It is way past time to put out a diary entry. No, they don’t seem to be useful or popular — but that is your business, not mine. My business is that I want this blog to reflect my doing mathematics, although its main focus is the presentation of mathematics.
The two latest posts — color, and surfaces — should suggest, correctly, that I have put down wavelets for a while. Well, I have put them down, but I am still working on them indirectly. It is way past time to move into the frequency domain for wavelets, and I have begin to do so.
There was a calculation that I wanted to be able to do. I’ve known a particular truth for at least a year and a half, but getting the details right for an example was challenging. I still don’t understand why such an example is not commonplace. Nevertheless, although people say it, and although they present sketches to illustrate it, I have found one — yes, one — actual example. I was tired of just sketches. And, not to put too fine a point on it, I needed some help in getting the details right.
Now, having found and modifed an example, I am looking forward to putting out a post showing the relationship between the Fourier transform and the Fourier series. (Not to be mysterious, here’s the truth: the coefficients in the Fourier series of a periodic function are samples of the Fourier transform of one period. Here’s the rub: get the details right.)
I have also picked up Cohen’s “visual color and color mixture” again. It makes a remarkable amount of sense, the second time around, because — of all things — the mathematics he needs is the mathematics Malinowski used for “target testing”. I am closer to putting out a few posts about the perception of light spectra, as well as a post explaining Cohen.
Those two, spectra and Fourier, are pretty much what I’ve been doing in mathematics lately. I had been looking at a wide range of subjects, and it was a pleasant meandering stroll through fiber bundles, submanifolds, and spinors — which ended in the “12 pentagons” post.
I love the connections in mathematics. That Malinowski and Cohen needed the same mathematics for two different purposes; that group representations could take me back to simplicial surfaces; that adjoints and color would come together.
For the blog, yes, I put out a summary post about color; and I have prepared a summary post about the matrix of an adjoint operator. (My comment at the beginning of the recent color post, that I needed to write better summaries, was motivated by rereading my posts both about color and about the adjoint.)
And that’s probably why I picked up Cohen again. Interestingly, Cohen will have us talking about adjoints and dual bases. I wish I could take credit for some sweet organizational skills, but I didn’t realize that when I wrote the color and adjoint posts! (And yes, although the numbers would be the same, we will finally want to focus conceptually on a dual basis rather than a reciprocal basis. And no, Cohen did not use this terminology.)
I have done at least one other interesting thing since the previous “happenings” post six weeks ago. I am embarrassed that I didn’t think of doing this a long time ago.
Once again I sat down to summarize chapter 3 of Bloch. Lord, my summary of chapter 2 dates from May, 2008; the latest post dealing with chapter 3 dates from December, 2008. So I’ve been thinking about a summary for at least eight months. Admittedly, the 12 pentagons post was made possible by what I had learned in chapter 3.
This time I did something different. I did not review my notes and I did not look at the book. I just tried to write a summary, out of my head.
Let me just say that it was an educational experience. Oh my, the things I could barely remember, and the details that totally escaped me. It wasn’t about what I remembered — it was really, wonderfully, embarrassingly about what I did not remember and what I did not understand. As I said, educational.
Boy, when I looked through the chapter after writing that summary, I knew what I was looking for and it all jumped off the page. That was back in the first week of August.
But did it stick? Did I understand it?
The proof of the pudding, of course, is going to be to pick up my ignorant summary again, and see if I understand enough this time around to get it all right — once again without the book and without any notes. Enough time has elapsed that doing it again should measure my understanding rather than my simple recollection of the answers.
(I wish I had a photographic memory, so that I could always call up something I didn’t understand in order to work on it. Well, I wish I had more talent and more skill, too, and that I worked harder….)
As I said, I can’t believe I haven’t been doing that kind of reviewing all my life. It seems so obvious in retrospect…. On the other hand, this blog is the first teaching of mathematics that I’ve done since graduate school, so maybe I can be forgiven for my lack of insight about teaching me myself.