \mu(A_i)\log\mu(A_i)$

Yes, I am still here

February 15, 2007 in flickr, geekiness, images, math, photos, thought, williams

Although you’d have good reason to doubt it. I’ve been engrossed in typical start of semester insanity, including two almost-all-nighters in as many weeks and my first tutorial meeting today. Speaking of teh tutorial meeting, it was absolutely wonderful, and all of my fretting over the last few days about writing the paper and doing philosophy and doing philosophy with other philosophers amounted to nothing. I had no problem presenting my ideas in the meeting, and I think I actually had something interesting to say, and really enjoyed the dynamic created by me, my partner and the professor.

I’ve taken math tutorials before, but they’re very different in the form (not to mention the subject matter). In “normal” Oxbridge-style tutorials, you have weekly meetings with your partner and the professor (like mine today) where one person has prepared a paper on some assigned (mostly optional, in our case) readings, and the meeting is spent discussing the paper and the readings. Math tutorials don’t really work that way, since writing papers is not a very common way of doing math, and so the format is usually up to the professor and somewhat non-standard. In one we just had a book of theorems and had to do as many proofs as we thought we would get through presenting in our bi-weekly meetings, and in another we would have a few sections from a textbook to read and a problem set to do and present in meeting.

So, point being, now that the big question mark that is a reading/writing tutorial has been resolved I’m feeling pretty good, and really looking forward to the rest of the semester. I’m really enjoying my other classes, especially my computer science class, which is nothing short of nerd-heaven with Unix and emacs and programming etc. Paresky (the new student center) is opening this weekend, which is Winter Carnival, which means we don’t have classes on friday. I can’t even tell you how excited I am about finally having a student center, and a snack bar that’s open all day, and a dining hall whose schedule doesn’t leave me without meals on a regular basis. Apparently the dining hall is going to be something like the one in the UMaine student center, with a la carte stuff and meal-equivalency money, which means that it can stay open continuously from breakfast to dinner, and that means that when I realize I’m hungry at 2 pm there’s somewhere I can go for lunch. That, and there’ll be some central, happy, inviting place to go and be all communal-like and revel in the small-school, Williams community. Huzzah!

(Link to photo’s page)

More ramblings

January 31, 2007 in friends, geekiness, links, math, science, thought

Jue‘s got a neat post today about an experience in a bio lecture that got him thinking about emergence again, and what he wrote certainly helped clarify some of the thoughts that have been swirling around in my head for the last few weeks. He talks about emergence as it relates to the alluring promise of reductionism in science, which is that, if we know enough about how things are arranged and interact at a low level (say, molecules, or even further, atoms or fundamental particles) we can predict what will happen at higher levels. He ends with this:

That’s why I wanted to say to my professor, well it just seems like we can’t understand cell motility from studying actin monomers — in fact, the futility of our approach stems from the fact that we can’t know EVERYTHING about individual actin molecules, and it is simply more efficient to observe them in the aggregate, in order to learn about them in the aggregate. Our inability to bridge levels of complexity is a practical failure, not a fundamental conceptual one.

Does this mean that in principle, some sort of deterministic link, in the manner of the soundly discredited 19th century scientific worldview, still exists between actin monomers and their network behavior? I understand just enough of 20th century physics to suspect that the “practical failure” I pithily dismiss above is actually something much deeper and crippling, but not enough to finish the connection. Guess I’d better take quantum mechanics. Ugh.

Seeing as I’ve had a teeeensy bit of quantum mechanics, and wrote a paper about entropy in dynamical systems for a project, I thought I’d take a stab at that last question. First of all, the relationship between amount of information we have about a system and how predictable it is has been studied pretty well in mathematics, under the name of entropy.

On a more personal note

January 9, 2007 in books, found, friends, geekiness, history, images, knitting, links, math, on..., other blogs, photos, williams

I spent the better part of this evening reading bits and pieces of Second Nature, the book about neuroscience and epistemology I mentioned earlier (and found out about here) and Gödel, Escher, Bach and drinking the last of my delicious delicious Darjeeling tea. The rest of the day was spent mostly in the LEGO lab, modding our robot’s ineffectual treads with duct tape (surprisingly effective!), getting it through the maze, finessing the obstacle avoidance program, adding light sensors, programming it to follow a line on the ground, and finally taking the whole thing apart and rebuilding the chassis to be a good 30% bigger.

What time I didn’t spend in the lab today was spent ice skating with Ruth, which was absolutely wonderful, and tomorrow I’ll be hiking and it’s supposed to be chilly and maybe a little bit snowy, so I’m looking forward to that quite a bit. Speaking of Ruth, as of yesterday it has been exactly one year since we entered into that most holy of contracts, the Facebook “In a relationship” status. We exchanged knitted gifts, and had Thai food, and it was lovely. Speaking of knitted items, over break I read a post on Boing Boing about knitting or crocheting mathematically interesting objects, like hyperbolic planes (picture at right) and moebius strips, and knew right away that I had to knit myself a moebius band. After much frustration and grunting and cursing, I finally succeeded in producing a somewhat silly looking thing that I then proceeded to wear as a headband for the better part of New Year’s Eve.

