Geeking out

I seem to be the only one around who thinks this is hilarious, or even gets it. I am the biggest geek in the department of mathematics.

Ring homomorphisms

There is a special kind of function between rings, just like we have in groups. Given rings and , a function is called a homomorphism if it preserves all the ring structure.

The sort of odd thing here is that we’ve got two different kinds of rings to consider: those with and without identities. If we’re considering rings in general, we require that

• 
• 

but if we’re restricting ourselves to rings with identities, we also require that

• 

where the on the left is the identity of , and the one on the right is the identity for . If we have two rings with identities but we consider them as general rings there will be more homomorphisms than if we consider them as rings with identity. It becomes important to pay a bit of attention to what kind of rings we’re really concerned with.

As an exercise, consider an arbitrary ring and see what ring homomorphisms exist from to . If has an identity, which of these homomorphisms preserve the identity?

Oh, and I probably should mention this: all the terminology from groups comes along for the ride. An injective (one-to-one) ring homomorphism is a monomorphism. A surjective (onto) ring homomorphism is an epimorphism. One that’s both is an isomorphism. A homomorphism from a ring to itself is an endomorphism, and an isomorphism from a ring to itself is an automorphism.

[EDIT: cleaned up LaTeX error and added comments at the end about terminology.]

KLV errata

I just got home from a long discussion with Dr. Zuckerman about this whole business. I’m not quite ready to say exactly what’s going on, but I want to correct a couple errors that I’ve made. Let it not be said that I don’t admit when I’m wrong.

Firstly, in my little added remarks about the Monster group in my “Why We Care” post, I was oversimplifying. First of all, the lattice is not the Leech lattice. The Leech lattice lives in 24-dimensional space for one thing (doh). Basically, you put together three copies of the lattice and then tweak it a bit.

Putting them together I can explain. The simplest lattice is just the integers sitting inside the real line. If you move to the plane, the points with integer coordinates sit at the corners of the squares in a checkerboard tiling of the plane. This is “adding two copies of the integer lattice”. For three copies of , we want 24-tuples of numbers so the first eight, second eight, and third eight are each the coordinates of a point in the lattice.

When you do this, it turns out there’s just enough room to squeeze in some more points to get a new lattice. That’s the Leech lattice. The Monster also isn’t quite just a group of symmetries of this lattice, so there’s still a few more steps to go, but it’s definitely related. So the connection isn’t quite as close as I’d implied, but it’s there.

The other thing is about real forms. I’d forgotten that not every choice of “realification” of the Killing form gives a Lie group, and further that not every choice that does work gives a unique Lie group.

What is true is that to every real form of a complex Lie group , there’s a largest compact subgroup . This means that its ends curve back in on themselves like the circle or the torus, and don’t run off to infinity like the line or the cylinder. Then we can “complexify” this group to get another complex group that’s really interesting to us. This group is a subgroup of , which will be important. In particular, if we take the compact real form of , its maximal compact subgroup is just itself, so its complexification is just back again.

Coloring knots

Today I’m going to be talking to the graduate students about various topics relating to coloring knots. I think I’ll leave you with a little project to play with.

First, go to Bar-Natan’s table of knots. Notice how all the diagrams seem to be made up of arcs meeting up where one strand of the knot crosses under another. Pick a knot diagram and try to color each arc either red, green, or blue, subject to the following rule: at any crossing, the three arcs that meet (two for the undercrossing strand and one for the overcrossing) must either be all the same color or all different colors.

Which knots can you color using all three colors at least once? If that’s too easy for you, how many ways can you color a given knot? If that’s too easy for you, you’ve almost surely seen this before.

To get you started, I’ve tricolored the trefoil knot using all three colors.

How to Play by Yourself

Sometime between dragging myself into my bed after the calculus exam and related activities last night and dragging myself back out of it in time to teach this morning, an article claiming to solve triangular peg solitaire went up on the arXiv. I’ve obviously not had time to read it, so I don’t know how good it is, but the subject matter at least should be pretty generally accessible.

