**Albert Tucker** was known as Al to his friends, colleagues and students. He was born in Oshawa, a small town on the northern shore of Lake Ontario and the family lived in various small towns on the northern shore as he grew up, never spending more than three years in any one place. His father, William B Tucker, had some mathematical training but it was his role as a Methodist minister which led to the many moves. Albert's mother was Florence M Huff. He was his parents' only child and did not begin his school education until he was six years old. He explained in [8] how he became interested in mathematics while in his second year at a small high school with only three teachers covering all subjects. In addition to these teachers, the Principal of the school taught science and mathematics:-

A few weeks into my Euclidean geometry course, the Principal decided to give us a test. For his own convenience, he used part of a provincial examination. This contained both questions of knowledge and "originals". He had not previously given us any originals and didn't expect us to answer them. Well, I didn't know this, so I answered the originals. That night the Principal came to see my father and wanted to know if my father had been coaching me, because my father had taught mathematics for a year of two. My father said "no", he had not. Then the Principal said: "I think your son must be a mathematical genius. I think he can have a very promising career as an actuary!"

Tucker wanted to study at university but as his family were poor this required him to win a scholarship. He repeated his final year at high school so that he could obtain a scholarship to enter the University of Toronto, coming first in the provincial scholarship examinations in both mathematics and Latin. He began his university studies in 1924, enrolling for the four-year Honors Course in Mathematics and Physics. Although the Principal of his High School had suggested that he aim at becoming an actuary, in fact Tucker thought at this stage that he would become a schoolteacher of mathematics. In his first year at Toronto he took a conic sections course taught by John Synge, who quickly recognised his great potential. As well as mathematics and physics courses, he also took courses in chemistry and astronomy.

As part of his physics course he was asked to write a report on quantum mechanics. The subject was in its very early stage at the time and only research papers existed. The chairman of physics was very impressed with Tucker's report and asked him to change from a joint mathematics-physics degree to a single subject physics degree. The chairman of mathematics, Alfred Tennyson DeLury, agreed that he should change to a single subject degree but, of course, thought Tucker should choose mathematics. It was a difficult decision for Tucker who really enjoyed physics but felt that he had a deeper understanding of mathematics. After considerable deliberation, he changed to a mathematics degree realising at the time that he was now taking a course which was not suitable for an intending schoolteacher.

In his final year of undergraduate study Tucker was advised by DeLury to go abroad for graduate study. DeLury suggested Paris, Göttingen or Bologna as the best places, with Cambridge as the best option if he felt he had to have teaching in English. Tucker immediately rejected the non-English speaking universities, and wrote to the University of Cambridge enquiring about their graduate programme. In fact the courses did not look too attractive to Tucker so, after the award of his B.A. in 1928, he remained at Toronto for a fifth year. He was employed as a Teaching Fellow during 1928-29 and also worked for his Master's Degree which he was awarded in 1929. DeLury still pressed him to go to Europe to do graduate work or, failing that, to go to Chicago or Harvard. Tucker, however, wanted to go to Princeton and eventually DeLury agreed. This was not the end of his problems since his application to Henry Fine arrived shortly after his death in December 1928 and was only later passed to Luther Eisenhart by which time applications for graduate study had closed. However there was a part-time Instructorship which Princeton was about to fill so, with Alfred DeLury's support, Tucker was appointed to this position for 1929-30. He was at last able to begin working towards his doctorate with Solomon Lefschetz as his thesis advisor. He spoke of the faculty members at this time in an interview [10]:-

When I was a graduate student - I started in1929and got my degree in1932- the chairman of the department and Dean of the Faculty was Eisenhart. He had become Dean of the Faculty in1924. I'll take the others alphabetically. This is the list for the academic year1930-31.(

James W Alexander, professor;

Alonzo Church, assistant professor;

William Gillespie, professor.Gillespie had been brought in from the University of Chicago in the1890s, but he contented himself with teaching underclass courses.)1920

Einar Hille, associate professor. He had come here in the earlys from Stockholm.1920

Morris Knebelman, assistant professor;

Solomon Lefschetz, professor;

Howard P Robertson, associate professor. We should have mentioned him as one of the people developed here during thes. He took his Ph.D. at Cal Tech and came here on a post-doctoral fellowship. He then was appointed to the faculty.20

Tracy Y Thomas, assistant professor. He got his Ph.D. in the 's with Oswald Veblen.

