The Unapologetic Mathematician

Last night I stepped outside around 23:00. There were some high, thin, even clouds and a full moon almost directly overhead. Together, they made for a stunningly clear lunar corona plus a 22º halo. The halo was very tight and the corona was amazingly well-defined, with a very distinct spectrum as it faded off at the edges. Marvelous.

But also very interesting. Atmospheric optics give wonderful insight into the field of optics in general, where light is modelled with either rays, bending as it passes from one medium to another and propagating in straight lines apart from that, or waves, spreading out like ripples in a pond. Each optical effect comes from a specific sort of atmospheric condition. My descriptions here are adapted from the site linked above, which also has pictures illustrating what I’m talking about.

The 22º halo is created by relatively large crystals of ice suspended in the air, which are shaped like hexagonal prisms. It occurs when rays of light from the moon hit these crystals of ice, which act as little prisms. If a ray of light hits one side of the hexagon it bends before passing through the crystal. When it hits another face it bends again as it leaves the crystal. If the entering and exiting faces are at 60º to each other, the total bending will be between 22º and 50º, mostly at the lower end of this region. Since the moon is so far away that we can approximate all the rays coming in parallel the crystals that bend light rays 22º and then send them on to our eyes will all lie on a cone with cone angle 22º. When we see thos crystals lit up, that’s the halo.

The corona, on the other hand, arises as a diffraction effect. Tiny particles, like water droplets or very small ice crystals in a cloud, behave like pebbles in the path of the onrushing light wave. Instead of an even, parallel wavefront, each point on the particle’s surface scatters a new wave starting at that point. As all these new waves rush over each other, parts of them reinforce and parts cancel out. The pattern of such “interference fringes” is familiar from the usual explanation of the quantum-mechanical “double-slit” experiment, which is another diffraction effect. Further, since the color of light influences the length of its waves the peaks of intensity for different colors will be in different places. Near the moon itself the fringes stack up and the whole cloud is illuminated, but out towards the edge the red fringes are further apart than the blue fringes, which leads to the spectrum effect at the corona’s edge.

It’s amazing that just by looking up at the right time I could see evidence of exactly what was going on in the sky above me. The clouds consisted of ice crystals in a somewhat heterogenous mixture. Smaller crystals directly overhead diffracted the moon’s light into the corona, while larger crystals refracted the light to form the halo.

The two effects also relied on two different properties of light. The halo comes from treating light as little particles flying along straight lines, while the corona depends on treating light as a wave. That means that the display wouldn’t have been possible without quantum mechanics!

I’ve said it before and I’ll probably say it again: Science. It works.

A good ramp up into abstract algebra is the idea of a group. Groups show up everywhere in mathematics, and getting a feel for working with them really helps you learn about other algebraic notions.

There are a number of ways to think about groups, but for now I’ll stick with a very concrete, hands-on approach. This is the sort of thing you’d run into in a first undergraduate course in abstract (or “modern”) algebra.

So, a group is basically a set (a collection of elements) with some notion of composition defined which satisfies certain rules. That is, given two elements and of a group, there’s a way to stick them together to give a new element ab of the group. Then there are the

Axioms of Group Theory
1. Composition is associative. That is, if we have three elements , , and , the two elements and are equal.
2. There is an identity. That is, there is an element (usually denoted ) so that .
3. Every element has an inverse. That is, for every element there is another element so that .

That’s all well and good, but if this is the first time thinking about an algebraic structure like this it doesn’t really tell you anything. What you need (after the jump) are a few
Read more »

This semester I’m sitting in on Jayadev Athreya’s course on billiards. “Billiards?” I hear you cry, “The game like pool but with all the red balls?” No, that’s snooker. Besides, the course is on (not surprisingly) a mathematical model inspired by balls bouncing around on a table.

When playing mathematical billiards, we place a ball down on a polygonal table and send it off in some direction. When it hits the edge of the table it rebounds, making the same angle as it leaves as it did when it came in. We simplify a bit by assuming that the ball isn’t spinning, has no friction with the table, and so travels at a constant speed in a straight line between bounces.

