ROGER L. COOKE, Ph.D., EMERITUS PROFESSOR OF MATHEMATICS
Retired as of May 31, 2003
Email: roger.cooke@uvm.edu
Areas of expertise: history of
mathematics, Fourier analysis
Applications: Both areas help
people understand the world.
This
picture of me was taken
News Bulletin: I
came out of retirement and taught MATH 161 (The Development of Mathematics)
during the winter-spring semester of 2011.
There were two sections of this course, and the enrollment began at 80
students. With a few dropouts, it was 76
by the last day of classes on May 4.
MY
PROFESSIONAL LIFE (
The
details are all here in
this brief summary of what I’ve been doing for the past three years or so, plus
a bare-bones outline of my entire academic career. Since I’m no longer seeking academic
advancement or employment, there is little point in posting more details than
that. The general features of my career
are described in the following paragraphs in any case.
My Education and Employment: After graduating from Northwestern
University in 1963
with a mathematics major, I attended Princeton
What I've Produced: My research was originally in multiple
trigonometric series. The best thing I did in that area was to prove a Cantor–Lebesgue theorem in two variables, back in 1970. It turned out to be a simple thing to do, but
no one had expected it to be easy, so I was lucky enough to get in ahead of the
others. After 1975 my motivation to do
pure mathematical research waned and I began to be interested in the history of
mathematics. In 1981 I took a sabbatical year to start working in this area. My
first effort was The Mathematics of Sonya Kovalevskaya (Springer-Verlag, 1984). Over the years I have written a number
of articles about her mathematical work.
I think my best work in this area was a history of the Cauchy–Kovalevskaya
theorem, which I presented at a conference in Lisbon in 2001, but which I
have never tried to publish.
I
spent part of a sabbatical year (1988–89) in Moscow studying the works of N.N. Luzin. One result of this work was a study of the relation
between uniqueness of trigonometric series representations and descriptive set
theory from 1870 to 1985, which appeared in Archive for History
of Exact Sciences in 1994.
My
major effort in the history of mathematics is a textbook intended for a first
undergraduate course (1997). The second, thoroughly
revised edition, whose cover you see below, is available from Wiley. The cover design came from a quilt bearing
the name "A Number Called Phi" that I saw at a show in Northfield,
Vermont back in 2003. The quilt's
creator, Mary Knapp of
During
my years at the 
professor at 
Norwich
University in Northfield, Vermont.
Gerard's dissertation was in almost-periodic functions and appeared as
our joint paper "A characterization of the Fourier series of Stepanov-almost-periodic functions" in the Journal of Fourier Analysis and
Applications, Volume 7 (2001), No. 2, pp. 127–143.
In my
retirement I had hoped to work on the kinds of hopelessly difficult problems
that young mathematicians dream of solving, such as the Riemann
hypothesis. I also wanted to continue my
work in the history of mathematics and science by learning the history of
superstring theory. I got distracted
from these projects during 2007 by another writing project that I couldn’t
resist, a book that I call Classical
Algebra: Its Nature, Origins, and Uses.
It has now been published, and I’ve looked at it enough to find three
misprints and one small error. So, I
guess I’d better put up a web page of
corrections. As long as I’m on the
subject, I’ll just boast a bit and mention that this book has won a CHOICE on-line readers award as one of nine outstanding academic books
in mathematics for 2008.
Since
I am now working on the third edition of the history textbook, it will
obviously be some time before I resume working on the story of superstring
theory. Since I have much to learn
before I can competently write about any of this material, I expect it will be
several years before I publish again.
But please stay tuned in to this website. I may find something interesting and decide
to blog about it.
And who knows what writing I may undertake if asked to do so.
Sidelines: In addition to teaching and
research, I have contributed, I hope, to the advancement of knowledge through
several ancillary projects, some of which are the following.
