Topology Atlas Document # topc-49.htm | Production Editor: Thomas M. Zachariah

## LEONARD GILLMAN; AN INTERVIEW (Part 2)

by Melvin Henriksen

Leonard Gillman
Austin, TX 78746-2236
Phone: 512-327-2277, Fax: 512-327-2274
Email: len@math.utexas.edu
March 1998
Interview from Volume 3, #1, of TopCom

### 9. Purdue university

In Spring 1952 we acquired a newborn puppy, whom we named "Tarski." At about the same time, I got a telegram from Ralph Hull, the department head at Purdue, offering me an instructorship to start in the fall. He said he did not have final budget figures yet, but the salary would be between $3600 and$4000. Reba and I were struck by the warmth of his letter. I wrote back to accept, but pointed out that because moving to Indiana would cost $400, I was hoping that the eventual figure would be at the upper endpoint of the interval, which it proved to be. At the 1952 summer meeting, I encountered Walter Rudin and we exchanged notes. He had accepted an assistant professorship at the University of Rochester for the same figure. We were not sure what to make of this. At the same meeting I met Ralph and Pauline Hull. The students at Purdue, and the atmosphere in general, represented culture shock. My teaching load the first semester there was 15 hours, which to someone brought up at Columbia and MIT, where professors taught 6 hours, was sheer barbarism. But coming from an 8-hour-a-day job, I found to my embarrassment that I had oodles of time to think about mathematics. In part this was because class preparation was minimal: those 13 hours con sisted of a 4-hour analytic geometry, a 5-hour college algebra (i.e., high school algebra), and a 6-hour college algebra; this last was the same as the 5-hour class but intended for the weaker students, on the educational principle that the presence of the corpse in the classroom implied that learning was in progress. The next term I taught 13 hours, and before long, 8 and then 7, which were close enough to 6 to satisfy me. My office was a long, narrow room that I shared with new instructor Melvin Henriksen, assistant professor Gordon L. Walker (later to become Executive Director of the American Mathematical Society (AMS), and a retired geezer named Stone who rarely showed up. Gordon did not spend any extra time in the office either, so Mel and I essentially had the place to ourselves. He had completed his doctorate a year earlier in the ring of entire functions and tried to interest me in the subject, but my head was still swimming in my own work. He had also been reading Edwin Hewitt's work in rings of real-valued continuous functions. More recently, he had read a paper by Kaplansky in which Kap pointed out that in the ring of real functions on a discrete space, every prime ideal is maximal, and Mel wondered whether conversely, this algebraic property for C(X) characterized X as discrete. Pretty soon we were working together on the question. We answered it (No) and derived the correct characterization: all zero-sets of continuous functions are open (as well as closed). We called these spaces "P-spaces", "P" for pseudo-discrete or for prime or, as we joked, for professor, as both of us had just been promoted to assistant professor (and our salaries increased to$5200). Our paper gave examples of P-spaces that are not discrete.

At about the same time, Mel organized a seminar in rings of continuous real functions. Our colleague Meyer Jerison and I joined as sponsoring faculty. Four graduate students participated: Joe Kist, Carl Kohls, Maynard Mansfield, and Bob McDowell. It was an exhilarating, no-holds-barred affair. We were all learning together. And learn we did. The results were four additional GH papers in the next two years, one of them with Jerison and one with Erdös, one or more HJ papers, doctorates with one or another of us for the four students, and, eventually, the GJ book.

The department had an assistant head, Harold Jonah, in charge of minutiae such as course schedules. He office was a tiny cubicle piled high with papers. Deciding who should teach what was difficult, as enrollment figures were not known until the day before classes started. That evening, department faculty would drift into the waiting room to find out their assignments. Jonah fashioned a grid on the immense blackboard and made entries as the news from the registrar trickled in. Often one had to wait until after midnight to find out that he had an 8:00 o'clock class the next morning and what the subject was.

Jonah was proud of his ability to accommodate faculty requests. Once I decided I needed an easy schedule, and phoned him to request two sections of the same course, on successive hours, in the same room. Jonah phoned back the same afternoon to report that he had two sections of Calculus 1 for me, one at 9 o'clock and one at 10 o'clock; then, in the voice of a crushed man, added, "But the first one is in Room 206 and the second one is in Room 207."

I was hoping my doctoral diploma would be dated 1952 so that it would be signed by Dwight Eisenhower, then the President of Columbia University; but the dissertation defense was scheduled in 1953, by which time Eisenhower had left Columbia to become President elsewhere. Incidentally, the dissertation defense was quite a dignified affair in those days. It was held in a mahogany-paneled room in the imposing Low Memorial Library. The committee was not just departmental but was intended to represent the larger Faculty of Philosophy and typically included two professors outside the department. My two were Ernest Nagel, an eminent philosopher of science, and a physicist who announced early on that he did not understand anything of the dissertation. But he did ask an intelligent question: why is it always powers of two that I talk about: why not ever 3 or some other number?

It was now 12 years since my undergraduate degree and I had reached the age of 36.

The triple papers that Mel and I wrote deserve comment. Jerry had conjectured a characterization of beta X (the Stone-Cech compactification of X) and the three of us had proved that it was true. Then he dug up a 1939 paper by Gelfand and Kolmogoroff that Hewitt, in his big paper, had referred to but apparently not appreciated, and there we found Jerry's characterization. The three of us sat around to decide what to do; we called it the "wake". Since the authors had not furnished a proof, we decided to publish ours. When the referee expressed himself strongly that a title should be informative, we came up with On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions. (This proved to be my second-longest title, and a nuisance to refer to.) Kolmogoroff died many years ago, but Gelfand is still living, a vigorous octogenarian now at Rutgers. A year or so ago, I met him at a dinner party in Austin and mentioned the 1939 paper. He remembered it very well and proceeded to complain that the only contribution Kolmogoroff had made was to point out that a certain result was valid for the complex case as well. I was intrigued to see how the giants grouse about each other just as we do.