Oswald Veblen, professor;

John von Neumann, professor, but on leave during the first term;

Joseph H M Wedderburn, professor;

Eugene Wigner, professor.

Now von Neumann and Wigner shared, at that time, a professorship at Princeton and an appointment in Berlin. One of them would be here for one of the terms, and the other for the other term.

Tucker received his doctorate in 1932 for his thesis *An Abstract Approach to Manifolds*. However, his first research paper came about because he found an error in Luther Eisenhart's course on Riemannian Geometry which he attended during his year as an Instructor. Eisenhart encouraged him to write it up for publication and *On generalised covariant differentiation* appeared in print in 1931.

Tucker was a National Research Fellow during 1932-33, spending the autumn term at the University of Cambridge working with Max Newman. He also attended the International Congress of Mathematicians held in Zürich in September 1932 where he heard talks from, among others, Marston Morse and James Alexander. In December 1932 he went to Harvard, where he worked with Morse, leaving in June 1933 to spend the summer at the University of Chicago. In 1933 he was appointed as an Instructor at Princeton and was promoted to assistant professor in the following year. He became an associate professor in 1938 and a full professor in 1946. On 17 December 1938, soon after he was promoted to Associate Professor, Tucker married Alice Judson Curtiss. They had three children; sons Alan Curtiss Tucker (born 6 July 1943) and Thomas William Tucker (born 15 July 1945) and a daughter Barbara Jane Tucker.

In 1954 he succeeded Emil Artin (who became Fine Professor at this time) as Albert Baldwin Dod Professor of Mathematics at Princeton, having become chairmen of the mathematics department at Princeton in the previous year. He retired in 1974 when he was named professor emeritus.

His early work was in combinatorial topology, continuing to develop the areas he had studied for his thesis. He published papers such as: *An abstract approach to manifolds* (1933); *On tensor invariance in the calculus of variation* (1934); *Non-Riemannian subspaces* (1935); *Cell spaces* (1936); *On chain-mappings carried by cell-mappings* (1939); and *The algebraic structure of complexes* (1939). During World War II Tucker was involved in war work of an applied mathematical nature in the Princeton Fire Control Research Project. Here 'fire control' refers to gunfire controlled by range or height finders. He recalled [20]:-

... from September1941until about1944I worked for the Princeton Fire Control Research Project, for which I was the so-called Associate Director. Merrill Flood was the Director. I did this in addition to carrying the normal teaching load at the University. ... In1943the Army Specialized Training Program started at Princeton, and somewhat later the Naval College Training Program. I had charge of the mathematics portion of the Army Specialized Training Program, and although I did no teaching in the Naval Program I had some administrative responsibility for that. The one somewhat unusual piece of teaching that I did was a mathematics refresher for Naval officers who were being trained in radar.

After the Princeton Fire Control Project ended he then worked as an assistant to Samuel Wilks on statistical projects, and spent some time working with von Neumann at the Institute for Advanced Studies on techniques that could be used on the high-speed computer that von Neumann was beginning to develop. He returned to his regular duties at Princeton University in 1946 but found it difficult to pick up his topology research after the forced break caused by World War II. This, in part, led to his involvement with game theory and linear programming for which his name is today most associated.