So let’s start with a square table. How do we determine how the path behaves as time goes on? Imagine the ball approaching a side of the table. Instead of the ball reflecting off the edge, let’s reflect the whole table through the edge and let the ball continue on its straight-line path. Whenever the ball’s about to hit an edge, we have a reflection of the table ready for it.

But we really don’t need all that many reflections. In fact, four will do it: the first table, a horizontal reflection, a vertical one, and one reflected in both directions. We can put these four squares together into one bigger square. Instead of reflecting the ball at an edge, we can just let it wrap around to the other side of the table travelling in the same direction, just like a game of Asteroids. That means we’re really just rolling a ball across the surface of a torus.

Now the whole picture is pretty clear: by Weyl’s criterion, if the tangent of the angle between the path and an edge of the table is rational the path eventually closes back up and runs over itself again. If not, the path covers every point on the table equally. By this I mean that if the whole table has area , then given a section of the table of area , the ball spends of its time in that section.

There’s a lot of interesting material here, and a lot of it can be broken down to bite-sized chunks, tying into all sorts of other areas of mathematics. I’m starting a category (in the WordPress sense) so if you’re interested, there will be plenty more.

Here’s a meme I’ve seen floating around the net. It’s a nice way to break tone, feel a little more personal, and get a cheapo post. I’ll take a track that has come up in the past week on my iPod and post about it.

First up is a song from Mary Prankster. Now I don’t expect many peopleanyone to recognize the name, but back in college I followed every time she played in the Baltimore and D.C. area. I wrote album reviews and even a few interviews for a “zine” (yeah, bloggers are the first to drop the beginning of words) a few of my friends ran back then. Nobody rocked out like Mary, and if you don’t mind the fact that many of her songs would be one long beep on the radio her albums are highly recommended.

Slightly edited, then, is “New Tricks”. It’s a cute little sing-songy number, all about the failure of childhood dreams, their replacement with more adult goals, and uncertainty about whether even those can be achieved. Lyrics behind the jump.
Read more »

Last night I watched the premiere of Alexandra Pelosi’s documentary Friends of God. No, there wasn’t any actual (or purported) mathematics in it, but parts reminded me of an observation I made a few years ago.

First of all, I want to make clear that I am not indicting religion in general. Religion does a great many things, some good and some bad. What I’m attacking is a worldview that is most clearly exemplified in certain religious traditions, but can arise almost anywhere.

I’ve been calling this worldview “gnosticism”, referring to (but not identified with) the early Christian movement. The central point of a gnostic viewpoint is some revealed truth superseding observed evidence. I contrast this with “empiricism”, which takes observational evidence as the ultimate arbiter of truth.

The easy target evangelical Christianity holds up is creationism. Other bloggers have tackled this argument better than I can, but I’m not trying to argue that point here. Instead, I’m just concerned with what I see as the fundamental split: at the level of epistemology the advocates of creationism found their beliefs on the revealed truth of the Judeo-Christian bible, while the advocates of evolution found theirs on the (sometimes literal) mountains of scientific observations. Neither side is going to convert anyone by their own arguments. To a creationist, the revealed truth is axiomatic — by definition, no observation can contradict it any more than any number-theoretic statement can contradict one of the Peano postulates. To an evolutionist, observations are the bottom line, and no revelation can overrule them.

Many evolutionists complain that creationists are being illogical or irrational. I think this misses the point. Creationism is a perfectly rational conclusion from the axiom of the Bible’s literal truth.

The same divide comes up everywhere. Policy debates are a great place to find it. In the run up to the Iraq war, those in power took as given that the American troops would be “greeted as liberators”, and that actual observations and predictions to the contrary were by definition false. Their problem was not that the naysayers were incorrectly interpreting the facts, it was that they were relying on facts in the first place. Facts and observations, to a gnostic, are only useful as a posteriori buttresses of the basic axioms.

