ChapterZero

I just realized that Frankel’s book, which has been sitting untouched on my shelf for a year or so, is the perfect complement to Marsden’s book. First, it is approachable– here’s a book that’s actually aimed at engineers and working scientists– and second, it is almost as comprehensive as Marsden’s book, in terms of concepts introduced, but less mathematically rigid. The feeling I’m getting is that I can read this to develop an intuition for dealing with the concepts (Lie derivatives, etc.) that Marsden’s book covers in too much detail to be digestible to first timers, then jump into Marsden’s text to see rigorous developments if I care to. Which I probably won’t, at this point.

But, the most important point, and the one putting a smile on my face as I type these words, is the fact that working in coordinates is not as much of an afterthought in Frankel’s book as it seems to be in Marsden’s. So, e.g. I can learn about tensors in a natural manner, where the coordinate manipulations are developed naturally with the more intrinsic operations. Considering that the geometry final is going to be almost exclusively on manipulation of coordinate formulas, this is a big plus for me.

I’m actively avoiding working on my geometry homework, which I got an extension on until Monday afternoon– so tomorrow’s going to be a busy day– since I have a project due for my numerical PDE course on Tuesday. The project is to implement a (2d) multiphase fluid flow solver using the level set method… it’s quite fun and the most visually appealing assignment to date, but I feel underwhelmed by the depth of my understanding of what is going on. For instance, choosing the timestep has so far been a magical process. So it’s only natural that I decide to spend my avoidance time looking at that stuff, and even more natural that I get derailed from that also– I mean, it just wouldn’t be natural for me to constrain myself to work that I actually have to do, now would it?

At some point, I decided to look for notes on Hamiltonian PDEs, to do some background reading. I couldn’t find any good introductory material, but I did come across some neat material on the numerical methods used to solve them. Not surprisingly, due to their close relationship to dynamics, a lot of geometrical ideas are associated with Hamiltonians. The relatively new field of geometric integration aims to find natural ways of integrating/solving differential equations that preserve natural geometrical properties. I don’t know much more than that high level overview, but it sounds like an interesting field, and more importantly, it helps assure me that what I’m learning in my 202 course may be applicable in the immediate future. If I ever have the time to delve deeper into this area.

But really, that’s besides the point. Just as learning numerical methods enlarged the mathematical arena I can work in, and the paradigms I have to attack problems, learning this geometry stuff that seems so annoyingly abstract at times may be useful. It’s reassuring to see that this is true in reality, as well as theory.

I came across a wired article which purports that science fiction is the last genre of fiction which explores the big ideas:

Which brings me to my point. If you want to read books that tackle profound philosophical questions, then the best — and perhaps only — place to turn these days is sci-fi. Science fiction is the last great literature of ideas.

From where I sit, traditional “literary fiction” has dropped the ball. I studied literature in college, and throughout my twenties I voraciously read contemporary fiction. Then, eight or nine years ago, I found myself getting — well — bored.

Why? I think it’s because I was reading novel after novel about the real world. And there are, at the risk of sounding superweird, only so many ways to describe reality. After I’d read my 189th novel about someone living in a city, working in a basically realistic job and having a realistic relationship and a realistically fraught family, I was like, “OK. Cool. I see how today’s world works.” I also started to feel like I’d been reading the same book over and over again.

Heh. I almost agree, and I’m certainly glad to see science fiction getting its props. It is true that a lot of the general fiction I’ve read lately (whatever that is) is more self-conscious than anything else– the emphasis seems to be on writing a novel about anything, perhaps even just a stylistic exercise, rather than letting a story tell itself in a way that happens to get across a point. But I’d hesitate to say that there aren’t any authors outside of the bounds of science fiction tackling the big issues. One that comes to mind is Percival Everett.

I think the more important issue is how engaging the novels are. Like Thompson says, if you read book after book all written in the same style, you tend to get bored quickly. One of the virtues of science fiction is that each story is intrinsically different from the others: the stories take place in literally different worlds. It takes much more reading to get bored that way.

BTW, am I the only one who thinks the word ‘novel’ carries with it an air of pretension?

I haven’t done too much this weekend. I read the first of four sections in the chapter on tensors for the geometry course, and predictably got lost in a welter of details and notation. Did I mention the horridness of the official course text? I’m going to do the homework for that section tomorrow– if I find myself lost, I remember a book in the library that covers exactly the same material in a more friendly manner, so I can go reference that.