And oh! our first entry-reunion broomball game of the year was a rousing success, with the Ice Vampires pulling through to defeat Dodd House 3-1. Huzzah! I’m also currently drooling over the iPhone…so sue me, I have a soft spot for tiny, black, shiny, light-up techno-thingies with smooooth user interfaces, especially when they’re being cradled by Steve Jobs.

Eep! A few more cool things: a site that estimates how many people there are in the US of A that have any particular name. Apparently there are somewhere around 31 David Kleinschmidts out there. I’ve also resolved to write my own script for generating knitting patterns based on cellular automata, inspired by the project described here. Last but not least, the site that I got my moebius band patter from and which has a bunch of cool math/knitting links, and also a blog called Math4Knitters (‘nuf said).

Filling in the cracks

November 15, 2006 in flickr, found, geekiness, images, math, photos, thought

We’re starting to get to the end of our book for my math tutorial and that means that this past week was a sort of “go back through and do all the problems we skipped in old chapters because they were too tricky.” It’s really cool to go back and review all the material we’ve already covered and realize just how much math was required to get to the place we are now, and to be able to appreciate a little better now how all the pieces fit together.

Oh, what’s that? It’s November?

November 14, 2006 in coffee, math, ordinary, thought, work

Yes, it is November! I am a little batty due to not getting a lot of sleep recently and having a paper due tomorrow (of all days!)

This past weekend New England reasserted itself, complete with with cold, biting, steady rain on Sunday that caught us all off guard. Honestly, I’m glad to have somewhat normal weather back, as physically uncomfortable as it may be. Last winter was so unbelievably bizarre weather-wise (complete with a few weeks of 50s in January and February) that I’m really jonesing for some good ol’ fashioned snowy, blustery, bitter late fall/early winter weather.

I had this whole awesome idea about how I was going to talk about the cool, really fundamental tension in mathematics between the ideas of grouping discrete objects and counting them. But then I got lazy. There’s an abbreviated version below the break if you’re really really curious, but my cup of Kenya AA coffee and my essay on reincarnate Tibetan lamas is calling! To wrath! To ruin! Dooo-eeeeee!!! (Riders of Rohan horn of Helm Hammerhand call thing, a la Peter Shin)

“so what impact exactly does that have on what you’re doing?”

October 30, 2006 in geekiness, math, music

To my immense, nerd-y delight I discovered today that Audacity (the cool and free audio editing software) will calculate the Fourier Transform of any passage of sound you give it, which is something I’ve always wanted to be able to do to music ever since seeing an awesome video giving a real-time graph of the pitch content of a piece of classical music a few years ago.

The Fourier Transform is a way of doing just that: given a function x(t) (like a sound wave, for instance) you take a certain improper integral and BAM you have a function X(w) where w is any frequency (i.e. pitch in the case of sound) and X(w) tells you how much of that pitch the soundwave you gave it contains…sort of…what it actually tells you is what the coefficient of the sine wave of frequency w would be in the Fourier Series expansion of x(t).

The important thing is: if you look at the image above, where the vertical line is on the graph corresponds to the peak at 88 Hz, or F2, and the next ones (in order) are: C3, F3, A3, and C4. Guess what chord is being played in the song at that moment? F major (which contains F, C, and A). w00t!

On skyscrapers and infinity

October 17, 2006 in a cappella, clarinet, flickr, friends, gq, images, math, on..., photos, williams

This afternoon was passed in the ever-delightful Tunnel City, taking shelter from the oh-so-typically-new-england-fall chilly, raining weather. I went there specifically to get a jump start on my math tutorial problem set, which has reliably been the bane of my mid-week existence since the beginning of the semester, but ended up running into Alex, and hanging out with him and Julian in the little nook next to the big garage-door windows (that they open up in really nice weather!) Hanging out with them, goofing off with Alex and talking about math versus analytic philosophy and infinity with Julian, really put me in a good mood.

So, despite getting the week of on a terrible foot (discovering on Sunday night at 10 pm, just as I was about to start what I thought was the work to do for Monday, that in fact I also had a computer science problem set due on Monday as well…) I’m feeling really good about it now. My tutorial meeting and clarinet lesson, the two big stressors for me during the week, both went quite well, probably. The later is probably due to the fact that I actually practiced this past week, unlike the previous week. Hooray, responsibility!

Furthermore, I think I’ve finally come to grips with the fact that I simply do not have time for frisbee this semester, which as sad as it is, is the reality. I essentially wasted the entire saturday this past weekend at a tournament that I didn’t play in (due to the sudden and unexpected resurgence of my asthma), and consequently wasn’t much fun.

But I got a really good start on my problem set today, and hammered out the details of a few of (what I hope will be) the trickier problems we have to write up. GQ is going amazingly, and we’re really digging into some new music. Our last two gigs went so so well, and the group really feels like it’s beginning to gel, thus allaying many of my fears from earlier in the semester. And now I must run off to a movie about Tibet, which I trust will be great just like everything associated with that class.

(Ruth took this picture!)