I first articulated the divide in the aftermath of the 2004 presidential election. Shortly before the election the Program on International Policy Attitudes published a survey outlining what George W. Bush’s supporters believed about his foreign policy attitudes and what John Kerry’s supporters believed about his. The former group was stunningly inaccurate about their candidate’s positions, despite four years of evidence. It seems clear from this study that while almost everyone in the United States wants the same things on the majority of issues, just over half of the country believed — contrary to the observational evidence — that Bush espoused these same opinions. Axiom: Bush is the right choice; Opinion: the United States should participate in the International Criminal Court; Conclusion: Bush wants the United States to be part of the ICC. The fact that his administration has become, if anything, more hostile to the ICC is to be discounted because it runs contrary to the axiom.

So, what does this all have to do with mathematics? Like all sciences, mathematics is inherently an observational enterprise. Yes, we speak in terms of axioms, but every statement carries a litany of unspoken qualifiers. This conclusion holds only if this collection of hypotheses (including a background such as set theory) does. Mathematics cannot say that anything is true or false in a vacuum, and it is very conscious of this limitation. What mathematics does is distill the art of reasoning from observations to conclusions. Observations are the raw stuff of satisfying the hypotheses of the mathematical statements scientists invoke to draw conclusions.

The empiricist takes these observations and feeds them into the machine of the scientific method to draw conclusions. This machine is not only constructed from mathematics, it is never assumed to be in perfect working order — mathematics constantly turns in on itself to question and verify its own validity. What comes out of this process is not the Ouroborean self-reference of the gnostic constantly reaffirming his own beliefs, but an outward-looking search to understand the observations we make, the near-miraculous ability to predict what we will observe in the future, and the framework necessary to design technologies that can take advantage of those future observations.

Not to put too fine a point on it, one cannot be a gnostic and a mathematician or scientist. The passive acceptance of fundamental beliefs and rejection of observations is simply anathema to the empirical methods that underly these fields.

To close, I think Randall Munroe has summed it up best in this comic. It’s slightly coarse, but it puts it as succinctly as anyone could hope to.

What? Has the UM gone in for tasseography? No, but that’s about the level of reliable information available in the job market.

Last fall I applied at 85 schools for something like 130 different jobs overall. Of course, each one only has a handful of possible slots, so the whole thing’s a long series of shots in the dark.

Here’s how the application procedure works. You see an ad for a job. You write up a letter of application and assemble the materials the school wants. You send them in, either over the internet or through the mail, and wait.

And you wait.

And you wait.

And there’s really nothing to do and almost no information comes back down the pipe unless it’s a confirmation that your materials were received or an early no.

So yesterday I was surprised to get an email from one place I applied. Their application deadline was the 8th, so this is two weeks along — a bit late for just confirming that they got my stuff. It didn’t say the words “short list” though, which would explicitly indicate passing the first cut, and it didn’t try to set up a job talk (tenure-track jobs, like this one, usually have you come out to the school to give a talk and interview with various people), so i have no idea what it means. Basically all I have to go on is a phrase I’ve heard other applicants mention last year: “Are you still interested in the position?”

I’ve asked around the department and signals are mixed. Does this indicate a shortlisting? Does the two-week delay? Ask two mathematicians and you’ll get three answers.

I’m left feeling like there’s a whole sub rosa protocol in place that somehow I’ve missed hearing about.

I was just telling one of the first-year students about my pipe dream of a paper. I would need to find a collaborator with Erdős number 1 and a good sense of humor. We would submit the paper to the Annals of Improbable Research. The paper would be titled, “John Armstrong has Erdős number 2″, and the paper itself would be the proof. Other than the difficulty of citing a paper in its own bibliography, I have no doubt that this would be a great idea.

The student, needless to say, was dumbfoundedawestruck. I pointed out to him, “There’s a thin line between genius and madness, and I walk all over that line.”

From The n-Category Café I’ve learned that there will be a conference on “The combinatorics, geometry, topology and physics of knot homology” in Faro, Portugal. I’d love to go, but this will take some financial juggling. I need to figure out how to add a PayPal link to this thing…

This year I managed to attend the annual Joint Mathematics Meetings, this year held in New Orleans, LA. It was greatly enjoyable, but one talk in particular irritated me. It manages to tie into a large number of useful subjects, so it will provide for many, many posts to explain what I’m talking about to the lay audience. Unfamiliar terms will eventually be defined in later posts and this one will be updated with links to the explanations.