Translations. I began translating Russian
mathematical articles for the American Mathematical Society in the 1970's. The AMS translations project
was taken over in the 1990s by the London Mathematical Society, for whom I translated a few articles from
the Soviet journal Математический Сборник (Matematicheskii
Sbornik). I did a great deal of translation (I
estimate some 10,000 pages) of Russian and Ukrainian articles and books during
the years 1986–1998, when my three children were in college. The next-to-last project I undertook in this
area was a translation of the fourth (2002) edition of the two-volume Математический Анализ (Matematicheskii Analiz), by Vladimir Zorich, for Springer-Verlag. This work is
the best rigorous, yet thoroughly applied work on real analysis for
undergraduates that I have seen. I am
very proud to have been the translator of these excellent textbooks. The very last project was to translate 20 of
24 essays on various aspects of twentieth-century mathematics with emphasis on
Soviet contributions, a collection bearing the title Математические События ХХ Века (Mathematical
Events of the Twentieth Century),
which has recently been published by PHASIS/Springer-Verlag. That's absolutely it as far as I'm
concerned. No more translating.
Fun Problems. As a
puzzle enthusiast, I like to take the challenge of each year's Putnam
Examination, administered by the Mathematical Association of America. If you'd
like to compare your answers with mine, click here to see my solutions to the 2010 Putnam Examination,
given December 4.
Under
this heading, I plan to add, from time to time, essays on mathematical topics
that interest me. Here’s the first, a
proof of the Steiner–Lehmus Theorem that I thought of some 25 years ago,
then forgot about until reminded by my friend Tony Trono,
who learned of my proof from a former student of mine. While writing it up for posting here, I
thought of an analytic proof and included it.
Here is an essay I wrote
as a review of the recent book Naming
Infinity by Loren Graham and Jean-Michel Kantor. I wrote it at the request of the editors of The Mathematical Intelligencer. However, I let myself go in this one and just
wrote what I wanted to write, so the end result was too long and rambling for
their purposes. I wrote them a second
review, which appeared in 2010.
Here is a problem
recently posed to me by my colleague Sheila Weaver. It gives a sequence resembling the Fibonacci
sequence except that the ratio of the even-numbered terms to their predecessors
approaches the square root of 2, rather than the Golden Number.
On
Monday, May 16, 2011, I gave a lecture at the annual Math Day celebration at
the Davis Center at UVM. My subject was
the infinite and the way mathematicians have thought about it. Here, in printed form,
is an expanded version of what I said. The
Power Point presentation that I used for the lecture is here.
In mid-September
2011 I gave a couple of guest lectures on ancient measuring instruments in Dean
Grasso’s special topics course. For this project, I wrote an extensive essay
on the technical and historical side of these questions (http://www.cems.uvm.edu/~cooke/measurement.pdf),
from which I excerpted two lectures (http://www.cems.uvm.edu/~cooke/measurement.ppt)
and (http://www.cems.uvm.edu/~cooke/navigaion.ppt). I also cobbled together a couple of toy
models of ancient surveying tools out of scraps I happened to have in my
garage.
Useful Problems. I am also happy to serve as a consultant to
the public and to my colleagues at the University, as these free consultations
often lead to interesting problems to be solved. Here are some
samples of my work. May I politely ask,
however, that you not send me your angle trisections, circle quadratures, and the like. I have examined many dozens of these over the
years (one sample is posted here), and I feel I have earned my retirement from
this type of work.
Free Stuff!! Over the years, I have written several textbooks in my own quirky
style. Because they are so
idiosyncratic, they wouldn’t have much commercial value, and I have not tried
to publish them. I offer them for free
here. They were written using standard
Latex and converted to portable document format for posting. You can download these pdfs
and use them any way you like. If you’d
like to modify them, write to me, and I’ll send you the original TeX files from which they were produced. I’m putting up two of them here, with a third
(on vector analysis) to follow soon. (I am currently proofreading and indexing
it.) So, here they are. The first is a practical-math textbook for
social science and environmental studies students, which I call Empirical
Mathematics. I have actually used it in the classroom
once. It wasn’t a great success, but the
audience was, after all, reluctant to be in any mathematics course. The second is a course in real and functional
analysis intended mostly as background for understanding the use of linear operators
in quantum mechanics. I call it Classical
Analysis. I have never used it in the
classroom, but you may find it a useful source of problems. Be warned that my definition of the Riemann–Stieltjes integral is not equivalent to the standard one
that you will find in “baby Rudin.” I have what I believe are cogent reasons for
preferring mine.
Have
all the fun you like with these books and papers. Just don’t write to complain about any mistakes
or other infelicities you find. You got
it for free, and it’s worth every cent you paid for it.