The triple paper with Erdös was a forerunner of nonstandard analysis (so referred to by Robinson). A classical theorem of Hausdorff states that any two $eta_alpha$-sets of cardinal $aleph_alpha$ are similar (i.e., isomorphic as ordered sets). The $eta_alpha$ property is that no subset of power $aleph_alpha$ is either coinitial or cofinal and that no two sets of cardinal $aleph_alpha$ are contiguous (i.e., if every element of one precedes every element of the other, then some element of neither lies between them). For example, Q is an $eta_0$-set.)

By the time Mel and I corralled Erdös, we knew that all nonarchimedean residue class fields C(X)/M were real-closed and were $eta_1$-sets, and that the existence of an $eta_alpha$-set of cardinal $aleph_alpha$ required that $2^aleph_alpha} =$aleph_alpha+1$(the "continuum hypothesis at level$alpha$''). We conjectured that all$eta_alpha$-fields of cardinal$aleph_alpha$were isomorphic (as fields). The main theorem in the paper is that this conjecture is true. In the late 1980s, I encountered Erdös at an MAA Section meeting, at dinner, eating his soup. An enthusiastic host asked, "Erdös, do you remember Gillman?" Continuing to eat his soup, without looking up, Erdös replied in a somewhat bored voice, "Yes, we wrote a joint paper in 1955." In the same year, Mel and I attended the AMS Topology Institute at the University of Wisconsin directed by R H Bing. I drove to Madison on my motorcycle, which by then was a spanking new one. We noticed at once that the participants were heavily represented by descendants of R. L. Moore: Dick Anderson, R H Bing, Mort Brown, Mort Curtis, Eldon Dyer, Ed Floyd, Kirk Fort, Burt Jones, Lou McAuley, Mary Ellen Rudin, John Slye, Gordon Whyburn, Ray Wilder, and Bob Williams. Non-Moore types included Richard Arens, Len Gillman, Ed Hewitt, Orville Harrold, Mel Henriksen, Ernie Michael, Sumner Myers (Jerison's thesis advisor), John Roberts, and Walter Rudin. The one whose heritage I am not sure about was Erik Hemmingsen. The Moores were using the word "compact" to mean what the rest of us called sequentially compact, and "bi-compact" for what we called compact, so conversations frequently required unraveling. I once proposed we all adopt "bi-compact" and "sequentially compact", saving plain "compact" for private conversations, but the Moore clan rejected the suggestion. Mel and I reported our results about F-spaces. Mary Ellen presented her startling result that Dowker's conjecture (paracompact Hausdorff spaces are normal) implies "Souslin's conjecture" (the positive solution of Souslin's problem), which I had reviewed for Mathematical Reviews and later expounded on in a seminar at Purdue. Ernie Michael was developing his research in selection theorems, and spoke on that topic. These were the presentations I concentrated on. There was also an enthusiastic seminar in "applied probability theory" --- i.e., a poker game. Leon Cohen, the first Mr. NSF Math, which funded the Institute, came to Madison to see how things were going. When he learned I had driven up on a motorcyle, he asked me, "Are those your original legs?" ("Yes") "Why don't you quit while you're ahead?" Bob McDowell and Carl Kohls became my doctoral students. Bob was the topologist, Carl the algebraist. Bob wrote about extensions of continuous functions from dense subspaces. (All spaces I discuss here are assumed to be completely regular Hausdorff spaces, and beta X denotes the Stone-Cech compactification of X.) Early in his thesis, Bob considers spaces X, Y, and E, where X is dense in E and such that completely separated sets in X have disjoint closures in E, and characterizes the set of points of E to which a continuous mapping from X to Y has a continuous extension. With the help of this result, he derives the standard properties of beta X obtained by Stone and Cech, but without the need to know that beta X exists. Later sections investigate extensions of mappings of a space into itself, and a chapter is devoted to metric spaces; one interesting result is that if$Phi$is a countable collection of (continuous) mappings from a separable metric space X into itself, then there is a metric compactification of X over which all the mappings in$Phi$can be extended. The final chapter considers problems involving weights of topological spaces. McDowell is a knowledgeable musician, and the possessor of a very pleasing tenor voice. He and Reba used to sing Schumann duets, with me at the piano, sometimes in public recitals at the Purdue Union. In due time, he married a Dutch girl who was living in town, then got himself a pre-doctoral Fulbright to spend a year in Holland. Before long, he had achieved good fluency in Dutch. While in Amsterdam Bob met Han de Groot, with whom he hit it off and from whom he got much helpful guidance. The later sections of BobUs thesis, which are purely topological, were worked out under de Groot's guidance during later visits to Amsterdam. Bob has been by far the most visible of my five doctoral students. After obtaining his degree, he spent two years in Berkeley as the associate director and then director of the Committee on the Undergraduate Program in Mathematics (CUPM) of the Mathematical Association of America (MAA), which in those days was being heavily funded by the National Science Foundation (NSF). Then he settled at Washington University in St. Louis, where he was department chairman for about 20 years. He recently resigned to organize a center for the improvement of teaching. Now with his seventieth birthday around the corner, he is preparing to retire. Carl Kohls wrote an outstanding thesis about prime ideals in C(X), following up with an article in Fundamenta Mathematica. In these works, he put forth just about every basic idea and result that appears in Chapter 14 of Gillman & Jerison (except for the very end of the chapter, which was devoted to Gillman and Henriksen's P-spaces and F-spaces): the chain of prime ideals containing a given one; upper ideals in C(X) and C(X)/P (where P is a prime ideal in C(X)) and their unions, and lower ideals in C and C/P and their intersections; the characterization of upper and lower ideals in C/P in terms of roots or powers of infinitely small elements; the minimal prime ideals in C containing a given z-ideal; intersecting maximal chains of prime ideals in C; z-ideals vs. upper or lower ideals, the successors of a lower ideal and the predecessors of an upper ideal in C or C/P; and the$eta_1$-like properties of C/P. It is Carl to whom we also owe the fundamental result (stated in Chapter 5) that prime ideals are convex. Purdue had no music department (probably still the case); as a result I was the musical bigshot, which meant among other things that I was often invited to play at President Fred Hovde's pre-Commencement dinner for the Board of Trustees. On one such occasion, Mrs. Hovde asked me, "How do you resolve your music and your mathematics?" I looked her straight in the eye and replied, "Sometimes I respond to a piece of music by thinking That's as exquisite as a beautiful theorem.' And sometimes when I contemplate a particularly artistic proof, I say to myself, That's as elegant as a Bach fugue." In a mellifluous voice and with a melting smile, she replied, "You love your work, don't you?" I also performed chamber music at the Purdue Union, notably with mathematician-violinist John Dyer-Bennet (brother of the eminent folk-singer Richard Dyer-Bennet) and sociologist-hornist Dwight Culver; our specialty was of course the Brahms Trio for piano, violin, and horn (the "Horn Trio"). Finally, I appeared with my student Carl Kohls, playing the Brahms E-flat sonata for clarinet and piano and the Schubert song Der Hirt auf dem Felsen ("The Shepherd on the Rock") for soprano (Reba) with clarinet obbligato (Carl) and piano accompaniment (Len). A month before the Commencement in which Carl was to be awarded his PhD, having not received an invitation to play at the dinner, I wrote to Mrs. Hovde to tell her of the Bob-Carl-Len-Reba program. The scenario I fantasized was that when Hovde handed Carl the diploma, he would interrupt the proceedings to remark "Last evening, I had the pleasure of hearing this talented young man perform on the clarinet in a trio by Schubert for soprano, clarinet, and piano, with his thesis advisor at the piano, and his thesis advisor's wife singing the soprano part." But it went the way of all fantasies when Mrs. Hovde wrote back that she had just made other arrangements, adding that she felt sick about it. There was also a Lafayette Symphony Orchestra, in which the concertmistress and associate concertmistress were math wives Erma (Mrs. Merrill) Shanks and Jacqueline (Mrs. Gordon) Walker, whose stature as violinists were modest. On the other hand, the first clarinet was excellent. He was a physician from a neighboring town, who legend had it would abandon a patient about to give birth in order to get to an orchestra rehearsal. (Carl played second clarinet in the orchestra.) The first flute was the wife of a young math instructor, and she too was excellent, as were the tympanist, who was a service administrator at Purdue, and Dwight Culver, the first horn. I appeared with the orchestra as one of the soloists in the Vivaldi-Bach concerto for four pianos. as soloist in the Chopin E minor concerto, and, at a pops concert, in the first movement of the Tschaikowsky B-flat minor concerto (the "Van Cliburn" concerto). Eventually, Purdue created a student orchestra, and I played with them too: the first movement of the Mozart A major concerto (K488), playing my own cadenza, and the second and third movements of the Tschaikowsky. We did the Tschaikowsky in a 6-day tour, taking us as far as Peoria, Illinois; I discovered that one really learns a piece when performing it in public several days in a row. During the year 1957-58, Walter Rudin persuaded Rochester to invite me for an interview for the chairmanship. I went, and things were quite pleasant. When I returned home I wrote back asking for a salary of$11,000 (10% higher than what we had talked about informally), and recommending offers to Henriksen and Jerison as well. All this scared the administration to the extent that they made us no offers at all. At the same time, Rudin happily accepted a job at the University of Wisconsin.

