# My Life as a Mathematician By Theodore W. Palmer

[[This essay was illustrated with pictures of the most significant mathematicians, but again, I do not know how to move them onto my blog. Darn!!]]

Mathematics has traditionally been divided (approximately) into two halves: **algebra** and **analysis**. Algebra is an enormous expansion of high school algebra. Analysis starts with calculus and is characterized by the fact that all answers are theoretically obtained by infinitely many steps of approximation.

In 1965 I had written a thesis on a part of analysis called operator theory. It was all finished and I was waiting for my good wife to type it up after her long days of work in a Nobel Prize winner’s laboratory at Harvard Medical School. I bought a used book on the recent subject of **Banach Algebras** by **Charles Rickart** and was reading it for fun.

A central part of my thesis was a truly repulsively complicated argument involving integration in infinite dimensional spaces. I had checked it many-many times and was sure it was correct, but it was beyond anyone’s (certainly my own) ability to understand it all at one time.

In a flash, I recognized that an argument in Rickart’s book could be modified to replace my horrible integration argument by a fairly transparent proof of an entirely different character. The new proof was also much shorter. This experience of my own is a lot like the original discovery of the theory of Banach Algebras.

**Stefan Banach** was a Polish mathematician who published a very

influential book in French in 1933. In it he put together a number of his fundamental discoveries to show that one could work effectively in an infinite dimensional linear space if one assumed that it had a property called **completeness**. The wonderful thing was, that it was perfectly reasonable to assume that the spaces that arose in “practical” applications were complete.

Banach’s ideas were being exploited widely during the latter half of the 1930s. In 1939 one of the greatest of 20^{th} Century mathematicians, the Russian Jew, **Israel Moisevich Gelfand**

published three 3-page papers together in the Russian journal ** Doklady**. In the middle paper he used Banach’s machinery to show that a very deep problem in analysis had an amazingly simple algebraic structure. Thus he re-united the two halves of mathematics in a completely new, very deep fashion.

Using this brilliant insight, he was able to prove **Wiener’s Theorem** in half a page. **Norbert Wiener** was a very famous

American mathematician who was a faculty member at the Massachusetts Institute of Technology for the latter part of his life. In a series of deep papers in analysis he had proved the amazing result that if a function, f, has an absolutely convergent Fourier Series and if the function f never assumes the value zero, then the function 1/f also has an absolutely convergent Fourier series.

Wiener was extremely proud of this fundamental result and of course it is called Wiener’s Theorem. Wiener had been an amazing child prodigy and wrote a best selling book about this, so many non-mathematicians knew his name. He also had extremely broad interests, and was eccentric, so most mathematicians know lots of amusing stories about him. But the chief point is that Wiener’s Theorem was almost universally known and considered to be a crowning achievement of analysis in the first third of the 20^{th} Century.

Thus proving this theorem in less than a page by a method which almost any professional mathematician could immediately grasp was a stunning development.

The timing of the discovery was more than a bit unfortunate. Adolf Hitler had been threatening every one around in increasingly menacing ways during the 1930s. Joseph Stalin, who ruled Russia as brutally as Hitler ruled Germany had believed he could keep Hitler at bay. For this purpose friendship with America and England was crucial. Thus Gelfand published his three short papers in English. A Russian Jew had to keep track of which way the political winds were blowing. However almost immediately Stalin decided to completely shift sides and signed a nonaggression pact with Germany to divide up Poland and do other terrible things.

The result was that Russian journals with the work of Gelfand and his growing group of brilliant students did not get to America and other places reliably or on time. Hitler brutally invaded Poland on September 1, 1939. When England honored its treaty obligation and declared war on Germany on September 3. When the USA was drawn into World War II after the Japanese attack on Pearl Harbor on December 7, 1941, all our mathematicians and scientists were recruited (very willingly) to work in the War Effort. [[Personal note: The last two of these dates are two of my three earliest dated (traumatic) incidents of my life—just before I turned 4 and just after I turned 6.]]

Stalin did things differently. All the leading mathematicians and scientists in Russsia were sent to a secure area in Siberia where they were given the best musicians, poets, chefs and plentiful food while millions of less privileged Russians starved. They continued research unaffected by the War. By 1945 America was hopelessly behind. I had the privilege of being in the second cohort of those beginning to get us caught up in Banach algebra theory.

