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
- Composition is associative. That is, if we have three elements , , and , the two elements and are equal.
- There is an identity. That is, there is an element (usually denoted ) so that .
- 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
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.
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.
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.
My claim staked, I suppose I should send up a few shotgun blasts to mark my presence.
My first motivation in getting (back) into the weblog game is self-promotion. I know that isn’t the most original, but I’m trying to get a program off the ground here, and this might help get more people to hear about it.
My second reason is to have a soapbox. Admittedly there aren’t as many news stories that set my blood boiling directly on-point with mathematics as there are in astronomy or biology, but there are some out there. If I throw in all the misused statistics, errantly-drawn conclusions, and cargo cult science, well I’m sure I can come up with a good rant once a week or so. Actually, I’ll set that as a goal.
Which brings me to my third point. I’ll admit to a certain self-congratulation on being a member of the rarefied priesthood of professional research mathematicians, but it fades quickly. The uncomfortable silence that follows telling someone you work in math is a direct result of the fact that we’re generally content to talk amongst ourselves and leave the public at large behind. Yes, there are benefits to teaching basic math and reasoning skills, but there is a supremely beautiful world we end up keeping to ourselves — one that can match all the wonders of, say, astronomy, though without the cool pictures.
The fact is, anyone can understand the basic ideas in even current mathematics, and I try to always keep track of how I can tell an interested outsider about them. Here in this forum I hope to aim the discussions at just such an interested outsider. Mathematics is not for just the few who devote their lives to it, but for everyone. It need not be talked around in casual conversation to avoid making non-mathematicians uneasy. It need not be apologized for.
Rather than try to start from the ground up, I’ll jump into the thick of things and backtrack to explain what I need later. If I say something you don’t understand and want more information about, let me know and I’ll likely be glad to explain. Eventually I should have a decent store of material to refer to when I need it in later posts. Accordingly, the style will probably fall a bit closer to James Joyce than to Nicolas Bourbaki.
So, all that said, welcome to the Unapologetic Mathematician.