When Jonathan was four, back in Washington, Reba had organized a nursery school with some other mothers and had taught in it. Now she taught in the Purdue nursery school and, while she was at it, got herself a masters degree in child development. She also performed numerous times in the summer operettas put on by the Lafayette Opera Guild, directed by the conductor of the Lafeyette orchestra, among other things singing the leading role of Phyllis in Gilbert and Sullivan's Iolanthe in addition to lesser roles in The Pirates of Penzance and the non-G&S operettas that the group produced.

### 10. The Institute for Advanced Study

Purdue granted me a leave without pay for the following year (1958-59). With Tarski's support, I also won a Guggenheim fellowship (25th anniversary of Juilliard) and was accepted by the Institute for Advanced Study as a visiting member for a year. Jerison was there at the same time --- in fact, was my immediate neighbor --- and we put the finishing touches on our book. The last chapter is devoted to the Czech mathematician Katetov's thrilling generalization of Lebesgue's dimension theory for metric spaces to an elegant theory for compact spaces. This work, which was presented in a series of papers, brought him a State prize. Jerison and I organized and simplified the material immensely. Wistar Comfort quotes Katetov as saying that he never would have believed his work could be simplified, let alone so elegantly.

Jerison and I also wrote two joint papers that year. Others in residence were my old friends Nat Fine and Bob Williams and my more recent friend Warren Ambrose. New friends or acquaintances among the visitors included Michael Atiyah, Eugenio Calabi, Sol Feferman, André Haefliger, Moe Hirsch, Dick Palais, Steve Smale, Ed Spanier, and Lester Dubins. Among the permanent members I met were Armand Borel, Kurt Gödel, Marston Morse, André Weil, and, most notably, Hassler Whitney, an excellent violinist, with whom I played a lot of chamber music. I already knew Deane Montgomery. (I never did meet Selberg.) I also met the psychologist Solomon Asch, who was a neighbor.

The standard emotional trauma endured by a visiting member was to realize after two weeks that he had not yet written anything memorable, if anything at all, and to perform the obvious extrapolation to the remainder of the year. I went through that too. Unfortunately there were no excuses: no classes to teach, no deans --- and most important, no committees.

