## "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

##### Galois

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.)
##### Bell

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".
##### Green

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.)

##### Noether

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.