George Dantzig was working for the Air Force at the Pentagon and visited von Neumann at Princeton in November 1947 to explain the simplex method to him. Dantzig returned to Princeton in early 1948 to ask von Neumann for suggestions about where a university project on the mathematics of linear programming might be set up. By chance, on this visit he met Tucker who asked him to explain what linear programming was. Tucker immediately saw it as a topic which would interest him particularly since he saw connections to ideas from his topological research. By the summer of 1948 the project had been set up at Princeton under the Office of Naval Research (there was no Office of Air Force Research) with Tucker as its director. Solomon Lefschetz already had a project going at Princeton with the Office of Naval Research so the two projects were linked. Tucker began work with two graduate students, David Gale and Harold Kuhn, on the connections between linear programming and matrix games. He also had John Nash as a research student from 1948: Nash's thesis *Non-cooperative Games* was submitted in 1950. From this time on Tucker published papers in conference proceedings, choosing to leave the regular journals for younger mathematicians. His publications after the start of this project include: (with David Gale and Harold Kuhn) *On symmetric games* (1950); (with David Gale and Harold Kuhn) *Reductions of game matrices *(1950); (with David Gale and Harold Kuhn) *Linear programming and the theory of games* (1951); and (with Harold Kuhn) *Nonlinear programming* (1951). The rest of Tucker's research career was involved in aspects of game theory and linear programming.

In 1949-50 Tucker took sabbatical leave which he spent at Stanford University. Asked to give an elementary account of game theory to graduate psychology students, he remembered a game Melvin Dresher had told him about. Dresher, along with Merrill Flood, had devised a game to use in a psychology experiment while working at the RAND Corporation. Tucker made up a setting for the Dresher-Flood game which he called the "Prisoner's dilemma". Today the Prisoner's dilemma is presented in a slightly more elaborate form, but as Tucker presented it to the psychology students in May 1950 it read:

Two men, charged with a joint violation of law, are held separately by the police. Each is told that(1)if one confesses and the other does not, the former will be given a reward ... and the latter will be given a large fine ...(2)if both confess, each will be given a small fine ... At the same time each has good reason to believe that(3)if neither confesses, both will go clear.

Each argues that, whatever the other does he has a better deal by confessing. So they both confess and receive a small fine. However, if they could cooperate, they could agree that neither will confess and they will both be cleared.

J J Kohn became a graduate student at Princeton in 1953 at the time Tucker became departmental chairman. He writes [4]:-

From the beginning it was clear that Al was the person most responsible for the smooth running of the department. The atmosphere in the old Fine Hall was exceptional; it was stimulating, and it had that rare mixture of friendliness and competitiveness. This optimal environment was largely due to Al's management of the department, his leadership, his sense of fairness, and his commitment to research and teaching.

In June 1961 Tucker was awarded an honorary D.Sc. from Dartmouth College. The President of the College, John Sloan Dickey, addressed Tucker with the following citation (see for example [7]):-

Nearly three decades ago you began an academic career at Princeton which became a mission to mathematics. In a field where scholarship scores only if the idea is both new and demonstrably true, your ideas have won their way in topology, in the theory of games, and in linear[and nonlinear]programming. But even in mathematics a mission is more than ideas; it is also always a man, a man who cares to the point of dedication, whose concern is that others should care too, and who can minister to the other fellow, as the need may be, either help or forbearance. Because you, Sir, embody in extraordinary measure both your profession's love of precision and man's need for conscientious leadership, mathematics in America at all levels is today higher than it was and tomorrow will be higher.

In 1967 the Mathematical Association of America presented their Award for Distinguished Service to Tucker. The citation reads:-

... the award is made for outstanding service to mathematics or mathematical education on a national scale. ... the Association has honoured brilliant teachers, statesmen of the mathematical community, creative scientists, and crusading reformers of the nation's mathematical curriculum. The man we honour today has played all these roles with dedicated enthusiasm. By his willing service to those ho would learn, teach, advance, or use mathematics, Albert William Tucker has added measurably to the vigour and quality of mathematics today.

Tucker died of pneumonia at the Presbyterian Home of Meadow Lakes, a nursing home in Hightstown, New Jersey, after a long illness. He and his wife Alice had divorced in June 1960 and Tucker had married Mary F Shaw on 26 February 1964. He was survived by his second wife Mary, his two sons Alan and Tom (both eminent professors of mathematics), and his daughter Barbara Tucker Cervone (Special Assistant to the president of Brown University).

**Article by:** *J J O'Connor* and *E F Robertson*