"Lecture" for Week 11

Modern algebra

There is quite a lot of reading this week, since Allen likes number theory, Stillwell likes group theory and quaternions, and I like vector spaces. I will try to hold the written problems to a minimum.

Linear algebra

Some years ago I wrote a textbook for our applied-analysis-oriented linear algebra course (Math. 311). I realized that textbooks on linear algebra usually contain very little history, unlike books on calculus, differential equations, Fourier analysis, abstract algebra, topology, number theory. In fact, I realized that I knew very little history of linear algebra. So, I read up on the topic and wrote a short history chapter for the book.

Prof. Allen has very little on linear algebra either, so in the 2003 course I inserted a rewrite of my chapter on the history of linear algebra into the syllabus. Stillwell brings Chapters 20 and 21 right up to the dawn of modern linear algebra but has little to say about the latter. It is characteristic of the gap in perspective between pure mathematicians (especially algebraists and number theorists) and applied mathematicians, physicists, and analysts that the former think of quaternions as a major development while the latter think of them as a fascinating evolutionary side track (like woolly mammoths) on the way to the proper understanding of vector spaces and Lie groups. More on this next week.

Some biographical remarks and references

Evariste Galois vies with Alan Turing to be the most Hollywood-worthy mathematician. This article is fun to read: T. Rothman, "Genius and biographers: The fictionalization of Evariste Galois", Amer. Math. Monthly 89, 84-106 (1982). This article reinforces my initial warning not to believe everything you read about the history of mathematics, especially when written by nonprofessional historians. In particular, the statement that Galois "hurriedly [wrote] out his discoveries on group theory" the night before his duel is an exaggeration. (He had been writing manuscripts for several years and trying to get them published, but for a combination of reasons they had not been published. What he did that night was to write a summary in a letter to a friend and also to go through the manuscripts and annotate them. At least that's what Rothman -- who is not a professional historian either -- says.)
Rothman is especially hard on the author Eric Temple Bell. Bell's books on mathematics and mathematicians were immensely popular around the middle of the 20th century; I read several when I was in high school. Today they are out of favor, being regarded as inaccurate and antifeminist (although I think no more sexist than most books about math history from that period and earlier). That's why they are not listed in any Math. 629 web pages, although more than one of them would qualify as math history books on the basis of content. It turns out that Bell himself was a strange character (he had some private demons, as they say). There is an article about him by Constance Reid, Am. Math. Monthly 108, 393-402 (2001). Rothman's title is a play on Bell's chapter titles for Galois, "Genius and Stupidity", and Abel, "Genius and Poverty".
George Green was briefly mentioned in Stillwell Sec. 16.3, but in my lectures he appears here because I regard vector calculus as a logical outgrowth of linear algebra.

Green has a somewhat unusual biography. He was a self-taught young man from a small town in England who wrote an essay of about 100 pages in which he introduced all the things that bear his name and are so important in partial differential equations and the physics of fields (such as electromagnetism): Green's theorem, Green's integral identities, Green functions. (The essay is available on the Web, but the notation is so archaic that it is difficult to read.) The value of his work was recognized and he was invited to Cambridge to become an undergraduate (at a rather late age) and eventually a college Fellow. Unfortunately, he fell ill and died soon thereafter. No photographs or portraits of him are extant.

This information comes from a nice, short article, L. Challis and F. Sheard, The Green of Green Functions, Physics Today 56 No. 12 (Dec. 2003) 41-46. (As usual, you can get this on line through TAMU Evans Library, but it is not legal or practical for me to link to it.)

Another amusing difference of perspective between mathematicians and physicists is that every physics student is taught to revere "Noether's theorem" relating symmetries to conservation laws and would be surprised to learn that Emmy Noether was primarily an algebraist. Stillwell (p. 465) gives the impression that this famous paper was part of Paul Gordan's program on the brute-force study of explicit algebraic invariants, but I see it differently. Noether's paper has been translated to English by M. A. Tavel in Transport Theory and Statistical Physics 1 (1971) 183-207, with a commentary. The best related pedagogical paper I encountered as a physics graduate student is E. L. Hill, Reviews of Modern Physics 23 (1951) 253-260. I thought I could paraphrase an explanation of the theorem from one of these, but now I see that I can't improve on Wikipedia's summary: "If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time. [More precisely,] to every differentiable symmetry generated by local actions, there corresponds a conserved current." The article goes on to give some definitions and examples.