After some conversations with one of my roomies, an aerospace engineering grad student, I decided to take a look at continuum mechanics, and see if that’s a good choice for my required course external to the applied math department. There are only a few subjects I think would be worth the time and effort at this point in my career, when I should be focusing mainly on my research: analytical mechanics, continuum mechanics and fluid dynamics, relativity, and quantum mechanics. With the exception of relativity and quantum mechanics, these areas are tied to applied mathematics at the hip– a bunch of techniques and the attendant physical intuitions from fluid dynamics have made their way into the collective psyche of applied mathematics, and it’s next to impossible to go a couple of days in our department without hearing about a lagrangian or hamiltonian.

In fact, given the way physics terms– viscosity, stress tensor, lagrangian, boundary layers, advection, shocks, streamlines etc.– (a weak selection, but this is just off the top of my head) are tossed about by our professors, I’m surprised we don’t have a comprehensive physics for applied mathematics course in our department. There’s probably a mathematics of physics course somewhere on campus, but the emphasis in such courses is on applying mathematics to physics problems, as opposed to seeing how physical problems have motivated mathematical techniques and how artificial embeddings of problems in a physical context can assist tremendously in developing helpful intuitions. A subtle but important difference.

As for relativity and quantum mechanics, I just think they’re natural subjects to study. You know, being the two most important physical theories developed in the past century or so. Relativity is to me, one of the sexiest branches of physics, and quantum mechanics seems to be of growing importance in applied mathematics– Schrodinger’s equation seems to be becoming a motivational force in the same way, if to a lesser extent, that the Navier Stokes equations indisputably are. I’ll admit that I base this entirely on my person experiences, seeing a load of papers that somehow trace their motivation back to Schrodinger’s equation. I imagine that someday we’ll teach sympletic integration in our introductory courses on the numerical solution of ODEs.

All of which brings me, in a roundabout manner, to the original reason for this post: a link to the awesomeness that is the National Committee for Fluid Mechanics Films. I’ve only viewed part of the cavitation video (the segment showing the bubbles spun off of a propeller is achingly beautiful), but already I feel like pestering someone in the aero- or mechanical engineering departments to let me see a live hydrodynamics experiment.

I’ve been looking at Lie algebras/groups for CDS202 tonight: I have to compute the coadjoint action of SO(3) on the dual of its Lie algebra, and find the coadjoint orbits. Exactly what all of that means, and why I care, has yet to be determined. In fact, I suspect that I don’t care.

So far, I’ve found the Lie algebra– the set of skew-symmetric matrices– but what exactly is the dual space? I can’t think of any other way to get a concrete representation of the dual space than to use an inner product, but can I just introduce one? I’m going to do that, and see what happens. Hopefully something conforming to intuition (i.e. geometrically meaningful) will pop out.

Here’s an interesting calculation that came up in my reading. Let be a mapping from the Lie algebra to the Lie algebra the set of skew-symmetric matrices with the usual bracket . You don’t have to know what all those terms mean (I sure as hell don’t have a firm grasp on this myself) to consider the question:

Show that for all and , it’s true that .

In math, there are two extreme styles of teaching: the prof can either develop the entirety of the subject, so essentially the students’ only job is to take notes, or the prof can give just definitions and ask guiding questions, allowing the students to develop the subject almost independently.
The usual compromise struck is that the teacher gives lectures that cover the bones of the subject– the named theorems and other big ticket items– while homework is assigned that allows the student to put the meat on the bones themselves. So ideally, homework covers the little indispensable tricks, helps the students rediscover the most useful (at least judging from history) paradigms for approaching problems that apply the theory, and if designed particularly well, even helps the students learn to spot problems where the theory can be applied.

Then of course, there’s the path that seems to be taken in teaching CDS202: motivate the hell out of the students in class, pump them up by making incisive analogies to the finite dimensional or linear cases… then relegate the raw meat of the course, the not quite as beautiful true face of the subject, to the reading.

I love the shock value of this one; Nina Gordon covers “Straight Outta Compton”

In the same vein, there’s Jenny Owen Youngs’s cover of “Hot in Here”.The video’s brilliant: singing the song in an igloo

Then there’s Alanis Morissette’s cover of “My Hump”. I don’t really get what she’s trying to convey– the the original song is vapid?–, but whatever her motivation, it made for good listening.

I just discovered Jonathan Coulton, who in addition to his folksy troubadour cover of “Baby Got Back”, which you can listen to for free on his music page has an original song “Creepy Doll” worth experiencing. The video below has a cut version of Creepy Doll as the soundtrack; the full version is available on the same page.

To round it off, here’s Richard Cleese’s faux lounge act cover of “Baby Got Back”; if you listen past it, he also covers Depeche Mode’s “Personal Jesus”. He seems to have a lot of these covers on Youtube.