The situation at Purdue had become intolerable. Ralph Hull had been fired as department head as a result of maneuvering by Carl Kossack, a fifth-rate statistician and an exemplar of the man with a small mind who sets grand goals for himself and to whom we are indebted for the contribution to general illiteracy furnished by the term "matricee". He endplayed Dean Ayres as well, persuading President Hovde to make him department head. Ayres, in presenting this appointment to the department faculty, repeated the phrase "on my recommendation" so many times that it was evident to even the dullest person in the room, of which there were many, that the correct phrase was "despite my disapproval".

I was hoping to land a job so that I could leave Purdue. One weekend I motorcycled up to Wesleyan U in Middletown, Connecticut for an interview; but nothing came of it, which was disappointing, as Wesleyan, though not strong mathematically was otherwise very attractive as a leading liberal arts college. A year later, U Conn invited me up for interviews for the chairmanship. I got no offer there either. This was less disappointing, as I soon discovered that their bureaucracy was beyond belief. I had driven up and back in my car. I knew enough to record the mileage when I left Princeton and when I returned. Well, it turned out that my expenses could not be handled by a mere university. Instead, one day I received a form in the mail from the Governor's office. It asked for the mileage when I left Princeton, when I got to Storrs, and when I got back to Princeton. Oh well, I knew how to compute an arithmetic mean. It took another six weeks for the check to arrive.

Fortunately, I was accepted for a second year (1959-60) at the Institute; this time my support was an NSF Senior Post-Doctoral Fellowship. Jerison had returned to Purdue, but Nat Fine was back as were several of the others. New ones included Noam Chomsky, who took over Asch's apartment 3 doors down, Emil Grosswald, a well-known number theorist, who moved into Jerry's vacated apartment next door, and Joachim Lambek, an algebraist from McGill. Others were Aldo Andreotti and Eugenio Calabi.

