In 1926, when Leonard was nine, the family moved to New York where he spent two years in a state school. He also began studying piano under the outstanding pianist Louis Kantorovsky who was awarded a Frederick Steinway scholarship in 1927. Louis :-
... and his wife were much younger than my parents but the two couples became best friends. We were living in Queens, and the Kantorovskys lived in the Bronx; every Sunday afternoon my parents would take their two boys on the long subway ride to Manhattan, then transfer to the train to the Bronx, where Bobby and I had our piano lessons, after which the Kantorovskys served dinner.Leonard's parents both took degrees from Columbia University, his mother Etta being awarded a Master's Degree in psychology while his father Joseph was awarded a Ph.D. in economics. After two years at the state school, Leonard entered The Walden School, a small progressive school on West 68th Street in Manhattan. He began his studies at High School in 1929. He was in the same class as Reba Marcus, who he married nine years later, and Ellis Kolchin who, like Gillman, became a mathematics professor. Gillman had always loved mathematics in addition to his passion for music. He had enjoyed spotting patterns with numbers from a very young age but, even at high school, there was little in the way of mathematics in the curriculum other than elementary algebra and geometry. Gillman and Kolchin, who were close friends, approached their teacher in their final year at High School (1932-33) and told him they wanted to study more mathematics than the school was offering. He gave them a copy of Raymond Brink's Plane Trigonometry (1928). This book "aims to gain the interest of the students through its practical values, believing this interest will carry over into a more analytical and critical study of the background of the subject." Their teacher :-
... suggested we work through all the problems on identities. That was first-class advice, and working the identities was fun. Of course we also had to learn how to interpolate in trigonometric and logarithmic tables, then slog through the requisite number of problems. (Not so much fun.)Still wanting to learn more mathematics, the two young students took a day off school to visit the New York Public Library where they read articles on magic squares. Back in school they experimented with constructing magic squares.
After graduating from High School in 1933, Gillman sat the scholarship examination for the Graduate School of the Juilliard School of Music in New York. He won a scholarship as a piano student and began his studies there. The training at this music school was all geared to producing concert performers. In the spring of 1934, in addition to his studies at the Juilliard School, Gillman began to take courses at Columbia University. He took courses in French and Analytic Geometry to start with but, in his second year took Differential Calculus and another French course. Next he took courses on Integral Calculus, Matrix Theory, and Differential Equations. It was not only mathematics that Gillman studied at Columbia, for he also took more French courses, as well as courses on Psychology, German, English Composition and History. In May 1938 he was awarded his Diploma in piano from the Juilliard Graduate School, and also in May was the soloist in the Liszt A major concerto at a performance of the Juilliard Orchestra. Later that summer he took the same role in a production conducted by Edgar Schenkman, the younger brother of the group theorist Eugene Schenkman.
In December 1938 Gillman married Reba Marcus who had graduated from Bennington College where she had majored in music. At first they made their home with Gillman's parents but soon were able to move into an apartment of their own near Columbia University. He gave some piano lessons and enrolled at Columbia for some advanced music courses. He thought about graduating from Columbia majoring in music but, following a suggestion made by his mother in September 1940, decided instead to take the necessary courses to major in mathematics. Outstanding results led, in January 1941, to his professors suggesting that he might like to continue to a Ph.D. after taking his first degree. A couple of months later he was offered an assistantship and began teaching a course on trigonometry. In 1942 he graduated with a B.S. in mathematics. He was advised to take graduate statistics courses since Harold Hotelling controlled the fellowship money. He took courses by Hotelling, Abraham Wald and Henry Mann, and he was awarded a Carnegie Corporation fellowship. Leonard and Reba Gillman's first child, Jonathan, was born in the autumn of 1942. In the spring of 1943 Hotelling and Wald recommended him for position with the Tufts College Navy Project. After being interviewed, he was offered the position and was happy to accept.
At Tufts he worked on military applications, in particular :-
... an analysis of ordinary and generalized pursuit curves and of countermeasures that a target ship can take to defend itself from an incoming homing torpedo.This work was eventually written up as a report entitled Mathematical Analysis of Ordinary and Deviated Pursuit Paths. He had already completed all the necessary course work at Columbia for a Master's Degree and he wrote up the theoretical part of his work on pursuit curves and submitted that to Columbia as his thesis. He was awarded a Master's Degree from Columbia University in 1945. In the summer of that year his daughter Miki was born and, with the ending of World War II shortly after that, the project at Tufts College came to an end. He was offered a job by the Operations Research Group in Washington and began working there in December 1945. This work was in addition to working on his Ph.D. dissertation, having a family life with his wife and two children, and trying to keep up his piano playing. In  he gave an example of the type of problem that he worked on for the Operations Research Group:-
... find the best strategy for a target submarine to follow when faced with traversing a hostile channel. ... Enemy searchers are lined along the two shores; from each point, the probability of detecting the submarine if on the surface is assumed known (e.g., it is low where the channel is wide, high when narrow); a submerged submarine is assumed undetectable, but the submarine's batteries can maintain it under water for only a limited total distance (also assumed known). What is the distribution of points at which the sub should choose to be under water?He began applying game theory to solve problems of this type. Eventually he realised that there was no way that he was going to be able to write a Ph.D. thesis given all the other pressures on him. He was offered a sabbatical year in which he could undertake research but the condition was that he spend it at the Massachusetts Institute of Technology. Although at first he was reluctant to spend the year at MIT, he eventually decided to take this opportunity. He also had to decide on the topic of his thesis and his friends advised him to write a thesis on game theory since he had already spent a lot of time working on this topic. However, he wanted to write a thesis on set theory and he gave himself a self-imposed limit of six months to get into the set theory research and, if things were not going well after this time, then to change to game theory.
