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.

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