One day I said to Nat, "Look, we can't tell people we were at the Institute together for two years without writing a joint paper." He agreed, and we got to work. He could get interested in anything, learning the ropes in short order. I had posed the following problem for myself. Let P be a point of. $beta N\backslash N$; is $(betaN\backslash N) {p} C*-embedded in$betaN \backslash N$? I had proved that the answer was no in case P was a P-point of$beta N\backslash N$(countable intersections of neighborhoods are neighborhoods), but was making no progress toward removing this restriction. (The existence of such points had been established by Rudin, using the continuum hypothesis.) But Fine and I together solved it. I had also posed the problem of generalizing Eberlein's example of a point in$\beta\R$that is not in the closure of any R-closed discrete subset of R, to an example of a "remote" point: a point not in the closure of any discrete subset of R. Again, Fine and I, with the help of the continuum hypothesis, subdued it. Later, the two of us joined forces with Lambek, who had been studying quotient rings of commutative rings, to write a long paper on quotient rings of rings of continuous functions. It was published as a softcover book by McGill University Press. In an introductory note outlining our results, we mentioned incidentally that some of them were new proofs of known theorems. Later we discovered that McGill's advertising blurb was proudly heralding the book for its new proofs of known theorems. The National Geographic Magazine featured an article about New Jersey in its issue of January 1960, apparently having run out of South Sea islands. It included a few paragraphs about the Institute for Advanced Study, and there on page 30-31, you will find beautiful color photographs of Robert Oppenheimer (the Director at the time), Oswald Veblen ( a famous geome ter and nephew of the even more famous economist Thorstein Veblen), Derek Price (a British historian of science), and George Kennan (former ambassador to the Soviet Union), plus a photo titled "Mathematicians relax with the music of Mozart", showing Calabi and Whitney playing the Bach double concerto (for two violins and orchestra), with me at the piano representing the orchestra. Julia Calabi and the Andreottis are there as listeners. Reba taught at the Institute nursery school both years, where she met the children of many of the famous mathematicians. During our second year, she sang the leading role of Josephine in the Princeton Savoyards' production of Gilbert and Sullivan's H. M. S. Pinafore (she had appeared in Iolanthe the year before). She also made a short visit to Lafayette where Mel, who was less ready than I to leave Purdue, reported the following conversation: MEL: Every university has its garbage. The advantage of the Purdue garbage is that I know it and understand it. REBA: Lenny feels quite differently. He's looking for fresh garbage. ### 11. The University of Rochester In the Spring of 1960 I got an offer of the chairmanship at Rochester, in response to prodding by Robert E. Marshak, their eminent physicist, on whom I had made a good impression during my visit two years earlier. They had a new dean of the College of Arts and Science: McCrea ("Mac") Hazlett ( Haize-lett), a delightful human being whose field of expertise was English literature. He was aware of my earlier visit and the level of salary discussed. This time, he said, they were thinking of$15,000. I replied, "Well, that might overcome my inhibitions." The official letter that followed stated a salary of $16,000. I thought that was a clever touch and used the same ploy often in my own recruiting. I set about hiring a lot of people, some junior, some senior The junior people I inherited consisted largely of fresh PhDs or unfinished PhDs. With one exception, I hired only people who had had post-doctoral experience: Wistar Comfort, a three-year Benjamin Peirce instructor at Harvard; Ken Ross, a post-doc fellow at Oregon; and Charles Watts, a two-year instructor at Chicago. Two or three years later we acquired John Dollard, a physics PhD from Princeton who came to us from a post-doctoral year there; and Newcomb Greenleaf, another Harvard BP. The one exception was Sanford Segal, a fresh PhD in number theory from the University of Colorado, who impressed us with his erudition. The senior people I inherited were John Randolph, the former chairman, who had little standing as a research mathematician and was a stodgy, unimaginative fellow; Bill Eberlein, who was a very good mathematician but a manic-depressive; and Norman Gunderson, in mathematical education. There was also Ralph Raimi, who had spent the preceding two years as acting chairman and had been granted tenure. He is a highly educated and cultured man with a quick wit and uncommon common sense; his stature as a research mathematician never lived up to his obvious intelligence; but he was easily the most interesting and most useful member of the department. The senior appointments my first year were Dick Johnson, a ring theorist at Smith College; J. H. B. Kemperman, a live wire analyst and applied mathematician whom I stole from Purdue; Arthur Stone, a world leader in general topology, and his wife Dorothy Maharam Stone, a highly respected measure theorist, from the University of Manchester in England; hiring both required persuading Mac Hazlett to waive the rule prohibiting married couples in the same department. Women were still routinely treated as second-class citizens in those days, and my hiring of Dorothy Stone as a full professor was singled out in Winning women into mathematics, a recent MAA publication, as an exception. A year later, we added Leopoldo Nachbin, a renowned functional analyst who was highly respected by the Andr\'e Weil crowd; and at about the same time, associate professors Norman Alling, an analyst temporarily at MIT, whom I had also been instrumental in getting to Purdue several years earlier; Norman Stein, an algebraic topologist; and Stanley Tenenbaum, an emotionally highly unstable citizen but highly respected logician, who with Thomas Jech had proved that Souslin's problem was unsolvable. The result of all this activity was to transform the mathematics department from a pipsqueak group to a vibrant force. I had become interested in MAA activities, and starting in my second year at UR (1961-62), I signed up as an MAA Lecturer, which involved traveling to colleges in New York and surrounding states to present lectures and chat with students about graduate school and careers in mathematics. I did this for eight years. On Wednesday evening, November 29, 1961, I boarded a sleeper train for a restful night's journey to New York to give several talks at St. John's University in Queens. One of them referred to the divergence of the harmonic series. As usual, the train was the square-wheel special, and I lay awake until 3:00 o'clock. While tossing around, I cooked up the following argument. Assume on the contrary that the series converges to a number r. Then  r = (1 + 1/2) + (1/3 + 1/ 4) + (1/5 + 1/ 6) + ... > (1/2 + 1/2) + (11/ 4 + 1/ 4) + (1/ 6 + 1/ 6) + ... = 1 + 1/2 + 1/3 + ... = r so that r > r, a contradiction. Every step looks fishy, but the proof is ice cold. I included it in my article Infinite processes and paint in the Association of Teachers of Mathematics of New York City Newsletter of May 1964. Some time in the seventies I sent it to Ross Honsberger, editor of the MAA's Dolciani Mathematical Expositions (DME) and told it to Ralph Boas; shortly afterward I sent them reprints of the article. Honsberger put it in Mathematical Gems II (DME 2) but forgot to mention me. Boas included it in his article Some remarkable sequences of integers in Mathematical Plums (DME 4), citing an undated oral communication from me, a 1975 article by someone else, and Honsberger's note in Gems II(!). Honsberger has subsequently inserted a clarifying note in a reprinting (in paperback) of Gems II. During my first year at Rochester, I wrote a letter to Joe Wilson, president of Xerox and of the UR Board of Trustees, outlining my vision of the university. It included a recommendation to do away with the evening school, a pitiful enterprise where every teacher, regardless of academic standing, was paid the same shameful wage. A few days later, Mac Hazlett said to me, "Leonard, do you realize that the money we get from the evening school helps pay professors' salaries in the College of Arts and Science?" I replied promptly: "No, but if you want to run an outside business in order to raise money, why not open a whorehouse? You could take in a lot more, and faster." When I relayed this dialogue to Arnold Ravin, the new dean (Mac having been moved up to provost), he commented "It's the same thing. The evening school is an intellectual whorehouse." Compared with Purdue, where Reba and I had had a close circle of friends, our reception at Rochester was almost cold, conceivably because a department chairman is never just one of the boys. At Purdue, whenever Reba was out of town for a few days, our closest friends: the Golombs, Henriksens, Jerisons, and Walkers, would divvy me up as a dinner guest until she returned. At Rochester, when Reba and Miki had gone off to Europe for a few weeks and none of my friends were paying any attention to me, I bumped into a nonmathematician friend whom I will call Kurt, primarily because that was his name. KURT: Your wife is away, isn't she? LEN (hopefully): Y..e..s..? KURT: When she gets back, we'll have you both over to dinner. The presidency at Rochester had been effectively vacant for several years, until 1962 when W. Allen Wallis was appointed to the position. He was a statistician-economist, a political arch-conservative who was dean of the business school at the University of Chicago when UR signed him up. I had met him and his sidekick Milton Friedman (later a Nobelist in economics) during the war, at Columbia, where he was the head of the Statistics project (corresponding to Mac Lane in the Applied Mathematics Project). It was during that period that Abraham Wald's book Sequential Analysis appeared. In it he credits Wallis and Friedman with suggesting the concept. He told me that he then worked out and wrote up the entire theory in two weeks. Wallis, to his credit, had a classical New England sense of fairness, as when he and I were chatting informally about the directorship of the Eastman School (which is part of the University). The man he had decided to appoint was known to have had a drinking problem, but Allen asserted that he was perfectly willing to accept a former alcoholic, just as he would be to accept a former communist, and remarked cheerily, "Well, if it doesn't work out, then it's back to square one." Unfortunately, he was insensitive to people as people, and seemed not to notice, as Reba and I and everyone else had, that the guy was subsisting on tranquilizers and always looked logy. And it was back to square one. Wallis seemed to regard this as oh well, another day another dollar, but in the meantime, Eastman students and faculty had to live another two years in a mess --- a situation he had no appreciation of, as it involved human feelings. Allen chaired the college faculty meetings, and in one of them provided an outstanding example of his good side. A motion was on the floor to form a department of statistics, which he was strongly championing. There was considerable discussion, but Allen scrupulously took no part in it whatsoever --- a virtuoso performance. Eventually, the motion was tabled, and only then did he speak up, telling us (somewhat patronizingly to be sure) that he had heard all the arguments before, and that if he had been participating he would have pointed out thus and so. In miserable contrast, some years later, I had to cringe in embarrassment when a speaker from the floor at an MAA meeting criticized something MAA was doing and was interrupted phrase by phrase by sharp rejoinders from the president, chairing the meeting. Purdue had had 10,000 students, an immense number in those days, and I relished the idea of signing on with Rochester, with its 3500 students on the entire River Campus (Arts & Science, Engineering, and Business). During my first week on the job I got a note from Mac that they had overshot the freshman class: there would be 420 students instead of the planned 410, and asking me to let him know if this would cause me any undue hardship. I threw back my head and laughed in utter joy. A few days later he asked how things were going and I mentioned that I was being inundated by mail. He replied, "Oh, you'll learn to recognize junk and throw it out on the spot." The next day, a huge, bursting envelope arrived from the dean's office, prompting the instant thought, "Oh, I see what he means by junk." Early on, I rented a postage meter for the department, for which I created the message: "Give to the mathematics department of your choice.'' (The friendly message in Texas, where I am now, is "State penalty for private use.'') During the first semester, I taught a class in set theory, two of whose students were Doris Wood, a senior (in fact, the valedictorian), and Martha Jochnowitz, a first-year graduate student. After Doris graduated I lost touch with her for several years, but I became a good friend of Martha, who was our crackerjack TA. She ask me one day if would better help her complete her PhD: to marry Chuck Siegel now or concentrate on her degree first and marry him later. I told her to marry him now. She did, and they enjoyed a very happy married life for many years, until Chuck died suddenly a few years ago. Another graduate student I got to know pretty well was Linda Hill (also a student in one of my classes); once Reba and I even had her over for Thanksgiving dinner. A dozen years later, when I was MAA Treasurer, I got a letter from Doris W. Schattschneider, then editor of Mathematics Magazine; off in the margin was a handwritten note: "Do you still mark comments about mathematics in red and comments about English in blue?" That stopped me until I remembered that women have a way of changing their last names --- and I was right. Martha Siegel also served a term as editor of Math Mag. During that time, Linda Hill became chairman of the Committee on Sections, which made her a member of the MAA Executive Committee. Martha was there representing the journal editors. And Doris, who was then 1st Vice President, was there as well. This trio, who were the only women on the Committee, rapidly became known throughout the world as "Gillman's Female Legacy on the MAA Executive Committee". I asked Martha for an explanation of the tri-incidence and got a heart-warming response: during those years in Rochester, when I was becoming involved in CUPM, as an MAA Lecturer, and in other MAA activities, the message came across to the students that involvement in professional activities was a worthy way of contributing to the profession. Doris and Linda are no longer on the committee, but Martha, who is now the Secretary of the Association, will be there for several more years. Doris by the way was the MAA Hedrick Lecturer at the Burlington meeting a couple of summers ago. I had three doctoral students at Rochester: Norman Bloch, who showed among other things that for any ordinal$alpha$, if$2^aleph_alpha > aleph_alpha+1$, then there exist two real-closed$eta_alpha+1$-fields of cardinal$2^aleph_alpha$that are not isomorphic. Earlier, I had established the analogue for ordered sets (sans algebra). Norman is now an associate professor at SUNY Brockport near Rochester. Mark Mandelkern, who obtained results about well-ordered (increasing or decreasing) chains of prime z-ideals in C(R), generally with the use of the continuum hypothesis: e.g., there exist increasing chains of every countable ordinal type, and there exist uncountable decreasing chains. A special result is that there exists a maximal ideal in C(R) containing no other prime ideal. Mandelkern later swung to the opposite extreme and became a constructivist á la Errett Bishop. He is now professor emeritus at New Mexico State in Las Cruces. Don Plank, who wrote about "$beta-subalgebras" of C(X): with each such subalgebra $A$ he associates a set of "A-points" of $beta X\backslash X$ and notes the following special results: the P-points of $beta N\backslash N$ are precisely the $C^*(N)$-points, and the remote points in $beta R$ are precisely the C(R)-points. Don is a professor and sometime dean at Stockton College in Pomona, NJ, near Atlantic City.