Now I shall go back and explain how I happened to be writing a doctoral thesis on operator theory at Harvard in 1965.

Through high school, mathematics courses were easy and boring for me. Many seemed to involve learning the names for many things: dividend, hypotenuse, *etc*. This seemed stupid. Towards the end of high school I bought a used rather poor textbook on calculus and read it without real understanding. At the time I was reading books on both inorganic and organic chemistry with deep understanding and fascination. I already had an undergraduate understanding of most of biology due to my family surroundings.

From Webb City Junior-Senior High School I went to **Johns Hopkins University (JHU)**. At the time, freshmen were expected to start mathematics with analytic geometry. This is a subject I felt I had fully mastered from the calculus book, so I tried to skip it and the first term of calculus. I was wisely prevented from doing so, but I don’t think that I learned anything useful in the semester of analytic geometry. That course reverted to teaching lots of names of, at least slightly more useful, ideas.

At the end of that term I chose to take Honor Calculus, because I was in favor of educational honor. [I also took Honors History and became good friends with the most famous historian at JHU.] I think the class started with almost thirty students. (JHU was an all male institution.) Within two weeks it was down to 8 of us who all lasted the three-semester course. The text book (loosely speaking) was the excellent, even famous, book by **Richard Courant**.

The course started with **Peano’s Postulate**s for the **Positive Integers**: 1, 2, 3,…. These are also called **Natural Numbers**. From these we constructed (easy) all the **Integers**, 0, ±1, ±2,…. We then constructed the **Rational Numbers**: fractions with

integers on the top and natural numbers on the bottom. The final step in this process (3? weeks in) was using **Dedekind Cuts** to constructed the **Real Numbers**. For most people the real numbers are all the numbers. In particular each point on the ordinary line (indefinitely long on both ends) is in one-to-one correspondence with a single real number after one chooses where zero and one belong. The real numbers are used daily by nearly everyone, particularly technological workers, engineers, carpenters and scientists. However, from a mathematical viewpoint, they are more complicated than any human being will ever fully understand.

Now we were in a position to begin developing calculus. This subject is about **functions** rather than numbers. A function is a rule that assigns one thing to another—in mathematics very often one real number to another real number. You already know lots of functions. The square function assigns the square of a number to each number: *e.g*. : 1 goes to 1, 2 to 4, -6 to 36, 0.3 to 0.09 . Probably you know the sine function which assigns to any real number x the number sin(x) which is always in the interval between -1 and +1, inclusive. I will not continue with the description of this course. It was taught for the last two semesters by **Phillip Hartman**, a significant mathematician. Rather soon after he took over the course, Hartmann, took me out for coffee and told me I should become a professional mathematician. This idea was new to me but becoming attractive. I had entered college with the fixed purpose of becoming a biochemist. From my second year I was taking roughly half a normal graduate load of courses in biochemistry in addition to my full load of undergraduate courses.

At the end of honor calculus, the beginning of my third year, I enrolled in the very standard beginning graduate mathematics course called **Modern Algebra**. We used the first and most successful textbook written for this curiously named course written by **Garrett Birkhoff** and **Saunders Mac Lane**. Again the course started out with a reasonable enrollment, and quickly shrank down to about a dozen students. I actually liked both instructors who were brilliant mathematicians. However, since I will say seemingly critical things, I shall not name them. First semester the instructor was often a few minutes late and always pretended that he did not remember where we had left-off at the end of the last lecture. He would ask us and then pretend to make up a lecture on the spur of the moment. Sometimes this was all too believable, but more often he described the ideas in the book quite accurately, thus losing credibility.

Second semester we had a new instructor, a newly famous very young Japanese mathematician who spoke and understood not a single word of spoken English. We all knew mathematical symbols and the logical symbols that go with them. He could actually print a few words of English which are important in mathematics. In this way he communicated his beautifully planned lectures until on rare occasions we became hopelessly confused. By then we knew each other well. So we would bow to him and then begin discussing among ourselves. Either we figured out what was going on or decided it was hopeless at the moment. We would then bow to him again and he would bow to us and resume writing logical symbols on the blackboard. He was clearly well prepared except for his in-facility in our mother tongue.

