So, what mathematics am I working on?
I work on so many different things – okay, I play with so many different things – that I need to begin the weekend by sorting through my toybox. Here are the things I am playing with.
The control of aircraft. I have a book which uses classical methods to study the control of aircraft; this is where I want to begin, rather than in state-space. Nevertheless, I have found other books to be essential. There are two equations that gave me pause. What I call the magic equation (which I have not shown you), and our old friend the rotating coordinates equation (which I have shown you). I mentioned the latter in the questions last week, but I also indicated that I knew the answer. I need to add a mechanics book to the bibliography. I’ve probably got four posts to put together at this point, essentially just laying out equations. Then we can start on an example.
Solar energy. The calculations are fascinating. Given a geographical location, and a solar collector, just how much energy will strike it? How efficient it is, and what it will cost, are other issues. I just want to know the starting point, how much energy we start with at the collector. I have been trying to figure out how best to present the results of the computations. Well, the utility company gave a presentation a week ago for interested residents, and as a result, I know what number – numbers – I want to present. Now I just have to do all the calculations. Then I can present the theory behind them.
Geometry and topology. I have already read chapter 4 of Bloch; it is old, familiar material, and writing it up should be easy. But first I need to summarize his chapter 3. The challenge is that I don’t want to simply summarize his presentation. The book is very good, and if the subject interests you, you should buy Bloch’s book. I am not trying to write a replacement for it. Instead, I want to talk about what I got out of the chapter, what I didn’t get out of the chapter, what startled me, and what else I looked at. (We have seen much of what else I looked at in the triangulations and Euler characteristic posts. In addition, I want to talk about the distinction between TOP, PL, and DIFF (that is, topological, piecewise linear, and differential structure on manifolds), and between simplicial, PL, and triangulations. Before I do that, I need to add some more topology books to the bibliography. Oh, and I have to go find the pertinent small subset of all the notes I took from a lot of other books.
Principal components and factor analysis. Although I have summarized my current state-of-the-art in doing calculations, I want to write an overview. In addition, I want to describe “classical factor analysis”, which is altogether a cat of a different stripe, and I’m still researching it.
Color. I want to describe some of the simple quantitative aspects, but before I do that I need to add color books to the bibliography.
Lie groups and algebras. I found another introductory book by an author I like in general, so I started looking at Lie stuff again. I don’t have anything to put out here yet, although my bibliographic notes on the subject are rather interesting, if I say so myself. (As usual, I have more than a couple of books on the subject. Because the subject is of interest to physicists, I categorized my books as “physics” or “mathematics”. In addition, the subject comprises “matrix groups”, “Lie groups”, “Lie algebras”, and ” representations”; but not every book discusses all four of those, so I chose to indicate the coverage. Oh, and of course some of the books are more elementary than others.
In summary, then, I should be doing selections from calculations for solar energy, the questions from Basilevsky and about barycentric coordinates in simplices – oh, and wavelets, which I haven’t talked about; I need to describe the equations for aircraft, for color, and for principal components. I need to investigate TOP-PL-DIFF and classical factor analysis. I need to put out bibliographies for topology, color, and a couple of additional books. And before I can tell you about wavelets, I would need to put out a bibliography for them.
One last thing. Where did that puzzle come from last Wednesday? Well, I do mathematics because I enjoy it. I have begun explicitly asking the little kid in me, “What do you want to do?” I do mathematics, ultimately, because he loves it. He doesn’t need much time to play, so after a little while I am free to move on to the things my grown-up wants to do. Two weeks ago the kid wanted to do matrix groups, but he was bored with them last weekend, and instead he looked through the newly found – but not new – graph theory book, and then he got caught up in the mixing puzzle. After that, my grown-up was delighted to construct a post in about 45 minutes, instead of the four hours that most of them seem to take.