I remember once doing some wheeling and dealing in the university that was pretty slippery (but have forgotten what it was) and describing it to Reba when I got home, remarking that I felt like a slimy politician. After pondering this for a moment, she said, "A politician, yes. But not slimy." Isn't that just about the nicest thing a loving wife can say to her husband when he gets home tired from a hard day's work?

During my stay at Rochester, I was a member of CUPM and several of its panels; we were paid travel expenses plus an honorarium to attend their meetings. Those were the days when CUPM was churning out a series of booklets of curricular recommendations. Several eminent mathematicians contributed to the effort, including Ralph Boas, Sammy Eilenberg, Alex Rosenberg, and Ed Spanier, and, if I am not mistaken, Izzy Singer. I also spent two summer sessions with the School Mathematics Study Group (SMSG) in Stanford, and later became a member of its Advisory Panel, though I was highly critical all along of the garbage associated with the New Math.

In April of 1968, two months short of his 80th birthday, my father underwent surgery for an enlarged prostate, followed by a 10-hour coma from which he did not recover. I was shaken as never before or since. As a result, I have been wary of prostate trouble, and I now keep my prostate in check with the herbs saw palmetto ( Serenoa repens) and pygeum ( Pygeum africanum).

I performed a number of times with the UR orchestra: in the Vivaldi-Bach four-piano concerto and a Bach three-piano concerto; I also played the first movement of the Grieg piano concerto with the brass ensemble as the orchestra (quite effective, I should say). The pinnacle occurred in March, 1969, when I played Brahms's great and totally nontrivial B-flat concerto with the orchestra.

