So much math, so little time. Another week has gone by, and I am at least as ignorant as I was last Friday.
“The known is finite, the unknown infinite; intellectually
we stand on an islet in the midst of an illimitable
ocean of inexplicability. Our business in every
generation is to reclaim a little more land.”
–T. H. Huxley
The problem is, as I step further into the ocean, I see more of it. As I learn more, I see more that I don’t know.
At least I have gotten out the three color posts which I had planned. Beyond that, my list of things to do for the blog has only gotten longer. I plan to add a few more books to the color bibliography, and also to make some comments about the first color post.
I did take some time to separate my mathematics to-do list from my blog to-do list. I moved my laptop into my library, and sat there looking at my mathematics bookshelves. For many topics, the number of books I own is a measure of how interested I am in it. (There are exceptions, notably statistics and geometry, both of which seem fundamental and isolated from the rest.)
When I stop to think about it, differential geometry — defined very broadly — is my primary interest.
Why? More to the point, why do I want to know more about it? Because it leads to the Yang-Mills equations, from which we get quarks.
“It was not until the early 1970s, however, that dawn broke and,
in the clear light of day, it was recognized that a gauge field
in the sense of the physicists is essentially nothing other than
the curvature of a connection on some fiber bundle.”
(p. vii, Naber, Gregory L.; Topology, Geometry, and Gauge Fields: Foundations. Springer,1997. ISBN 0 387 94946 1.)
At present, my goals are simple. I should be moving along in both Bloch and Baker (Baker, Andrew. Matrix Groups, An Introduction to Lie Group Theory. Springer, 2003 (2nd printing with corrections). ISBN 1-85233-470-3.). I’ve let myself get distracted by too many other things.
And I know perfectly well that I will continue to get distracted by other mathematics.
I have two secondary interests: control theory and time series analysis. For control theory, I should be working on aircraft. For time series analysis, I am working on wavelets.
To be more precise, I am stuck on wavelets. I have seen a beautiful and rare calculation — it seems to be in just 1 of my 20 books — and I can duplicate it (translating MATLAB code to Mathematica), but I don’t understand why it works, yet. I have learned more, and discovered more that I don’t understand.
These I call the big three. One pure, two applied. (Compared to control theory and time series analysis, differential geometry looks pretty pure, despite my interest in its applications to elementary particle physics.)
At the level of tertiary interests, I have too many to list. I begin by making the simplest possible distinction: pure mathematics and applied mathematics.
Right now, in pure mathematics, I’m still reading Simmons (Simmons, George F. Introduction to Topology and Modern Analysis. McGraw-Hill, 1963 (reprinted by Krieger, 2003 ISBN 1575242389) ). As I said when I first mentioned it, I consider it an introduction to functional analysis, and a very, very fine one it is: Simmons is an excellent teacher.
Also in pure mathematics, I have just picked up an introduction to category theory (Lawvere, F. William & Schanuel, Stephen H. Conceptual Mathematics, A first introduction to categories. Cambridge University, 2000 (reprinted with corrections, twice). ISBN 0-521-47817-0).
My internal kid is responsible for both of those choices, and I sincerely hope he stays with his latest choice for a while. I do want to follow up whatever he picks up, but I can’t very well do that if he picks up too many different things. On the other hand, I don’t want to constrain him; as I said, the bottom line is that I do mathematics because I love playing with it, and I want to encourage myself to do that a little bit every day that I do math for myself.
In applied mathematics — apart from what’s in the big three — I should get down to the nitty-gritty of human color vision, aiming to understand the CIE chromaticity diagram. Incidentally, we will be doing linear algebra for this.
In other words, I have plans, as usual. We’ll just have to see what happens.
Oh, I have put two posts out on newsgroups. For one, someone asked how to find a curve given its curvature as a function of arc length, specifically for powers of arc length. It’s a beautiful question, but the answer is pretty complicated. That was “Curvature and arc length” out on comp.soft-sys-matlab. (That’s where I went to get some help translating from MATLAB.)
For the other, someone wanted to know more about the constant in the equation ; as I interpret it, he was asking about “constant” rather than about . The equation was the title of the post, in sci.math, but you should search on the English “gamma”, if you’re interested.