Prinarily, the talk claimed to be motivated by a problem in DNA splicing. A strand is cut and one end is joined to a nearby free end of another DNA strand. In the middle of the splice, we have three strands of DNA all meeting in one central area. You’ll remember that the basic structure of DNA is basically a twisted ladder. Here we ignore the rungs of the ladder and just consider the rails, which normall twist around each other. Ath the splice, though, the six rails — two for each of the three strands — tangle up in the center somehow. The problem, as I understand it, is to determine how they are tangled.

The talk completely did not address how the knot theory shown was supposed to determine the biology, nor how the inputs to the mathematics were to be read off from the biology. Despite the fact that this connection is a — if not the — major question in any work of applied mathematics, I will not deal with it now. I am not, nor have I ever been, a mathematical biologist.

The approach presented was to attempt to color the tangle in a similar way to more classical knot colorings. Specifically, we label each arc in the diagram with an element of the group so that when arcs labelled and meet at an overcrossing arc labelled , the labels satisfy . Rather than finding solutions for various values of , the authors wrote down the relations — one for each crossing — that any coloring would have to satisfy. These (linear!) equations are grouped in a matrix and the authors searched for various matrices that could arise from various tangles.

In particular they noted that for knots there are as many crossings as there are arcs (prove it!), so there are as many relations as there are variables, and the presentation matrix is square. With free ends, however, there are more variables and so the matrix is rectangular. The authors broke the matrix into four blocks and argued that when the whole matrix was put into a certain form one block was zero and two of the others were invariants of the tangle. The last they seemed to disregard.

Really what they were doing is determining a presentation matrix for the “fundamental involutory quandle” of the tangle. I was able to determine with a few questions that the speaker did recognize this phrase, at least in the context of knots (no free ends). The problem I have here is twofold.

Firstly, matrices are horrible, horrible things until the very end when you absolutely must do some calculations. Matrices depend on a lot of essentially arbitrary choices, and different choices give rise to different — though in some sense “equivalent” — matrices. If you want to study the quandle, study the quandle already! Don’t put so much stock in some array of numbers that introduces all sorts of artifacts to cloud the discussion.

Secondly, extending the notion of the fundamental involutory quandle to tangles with free ends has already been done. It’s a special case of part of my dissertation. The cospan construction handles the fundamental involutory quandle beautifully, and it was depressing to see it shackled into the ugly dungeon of matrices.

I’ll go into it more another time, but the idea of the cospan construction here is that we can talk about the fundamental involutory quandle of the edge of the tangle where no tangling goes on and consider how that edge includes into the whole tangle. Since we’re always considering the case of six points on the edge the quandle there is uniquely determined by that fact. In fact, if you insist on drawing out matrices to talk about the fundamental involutory quandle-cospan you get three matrices to consider: one for the edge, one for the whole tangle, and one for the inclusion mapping. These are exactly the three non-zero blocks the authors (should have) considered, but each one of them now has a meaning. One is not just an invariant, though, it’s completely determined by the fact that there are six strands coming in.

It’s one thing to recognize in a talk a bad reconstruction of a result you know exists. It’s another level to recognize a bad reconstruction of something you did.

I suppose I should say something about myself.

I’m a recent (May 2006) Ph.D. in mathematics from Yale University, staying on for a year as a visiting instructor while I look for a job. I’m not above some self-promotion, so if you happen to know someone on a hiring committee please pass on the link.

My primary field is knot theory, but I approach it with a heavily algebraic slant. Accordingly, I’m interested in all sorts of different fields — category theory, representation theory, theoretical physics, and more — as well as in making mathematics more accessible to the general public. I’m sure I’ll talk about my work in particular as time goes by.

This space is mostly about publicizing mathematics and in part about publicizing my own interests. I believe that mathematics does have an abstract beauty which can have universal appeal once the surface is peeled back. All too many people have been scared away from it by horrible teaching experiences, to the point that it can be discomforting to tell someone what I do. But at the end of the day I am a mathematician, and I don’t apologize to those who find that alien.

There’s a hell of a universe over here. Let’s go.