Different kinds of rings

There are a number of different kinds of rings differentiated (sorry) by properties of their multiplications. Most of them lead into their own specialized areas of study. I mentioned that a ring may or may not be commutative, and it may or may not have an identity, but there are a few more that will be useful.

One initially counterintuitive idea is that it’s entirely possible that a ring has “zero divisors”: two nonzero elements that multiply to give zero. Imagine starting with two copies of the integers, and , writing elements of the second copy as integers with a bar over them. Now consider pairs of elements, one from each copy, . Add pairs by adding the two components, but multiply them like this:

Notice that the product of any two elements of is zero! Weird. Eerie.

To be explicit: an element of this ring coming from is . We calculate the product:

So, any element for which there is a so that is called a left zero divisor. Right zero divisors are defined similarly. If a ring has no zero divisors, so the product of two nonzero elements is always nonzero, we call it an “integral domain”. The integers are just such an integral domain, fittingly enough.

Now if a ring has a multiplicative identity we can start talking about multiplicative inverses. We say an element has a left inverse if , or a right inverse if . If a ring has both a left and a right inverse they’re the same, since

In this case we call a unit and write its inverse as . We can also see that an element having a left (right) inverse cannot be a left (right) zero divisor:

If every nonzero element of a ring is a unit, we call it a division ring.

In the case of commutative rings, all these distinctions between “left” and “right” (zero divisors, inverses, etc.) disappear, since multiplication doesn’t care about the order of the factors. We actually have a special name for a commutative division ring: we call it a “field”, though everyone else in the world except the Belgians seems to call it a “(dead) body” (körper, corps, поле, test, lichaam, …).

[EDIT: added explicit calculation verifying that elements from in the example are zero-divisors.]

The ring of integers

As I mentioned before, the primal example of a ring is the integers . So far we’ve got an ordered abelian group structure on the set of (equivalence classes of) pairs of natural numbers. Now we need to add a multiplication that distributes over the addition.

First we’ll figure out how to multiply natural numbers. This is pretty much as we expect. Remember that a natural number is either or for some number . We define


where we’ve already defined addition of natural numbers.

Firstly, this is commutative. This takes a few inductions. First show by induction that commutes with everything, then show by another induction that if commutes with everything then so does . Then by induction, every number commutes with every other. I’ll leave the details to you.

Similarly, we can use a number of inductions to show that this multiplication is associative — — and distributes over addition of natural numbers — . This is extremely tedious and would vastly increase the length of this post without really adding anything to the exposition, so I’ll again leave you the details. I’m reminded of something Jeff Adams said (honest, I’m not trying to throw these references in gratuitously) in his class on the classical groups. He told us to verify that the commutator in an associative algebra satisfies the Jacobi identity because, “It’s long and tedious and doesn’t add much, but I had to do it when I was a grad student, so now you’re grad students and it’s your turn.”

So now these operations — addition and multiplication — of natural numbers make into what some call a “semiring”. I prefer (following John Baez) to call it a “rig”, though: a “ring without negatives”. We use this to build up the ring structure on the integers.

Recall that the integers are (for us) pairs of natural numbers considered as “differences”. We thus define the product

Our life now is vastly easier than it was above: since we know addition and multiplication of natural numbers is commutative, the above expression is manifestly commutative. No work needs to be done! Associativity is also easy: just set up both triple products and expand out, checking that each term is the same by the rig structure of the natural numbers. Similarly, we can check distributivity, that acts as an identity, and that the product of two integers is independent of the representing pair of natural numbers.

Lastly, multiplication by a positive integer preserves order. If and then . Together all these properties make the integers as we’ve defined them into a commutative ordered ring with unit. The proofs of all these things have been incredibly dull (I actually did them all today just to be sure how they worked), but it’s going to get a lot easier soon.

Rubik’s Cube Wrapup

I want to tie up a few loose ends about Rubik’s group today.