The History of
Mathematics at UVM. Around 1990, in connection with the UVM
bicentennial, I wrote a history of mathematics at UVM. I have recently looked at it again and added
a few endnotes to update it. I’m putting
it here in several forms, so that you can have your choice of format: (1) web page (.html); (2) Word 1997–2003 (.doc);
(3) Adobe Acrobat (.pdf). I also have a
Microsoft Word version (.docx) that I’ll be happy to
send to anyone. I don’t include it here,
since the .docx format doesn’t download very
well. Microsoft Internet Explorer
regards the file as a zipped file and handles it accordingly. Finally, I also have a plain TeX version, which I’m also willing to send. I’d be happy to rework this piece if any
ambitious historian of American mathematics out there wants to compile an
encyclopedia of what was going on at all the centers of activity, great and
small, during the early years of the Republic.
I think 1950 would be a good terminal year for such an encyclopedia. This is the one exception I would be willing
to make to my sworn intent (see above) not to get involved in any more projects
outside my main interest.
Apologia Pro Vita Mea: The immediate practical value of what I do is
very limited. My whole background is
"liberal artsy," and I regard simply understanding the world,
independently of any personal or economic gain, as being practical. I'm very much in sympathy with the ancient
Greek ideals enunciated by Plato and Aristotle that education should be aimed
at this kind of understanding. At the
same time, I am enough of a realist to recognize that this kind of education
has an economic cost to society, and, as sardonic old Henry Mencken wrote, one
should not expect to be supported because he knows Sumerian. Professors with my outlook owe it to society
to be good and dedicated teachers. We
should not adopt the arrogant attitude of Godfrey Harold Hardy, whose 1940 book
A Mathematician's Apology argued that, even if mathematics is a waste of
time, Oxford dons should be allowed to waste their time pursuing it. Such a view is self-serving. Why should
others work and be taxed or charged tuition in order to support the production
of papers that appeal only to a small elite?
If we expect such support, we should honestly say why such knowledge is
of value, and "sell" it like any other commodity. The goal should be to persuade others that
understanding, without regard to economics, is of value. In other words, we should either perform a
useful service for the community as a whole or enlarge the elite who appreciate
scholarship and willingly support it.
Burlington Scenes: While
walking, bicycling, and cross-country skiing in Vermont over the decades, I’ve
taken a lot of pictures. Here’s a small sampling
of pictures that I’ve taken around Burlington, arranged as a four-season album. It’s a fairly large one, as I deliberately
did not compress the images. You can
download them if you wish, and you’ll have high-resolution pictures in most
cases.
My Hobbies: Besides regular running for exercise,
gardening, and keeping up my languages (Russian, French, German, Japanese,
ancient Greek, Latin), I like to play the piano. With the Yamaha P-80 keyboard that my
colleagues so generously gave me when I retired, I have recorded some of my
favorite music. Here are two pieces by
Chopin that I particularly like, played by me with all the amateurish clinkers
you'd expect. I still regard it as a
great blessing to have been able to play these pieces, even very imperfectly,
and I rejoice that there are people who play them much better than I do, both
technically and artistically.
What
do you picture as you listen to them?
I’ve always been intrigued by the analogies between different
senses. How does hearing music create
pictures in our heads? There is
something that sounds and pictures have in common, I’m sure, so that they “go
with” each other. If not, sound tracks
would not so often fit the action in a movie.
In these two examples, from the romantic period, I “see” plenty of
storms. The Polonaise opens with several
lightning strikes, interspersed with rolling thunder. This builds up and finally bursts in what I
cannot help picturing as rain falling out of the sky, slowly at first, then in
a great flood. After that, it’s
anybody’s guess what Chopin was picturing.
However, the middle section in E major with the rapid octaves suggests
either rapids in a river or a cavalry charge.
I’m inclined to see the latter because of the hoofbeat-like
sounds in the bass.
The E
major étude is much simpler to picture. The beautifully lyrical and calm first part
suggests (to me) a clear, calm summer day, with puffy clouds floating across
the sky. The more agitated middle
section suggests first rising winds, then sudden flashes of lightning and
thunderclaps, again finally resolving itself in a furious rainstorm, which
gradually blows itself out, and brings in, once again, clear, calm weather as
the sun sets.