Happenings Jul 11

I sat down to write a happenings post this morning, and found myself looking at a wavelets post when I was finished with it.

Since the first draft turned into a very nice technical post, let me try this again. When I began, what else did I think I might write about?

I certainly do not mind that I ended up with a post different from what I intended. One of the secrets to writing is: don’t edit yourself while it’s happening. Of course, drafts need to be edited — but that is a different process. By giving myself free rein, I am often surprised at how well things turn out.

There is also something to be said for a piece of technical writing which flows, as opposed to one where I have to look up a technical detail every five minutes. If it flows, it makes some sense in my head. If I’m looking up lots and lots of details, I can’t really be said to know it.

My major focus since the last happenings post has been wavelets. I am struggling.

How are things going? The simplest answer is: I’m happy to just reproduce a drawing in a book, even if I don’t understand why the method works or where the function came from.

The next wavelet post — the one I just wrote — will show you a wonderful function that came from I don’t know where.

Sometimes I get so confounded frustrated that I bail on mathematics. Then sometimes I pick up that waste-of-time computer game and conquer the galaxy; if my frustration level isn’t that bad, I pick up a foreign language.

Sometimes, fortunately, I just go look at other mathematics. But that works best in the morning, when the day before didn’t go too well.

I picked up modules. We can think of them as “vector spaces” where the scalars come from a ring rather than from a field. One of the major properties we lose is the very idea of “dimension” (so they certainly aren’t actually vector spaces). Anyway, I happen never to have studied them. And, as it happens, I didn’t get much further than choosing a book to study them out of. I am being distracted by other mathematics.

I also picked up fiber bundles. They generalize the tangent bundle, which is the set of all tangent spaces to a manifold. What you should visualize is a circle with a whole lot of tangent lines drawn on it. And if you draw enough of them, you can dispense with the circle itself, which suggests that we can study a manifold by studying associated fiber bundles.

In contrast to wavelets, where I have to go through all 20 of my books on a regular basis looking for specific topics, for fiber bundles I grabbed about 10 of my 140 or so differential geometry books, and whittled that down to three. I may have missed a couple of excellent texts, but these three look to suffice.

I wasn’t looking for it, but one of them had a very nice discussion of immersions, submersions, embeddings, and related things. These distinctions are crucial when we study submanifolds. That’s another one of those things I never had to learn. So I’ve been distracted from fiber bundles and am looking at these.

I had gone through an undergraduate book about field extensions, which culminated in proofs that the three classical construction problems (square the circle, trisect an arbitrary angle, and double the cube) simply cannot be done.

Interestingly, this specific book — far more than most — lends itself to the following: I should try to briefly summarize the book. For most books, one can barely hope to briefly summarize one chapter. I still hope to write such a summary for the construction problems, but I have found one elsewhere, so it is no longer necessary that I write one myself. But it would be such a good thing for me to do, for my own understanding…. (And yes, I have in mind something a bit more detailed than “Don’t do it! I already tried it. It doesn’t work!”)

I had gotten to that book from finite fields. I tried to pick up finite fields again, but got distracted by the other thing that field extensions do for us: they show us that we cannot solve the general polynomial equation, in radicals like the quadratic equation, for a degree greater than four.

I had also picked up geometric algebra, which combines an inner product with an “outer product” (e.g. the vector cross product); and Lie algebra, which is related, and which led to the eightfold way and the classification of elementary particles.

I didn’t get very far. It’s as though what I know about them is archived on a backup hard drive, and all the work I had done on them looks very mysterious when I pick it up again. I’m not talking about mysterious textbooks — I’m talking about mysterious Mathematica notebooks that I myself wrote!

Well, it could be a very good thing, looking at what I did, trying to figure out what I knew back then. Here, instead of summarizing a book, I think I want to find a great example, one that embodies, as much as possible, the knowledge I want to recover.

Hmm. On second, thought, perhaps I should call that a “thorough example”. I have something different in mind for a “great example”.

As I said two weeks ago, I am always thrilled to find “a great question”, one that focuses one’s attention on new material. The three classical construction problems, while great, suffer from needing an entire book for their answer. Still, they help a lot in keeping me motivated.

But not everything great is a question. I know of at least one “great answer”.

Why should I study matrix algebra? Because the general solution to the problem of doing a least squares fit can only reasonably be expressed using matrices.

(Look, I can think of a lot of beautiful things that linear algebra and matrices give me. But if a student, or non-mathematician, is at the stage of wondering, “Why should we study this?”, I’m perfectly willing to show them something they can’t do without matrices. But the simple truth is, beauty aside, I really like being able to say, “Because without it, you can’t do this thing here.)

And maybe not everything great is a question or an answer. There ought to be “great examples”, too. One could argue that every counterexample is great, but I’m looking for more. Or less. “Counterexamples” are very important, probably deserving of more emphasis than they get, but I want counterexamples which are special.

OK, here’s a possibility, one that I’ve mentioned briefly before. I think that any discussion of “unique factorization domains” should begin by exhibiting a number system for which unique factorization fails: working in the set of numbers

a + b\ \sqrt{-5}\ , with a and b integers,

we have

21 = 3*7 = (1 + 2\ \sqrt{-5})\ (1 - 2\ \sqrt{-5})\

which would be no big deal (after all, 12 can be factored in more than one way, 3*4 and 2*6), except that in both factorizations of 21, the factors are irreducible. Not one of 3, 7, (1 \pm 2\ \sqrt{-5})\ can be factored.

(If unique factorization holds, then “prime” is equivalent to “irreducible”; one could argue that it is the failure of that equivalence that allows unique factorization to fail.)

Well, maybe that’s a great example, not just a run-of-the-mill counterexample; but let me keep thinking about it.

And I should look again to see if it is in fact a thorough example: is there anything in the study of non-UFDs that is not in that example?

I may have just convinced myself that I’m interested in finding thorough examples. Can an example count as “great” if it’s not “thorough”?

Let me start looking at my examples with a more critical eye….

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