He began to get some results on set theory and wrote to Alfred Tarski, who was at Berkeley, asking him if these results were new. Tarski replied quickly giving him ideas how he could extend the results he had already proved. Tarski :-
... suggested additional reading, which led me to some additional results. Still, he lived up to his reputation by detailing just who had sent which result to whom at which time. My quip was that I was in residence at MIT, on leave from a job in Washington, writing a dissertation for Columbia, with a thesis advisor in Berkeley.He wrote to Columbia telling them that he was writing a thesis on set theory and received a reply that nobody at Columbia was an expert on set theory but Ray Lorch had agreed to satisfy the university regulations by going down on the form as his official thesis advisor. Returning to Washington at the end of his sabbatical year, he did not feel like returning to the work he was doing with the Operations Research Group and handed in his resignation a couple of weeks later. In November 1950 he submitted his paper On intervals of ordered sets for publication. He revised it in the following October and it appeared in print in 1952. The paper contains the following acknowledgement:-
Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy, in the Faculty of Pure Science, Columbia University. This paper was written under the supervision of Professor E R Lorch; thanks are also due Dr R L Taylor for a number of suggestions. Special thanks are due Professor Alfred Tarski (University of California, Berkeley) for his encouraging interest and generous advice.In 1953 he published, in collaboration with John M Danskin, a paper on game theory entitled A game over function space. Harold W Kuhn writes:-
This paper is concerned with the solution of an example of a zero-sum two-person game with essentially bounded measurable functions on (0, ∞) as pure strategies. The existence of a saddle-point is established by the fixed-point statement of Glicksberg and the unique optimal strategies are characterized.In the autumn of 1952 he took up a position as an instructor at Purdue University and submitted his Ph.D. thesis to Columbia around the same time. Early in 1953 he defended his thesis and was awarded a Ph.D. by Columbia. Over the next couple of years he published: Concerning rings of continuous functions (1954) with Melvin Henriksen; On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions (1954) with Melvin Henriksen and Meyer Jerison; and An isomorphism theorem for real-closed fields (1955) with Melvin Henriksen and Paul Erdős. Gillman recalled :-
In the late 1980s, I encountered Erdős at an Mathematical Association of America Section meeting, at dinner, eating his soup. An enthusiastic host asked, "Erdős, do you remember Gillman?" Continuing to eat his soup, without looking up, Erdős replied in a somewhat bored voice, "Yes, we wrote a joint paper in 1955."He remained at Purdue until 1960 being promoted to Assistant Professor in 1953 and to Associate Professor in 1956. While he was there he continued his musical career performing chamber music, playing with the Lafayette Symphony Orchestra, and playing with the student orchestra. On one occasion he was asked by the President's wife :-
"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?"He spent 1958-59 at the Institute for Advanced Study at Princeton, supported by a Guggenheim fellowship. While there he completed writing his book Rings of continuous functions coauthored with Meyer Jerison. The book was published in 1960 and reviewed by Jean Dieudonné who writes:-
Modern "abstract" mathematical techniques (algebra, topology, order structures) were created as new ways to attack the problems of classical mathematics, and have amply demonstrated their fertility with spectacular success, especially in recent years. But these theories have also become for some mathematicians "ends in themselves" and have been developed without any concern for application to other issues, and this has naturally led to a host of new problems, often very difficult, and some of which have not been resolved at the cost of long and ingenious efforts. The work of Gillman and Jerison relates to this trend: the single point of contact they may have with other parts of analysis is the theory of normed algebras and Gelfand transformations; but these issues are absolutely not mentioned by the authors ... This book will certainly be a reference work for a long time, containing virtually all currently known results on these issues, including numerous examples and counterexamples. The style is excellent in its clarity and conciseness. ... It is obviously true that the authors do not write for mathematicians in search of tools and eager to reach the goal as quickly as possible, but for real "aficionados" of the theory, willing to devote all their time and to whom three different proofs of the same theorem will triple the delight they find there.Gillman spent a second year (1959-60) at the Institute for Advanced Study at Princeton supported by an NSF Senior Post-Doctoral Fellowship. Then he went to the University of Rochester where he was appointed as Professor and department chair. In 1969 he moved to the University of Texas at Austin where he was Professor and departmental chairman 1969-73. He remained at Austin until he retired in 1987 when he was made Professor Emeritus. During his years in Austin, he worked in several different capacities for the Mathematical Association of America. He was treasurer of the Association from 1973 to 1986 and on the board of governors from 1973 to 1995. He was President of the Association in 1987-1988. The Mathematical Association of America published Gillman's booklet You'll Need Math (1967), aimed at high school students, and his booklet Writing mathematics well: a manual for authors (1987). This gave a good outline of the fundamentals, covering the basics of organisation, presentation and preparation of manuscripts.
During his career, Gillman received several honours for his contributions. We particularly mention the two Lester R Ford Awards he received from the Mathematical Association of America. The first in 1994 was for his article An Axiomatic Approach to the Integral (1993) while the second was for his paper Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis (2003). He also received the Yueh-Gin Gung and Dr Charles Y Hu Award for Distinguished Service to Mathematics from the Mathematical Association of America in 1999. He continued his musical career throughout his life, performing with the Gilbert & Sullivan Society of Austin and well as being the pianist at four national AMS-MAA meetings, two with Louis Rowen, cello - 1976 (Bach, Brahms, Beethoven, Chopin) and 1980 (Bach, Franck, Beethoven, Rachmaninoff, Chopin) - and two with William Browder, flute - 1989 (Bach, Poulenc, Schubert, Gluck) and 1992 (Bach, Schubert, Franck, Mozart).
Article by: J J O'Connor and E F Robertson
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