In 1967 I got an offer of the chairmanship from the University of Texas but turned it down, lacking an incentive to move from a life of peace and quiet to a hornet's nest. In 1969 I got another offer from Texas. This time Rochester was the hornet's nest, and I accepted the offer. The chairman who recruited me was Woody Bledsoe; a finer, more level-headed citizen you will never meet. The dean who hired me was John R. Silber, who had shown himself to be a man of towering intellect, a terror or a charming gentleman as the occasion demanded, and a friend of the mathematics department, for whom he was reported to have personally obtained $300,000 directly from the State Legislature. There was a rule at the university that after retiring at age 70, a professor could continue to teach half time year to year upon the recommendation of the departmental "Budget Council" that he was still "fit to teach". Only a faculty committee could have formulated such a montrosity, requiring the chairman to eventually have to say to the man, "Sir, your colleagues have just voted that you are no longer fit to teach." R. L. Moore had been reappointed 17 times, so he was still teaching at the age of 87. But his age was not the significant problem; rather it was his stubbornness about his teaching methods. He was famous for a Socratic-like procedure now known as the "Moore method", in which one gives the students some axioms and has them derive all further results by themselves. When one student had a proof ready to present in Moore's class, those who had not yet worked it out would leave the room so that they could continue their own attempts uncontaminated. Consistent with this was Moore's firm rule that no student was permitted to ever consult a book. This method was very effective for learning point-set topology, Moore's specialty, but most mathematicians agree it would be hopelessly inadequate for learning subjects based on more intricate ideas. Moore took the extreme tack that there was no need for the mathematics library to acquire books in any subject at all, not just his own, and such was his power that he was able to enforce this precept. This of course played havoc with the hopes of eager young mathematicians to get a program going in, say, functional analysis. (Moreover, I was told that most of Moore's graduate students had private offices while most faculty were doubled up.) Silber understood the situation and realized the danger inherent in it; somehow, he saw to it that the Budget Council did not renew Moore's appointment for an 18th year. (Talk about politicians!) He and I hit it off and have remained on friendly terms, although he soon left to become President of Boston University (where he had a volatile career). I brought John Dollard and Newcomb Greenleaf with me from Rochester, and hired several good people, including Jim Daniel, currently chairman of the MAA Publications Committee; Jim Vick, winner of every teaching award on the books; and, the big prize, R H Bing (one of Moore's three most eminent students, the other two being Gordon Whyburn and Ray Wilder). I also rehired Efraim Armendariz, the current department chairman, who had been here but left. Moore had an army of loyal mathematical descendants, most of whom considered me an ogre for forcing him to quit teaching. I had done nothing of the sort (nor had I the power) but did insist that I would not accept the chairman's job until the battle over MooreUs eventual retirement was settled. Actually, he and I were on decent personal terms. In fact, I suggested and successfully pushed for naming the Physics-Mathematics-Astronomy building the Robert Lee Moore building. Retired professors are entitled to offices, and Moore had had a huge one, but when the department moved into a new building I noted that the new offices were much too small to accommodate his things. So I assigned him two offices. Shortly afterward, a group of his former students complained to the president of UT that Moore had not been given any office space in the new building. ### 13. The Mathematical Association of America In 1969, at Rochester, I had been appointed to a two-year term as AMS Associate Secretary for the Northeastern region. I introduced three innovations that virtually rocked the Society to its foundations: Scheduled the April Cambridge meeting for March and had the two invited speakers (Hyman Bass and John Tate) speak on the same topic ($K$-theory); and had the two invited speakers at the Washington meeting (Jim Ax and Alan Baker) introduce each other. Having moved to Texas in Fall 1969, I had to give up the appointment after the two years had expired. I had hardly arrived in Austin when Gordon Walker, the AMS Executive Director, phoned to ask me to meet him in San Antonio to look over the relatively new convention center as a possibility for the January meeting a mere two months away. The meeting was already set for Miami but had to be cancelled when the teamsterUs union, who owned the facilities, decided they needed them for someone else. Fortunately, the San Antonio site, which Gordon tagged "the best convention facilities this side of the Vatican", was still available, and we grabbed it. At the 1972 summer meetings at Dartmouth, I suggested to Henry Alder and the other MAA bigwigs that the 1976 meeting in San Antonio be centered on the US Bicentennial, and I outlined a possible program. My punishment was canonical and swift: immediate appointment as chairman of the program committee. At the same meeting, Nathan Jacobson told me about a doctoral student of his he thought I should know about: Louis Rowen, who was such an outstanding cellist that they wondered whether he really could hack it as a mathematician. I went back to Henry to suggest that Louis and I present a musical program at the San Antonio meeting, pointing out that as an Austinite, I could be considered local talent. To my pleasant surprise, he and the others were delighted. AMS co-sponsored the concert. We performed in the theater for an audience of close to 1000, and it was a smashing success. Our program was decidedly nontrivial: BACH: Sonata No. 2 in D major BRAHMS: Sonata No. 1 in E minor BEETHOVEN: Sonata No. 3 in A major CHOPIN: Sonata in G minor (Encore) POPPER: Elfintanz In January of 1973 I was elected MAA Treasurer. At about the same time, I resigned the chairmanship at Texas, where I was succeeded by Bledsoe and then Bing, Daniel, Dollard, and Armendariz (all of them appointments of mine). Greenleaf left UT after a few years for a Buddhist school in Colorado, later accepted a 5-year associate professorship in Computer Science at Columbia, and is now teaching at the Friends School in New York; Dollard is now associate dean of graduate studies at UT; Vick is now UT Vice President for Student Affairs. Tragically, Woody Bledsoe, who had received the prestigious Milestone Prize from AMS in 1991 for his work in artificial intelligence, died in 1966 of Lou Gehrig's disease. In 1978, the MAA, which had been renting office space in Washington, bought its own building --- actually, it was a complex of two connected former residences, plus a carriage house. The price was$750,000, and we inaugurated a fund drive to raise $300,000 before signing a final contract. We started with gifts from the Vaughn Foundation and the Dolciani Foundation,totalling$150,000. I have been credited with securing the one from Vaughn. I also organized a drive to raise $15,000 to name the beautiful conference room in memory of Ed Begle, former director of SMSG, who had died earlier that year. In fact, that drive netted us over$30,000. I also contributed a few thousand dollars of my own. The Secretary of the Association pointed out in one of his reports as director of the fund drive that I was responsible for bringing in over \$100,000, one-third of our goal.