We can fit Rubik’s group into a sequence that more clearly shows all the structure I’m talking about. Specifically, it’s a subgroup of the bigger group I mentioned back at the beginning. We can restate the three restrictions as saying the maneuvers in Rubik’s group are those in the kernel of a certain homomorphism. So, first let’s write down the big group.

The unrestricted edge and corner groups are just wreath products, which I’ll write out as semidirect products. Without restrictions, these two groups are independent, so we just have a direct product to give the unrestricted Rubik’s group.

I’ll write for a generic element of this group. Each part of this list corresponds to part of the expression for above.

Now we want to add up all the edge flips and make them come out to zero. We can write this sum as a homomorphism:


where the sum is taken in the group . You should be able to verify that this actually is a homomorphism. Similarly, we want the sum of the total twists as a homomorphism:


where the sum is taken in .

Finally, the permutation condition uses the “signum” homomorphism from a symmetric group to . It assigns the value to even permutations and the value to odd ones. We use it to write the last restriction as a homomorphism:

Now we assemble our overall restriction homomorphism as the direct product of these three:

and get the short exact sequence:

Commenter Dan Hoey brought up where my fundamental operations come from. To be honest, these four are just ones I remember off the top of my head. He’s right, though, that there are systematic ways of coming up with maneuvers that perform double-flips, double-twists, and -cycles. I’ll leave you to read his comment and work out yourself that you can realize four such basic maneuvers as commutators — products of elements of the form . This means that the commutator subgroup of Rubik’s group is almost all of itself. It just misses a single twist. In fact, — Rubik’s group is highly non-abelian.

Incidentally, this approach to the cube is not the first one I worked out, but it’s far more elegant than my pastiche of particular tools. I picked it up back when I was at the University of Maryland from a guy who had worked it out while he was at Yale as a graduate student back when the cube first came out: Jeff Adams.

That’ll learn me

I wanted to see how a book I’d checked out from our library treated a certain topic, hoping that it might have a theorem all ready for me to use. Unfortunately I didn’t remember the authors nor exactly what it was called, but I did remember what it looked like. So I went to the library and tried to find it with no luck. As a fallback, I asked Paul.

Paul Lukasiewicz is our librarian, and has been around forever. People I know who were students here in the early ’80s thought of him as omniscient already. You can give him a title and author of any book in the library and he can tell you what color it is off the top of his head.

Unfortunately it doesn’t work in reverse, so I was driven back to the online directory of the library system here to search through hundreds of books on the topic to find the one I remembered. They really need to make those things searchable on appearance.

Math and science testing in public schools

What do you do when 85% of your state’s high school students pass a comprehensive English test, but only 56% pass math and 38% pass science? Stop testing math and science of course! Washington State’s legislature is trying to remove the comprehensive mathematics and science tests required for graduation and replace them with end-of-course tests. It’s clearly more important that students be able to cram in exactly what they need for the exam and forget it the day after than they actually remember anything.

I went to public schools in Maryland, where we had similar exams. Those students sharing my cynical bent referred to them as the “functional idiot tests”. I really do mean that it mostly tested respiratory function. I passed two of them by filling in pseudorandom bubbles and never actually opening the test booklet. Those tests I took seem to correspond to the current level of the WASL tests.

There’s a story about a scout for the US Olympic archery team. He’s driving through the midwest and starts coming on these barns with bullseyes painted on their sides, and an arrow dead-center in each one. As he comes to a town, he stops for lunch and asks around about what he’d seen. The locals tell him it’s one of the farmhands around the area and give him directions to the place the kid works. The scout tracks down this miraculous archer and introduces himself, asking what trainer led to such amazing results. The farmhand says, “Naw, no trainer. I just shoot the arrows and then draw the target.”

This is exactly what Washington state is about to do. The important thing is evidently not to teach mathematics or science to its students. It’s to draw the target after we know where they land — give them a test we know they can pass. An examination like this used to be used to verify a certain level of knowledge, but now the exam has become the important thing — exactly the opposite of the natural order.

It’s a fool who looks at the pointing finger instead of at what it points to. Washington state is about to become that fool.