At the end of this year I had decided to switch from biochemistry to mathematics. I was scheduled to receive a Master of Arts as well as my Bachelor of Arts in biochemistry at the end of the next year. I needed to write a thesis for both and was deeply involved in a nontrivial research project. Thus I only had time to take one more math course. It turned out it was a dud and I only attended one lecture and the final exam.

You can imagine that when I mentioned my changed plans to the biochemists there was consternation. I was told I could have my Ph. D. in one year based on a bit more work on my then current research project. However they had a plan for revenge also. I had to take an oral exam. They recruited an old mathematics professor and told him to find a mathematical topic about which I was totally ignorant and then grill me on that topic for half an hour. It was easy to do and was as humiliating as intended. This could not go on forever so eventually a biochemist asked “What is the cause of progress in the world?” After ascertaining that general progress, not technological progress was meant and that a time span of hundreds or a few thousand years was appropriate, I was on the point of beginning an extemporaneous answer when the old mathematician declared very positively that there had been “NO PROGRESS”. The committee argued for awhile while I prepared a truly wonderful answer in my head. That was my biochemistry oral exam!

When I got to Harvard as a very poorly prepared mathematics graduate student there were about 30 of us. (I used to remember the precise number that was divisible by 3.) Since I had been so far from interest in mathematics as a college freshman, I interviewed all my fellow beginning graduate students. Only one had entered college with an interest in mathematics as a major. As it happened he was the only one to drop out in the middle of the first year.

I am a hard worker who loves study above nearly anything else. However, I have never worked so hard as I did that first year of graduate study. I had taken four math courses (counting analytic geometry and the one I only attended once). I suppose the other students had had a minimum of a dozen and most had many more. That first year as a mathematics graduate student, besides the courses for which I registered and for which I was more-or-less prepared, I sat in on two other seminars attended mainly by famous faculty members, many visiting: “The Arithmetic of Elliptic Curves” by John Tate

and “The Cohomology of Fibre Bundles” by Serge Lang. (The names of the courses attracted me and the fascinating subjects kept me involved for most of a year.) On the first day Tate announced that, after a short introduction, he would only discuss results he had proved since the last lecture. If he had no new results he would go over what the problem seemed to be. It was his way of forcing himself to work! Pretty effective, I guess, since at least 5 of the world’s most famous mathematicians were in the audience.

Graduate students had to take three half days of written qualifying exams at the beginning of their second year. Something similar, often a year later, is true in most graduate mathematics departments, but usually there is a syllabus of subjects about which one can be asked. Harvard had a syllabus: “mathematics”. As I learned during my second and subsequent years, over several years the number of new graduate students was divisible by 3, and precisely 1/3 passed the qualifying exam. This was a terrible system. Most of the first year graduate students had been the best mathematics major at their undergraduate university by the time they graduated. The 2/3s who flunked the qualifying exam, graduated with a Master’s Degree at the end of their second year and often earned a doctorate elsewhere faster than those of us who stayed. Indeed the requirements for a Master’s Degree were word-for-word the same as for not flunking out in the second year.

My route to a thesis was far from straight. My first mathematical love was number theory. Number theory was originally primarily about the fascinating, beautiful properties of integers primarily under multiplication. About a century before my graduate study the subject had begun to consider new number systems called **algebraic number fields **and** algebraic integers**. My first year at Harvard I took a wonderful full year course on this gorgeous subject from the very great practitioner, **Richard Brauer**.

**Class field theory** was a relatively recently developed part of algebraic number theory involving **Galois groups**. So far it only dealt with commutative groups. However just when I entered graduate school a number of complicated methods were being developed to extend some results from commutative to non-commutative groups. Thus I proposed to John Tate that I should try to use these new methods in class field theory. He loved the idea and for two years I made tiny progress which he claimed to find very encouraging. However I became increasingly uncomfortable. One of the great joys of mathematics is that one can totally understand what one is doing without any shadow of doubt. I was getting terribly uncomfortable because I was losing this feeling of absolute certainty.