In the same year, when the Austin Gilbert and Sullivan Society started rehearsals for Iolanthe, I went down to volunteer as pianist. During a lull, the director suddenly wheeled around and said "You look like the Lord Chancellor" (the leading role). I protested that I had never appeared on the stage in my life and that I couldn't sing my way out of a paper bag. This bothered him not at all. He won, catapulting me to stardom. I turned in six flawless performances (except for a speech in one of them when I said "so" instead of "therefore").

Louis and I followed up at San Antonio in January 1980 with another nontrivial program:

BACH: Sonata No. 3 in G minor
FRANCK: Sonata in A major
BEETHOVEN: Sonata No. 4 in C major
RACHMANINOFF: Sonata in G minor
( Encore) CHOPIN: Introduction and Polonaise

In 1981, we performed at the Jacobson Conference at Yale, playing selected works from our two programs.

I was musical director for the Austin G&S Society's productions of Trial by Jury in 1982 and The Gondoliers in 1983.

I was MAA Treasurer for 13 years, then President Elect --- President --- Past-President for the canonical four, followed by the canonical additional five years on the Board of Governors. I was thus an officer for 17 consecutive years and a member of the Board for 22 --- both of which are records. (Baley Price and Henry Alder amassed larger totals but not consecutively.)

My forte was behind-the-scenes maneuvering. In 1984, while still Treasurer, I tagged Don Albers to be the next Chairman of the Publications Committee; he converted the publications program into a bustling, profitable activity.

Next, I persuaded Ken Ross to turn down any deanships that might be suggested to him, as Dave Roselle was planning to resign as MAA Secretary, having been appointed Provost at Virginia Tech, and we were going to need Ken to take his place; I then arranged for me to be appointed chairman of the nominating committee. (Slimy.) Later, Ken became President of MAA.

At about the same time I met Ann Watkins at a cocktail party at MAA headquarters. She made such a favorable impression that I maneuvered to get her appointed Chairman of the Committee on Two-Year Colleges in order to give her greater visibility in preparation for her being nominated for 2nd Vice-President (the traditional slot for a two-year-college teacher) and then winning the national election (slimy3) --- all of which came to pass.

In January 1988, at the AMS Centennial banquet in Atlanta, I led 1900 mathematicians in singing "Happy Birthday, Dear American Mathematical Society." I have not yet submitted the item to Guinness.

My Presidential term was 1987-88, which according to the rule then in force meant from the end of the Business Meeting in January 1987 to the end of the one in January 1989. William Browder was to become AMS President earlier in January. He is an accomplished flutist who over the years had played sonatas with me informally in our homes. We agreed to put on the AMS-MAA Presidents' Concert at the January 1989 meeting in Phoenix. But the concert was scheduled for the day after the MAA Business Meeting, and the instant I adjourned that meeting I would no longer be President. Well, I figured out a solution, and when I presented the problem to Lida Barrett, who was to succeed me as President, she immediately came up with the same solution. So that's what we did. Instead of adjourning the meeting at the scheduled time, I suspended it for a day, reconvened at it the end of the concert, asked for further discussion, got none, and then in the hallowed participle-dangling tradition, announced: "There being no further business, the meeting is adjourned."

Our program was the Bach G minor sonata, the Poulenc sonata, and Schubert's magnificent Introduction and Variations. For encores we played the Donizetti sonata, one of the Brahms Intermezzos (solo piano) and Gluck's Dance of the Blessed Spirits, a flute solo from the opera Orfeo ed Euridice (with my piano representing the orchestral accompaniment).

The night before, at a retiring dinner for me, I accompanied Bob McDowell and Reba in the Schumann duets, Bill and I played the Donizetti sonata, and I played the Rimsky-Korsakoff Flight of the Bumblebee (arranged by Rachmaninoff) and the finale of Liszt's Hungarian Rhapsody No. 6.

At the January 1990 meeting in Louisville, I gave my Retiring Presidential Address on Teaching methods that work, which has been very well received. At the January 1992 meeting in Baltimore, Bill Browder and I presented the AMS-MAA Past Presidents' Concert, playing the Bach E-flat Sonata, the Schubert Introduction and Variations (again), and the Franck Sonata in A major. For an encore we played the first movement of the Mozart violin sonata in E minor (K304 I don't remember what tha encore was the Mozart violin sonata in E minor (K304).The Franck is written for violin and piano. I performed it with a cello(Rowen) and with a flute (Browder) but never with a violin. In the mid-1970's, I started to concentrate on articles for the MAA journals. One of them, "An axiomatic approach to the integral" in the January 1993 Monthly won a Lester R. Ford award. I also wrote "Writing Mathematics Well", an MAA booklet, published in 1987. I have mentioned that my parents lived for their two boys; we were their be-all and end-all. In 1980, when my mother became 88, she still had all her marbles, but as the year progressed, she displayed bouts of forget fulness. In a conversation with me that summer, she asked, "Leonard are you satisfied with your choice of career?" I assured her I was. "Are you making progress toward your PhD?" Somehow managing to keep my voice from breaking, I again told her Yes. She died two weeks later.