The Department of Mathematics was housed upstairs in a lovely little building at **2 Divinity Avenue**. The ground floor and substantial below surface level was occupied by the **Harvard-Yenching Institute**. (Yenching is a long obsolete name for Beijing, the Northern Capital of China.) This is one of the most venerable western academic institutions for the study of China. Since I had been fascinated by China since I was 5 years old, I made friends with some people downstairs including the Librarian, who oversaw some of the oldest Chinese books preserved anywhere in the world. As I became increasingly uneasy with my mathematics, an idea I had entertained since childhood began to percolate again. I would learn to read and write Mandarin. I had no particular desire to speak or understand the language, but it would be silly not to do this at the same time. It turns out there was an intensive Mandarin course developed during the War which met 5 hours a day 5 days a week and required one to commit to spending twice that time out of class. I signed up, knowing full well that this would leave little time for mathematical research. I think I was happy for the first semester. Then I finally understood that the greatest scholars of Classical Chinese often spent about a year understanding a single poem written on a Chinese painting. The motivation for learning Mandarin had been my ignorant desire to be able to read these poems. I finished the course but decided to drop out of the graduate program in mathematics, at the end of the year. I also stupidly dropped Mandarin use and study right away.

I spent the summer consulting on Route 128, the belt line around Boston which was the original version of Silicon Valley. In real terms I made more money that summer than I have ever since. I had married my wife, Laramie, when she was one semester away from finishing her BS at the University of Wisconsin. I had promised her (and her dad) that I would arrange for her to graduate. Thus she did so that fall semester, earning a Phi Beta Kappa key. I thought I would be able to support us consulting in Madison. I found the situation was very different from Boston. I did find an interesting job consulting, but at a tiny fraction of what I had been paid on Route 128. The job did result in my first publication which proved to be important in its field of quantum chemistry.

Second semester I had arranged to teach as a Woodrow Wilson Teaching Fellow at the historic black college, **Hampton Institute**. This is another fascinating story, for different time.

I realized that the only sensible thing to do was to go back to Harvard and finish my doctorate. My consulting in Madison had directed my interest towards operator theory. I had read a paper by the Romanian mathematician **Ivan Vidav**. Any mathematical theorem, when fully stated, says that if something is true, then something else is a logical consequence. The condition is called the **hypothesis** and the result is called the **conclusion**. Vidav’s theorem had a strong hypothesis (undesirable) and a strong conclusion (desirable). I immediately realized that it was possible that the conclusion of the theorem would remain true with a **much** weaker hypothesis. Several years later I discovered that a number of other mathematicians had had the same insight.

When I returned to Harvard I started to work very hard on research in operator theory. At Christmas break I decided my research was not progressing well enough and switched to working on an idea loosely related to Vidav’s theorem. This thesis progressed rapidly so that I was waiting for Laramie to type it by early fall the next year. The application of Banach algebra with which I began this essay, directed my attention back to Vidav’s theorem as originally stated. In a few months while I was walking in the woods back in Madison, Wisconsin in the fall, I found a very deep proof. Due to a nefarious referee the publication of this result was unreasonably delayed. While walking to the university on a very bright cold February day in Lawrence, Kansas I suddenly realized how to further improve the statement of my new version of Vidav’s theorem. I then managed to get around the dishonest referee. I published my result, since called the Vidav-Palmer Theorem, in a three page paper. (Remember Gelfand.) The well known analyst **Paul Halmos** chose this paper as one of the ten most important of the 1970s. Since then my deep proof has been replaced by a beautifully simple, elegant proof from a seemingly unrelated field of mathematics.

End notes. The point of this essay is to let readers know how a medium level mathematician experiences and understands his craft. It is unfortunate that even well educated people are not aware that much more than half of what we know about mathematics was discovered in the 20^{th} Century. Since most college level courses hardly get beyond what was reasonably well understood about 400 years ago, hardly anyone understands the enormous excitement of present-day mathematical research. Even if you know that Fermat’s theorem was stated in 1637 and proved (using some of the ideas from Tate’s seminar) by **Andrew Wiles** in 1994, you almost surely do not know how

many extremely famous mathematicians, developed deep and fascinating new ideas in their attempt to prove this simple sounding result. You are even less likely to be able to grasp the amazing new ideas which were needed at the very end of this three and a half centuries long quest.