**William Moser**is known to his friends and colleagues as Willy. His parents were Laura Feurstein and Robert Moser from Austria. They married there and their oldest child Leo was born in Vienna in 1921, before they emigrated to Canada. They settled in Winnipeg and on 5 September 1927 twin sons were born, one of the twins being Willy the subject of this biography. He grew up in Winnipeg where he attended both primary and secondary school. His biggest mathematical influence as he grew up was his older brother Leo who went on to become a leading mathematician with a biography in this archive. In 2005 Moser edited a volume of collected papers of his brother Leo Moser He wrote:-

Moser entered the University of Manitoba, following the career path of Leo, and graduated with a B.Sc. in 1949. In August 1949 he participated in a meeting of the Canadian Mathematical Congress. He wrote about the influence of this Meeting on at least two separate occasions. This first quote is from [2]:-Leo died more than35years ago, before his49(^{th}birthday, and he has been in my thoughts and memory all these years. Leo was my first mathematical mentor. When I was a teenager he was already showing me the pleasures of mathematics. Leo's enthusiasm for mathematics was boundless, his creativity was ingenious, and creating problems was his specialty. In this respect he was like Paul Erdős, whom Leo greatly admiredas did I); and like Erdős he was completely open and generous with his knowledge.

Further details of the influence of this event on Moser's mathematical life is given in [1]:-In August1949I participated in a meeting of the Canadian Mathematical Congress in Vancouver. This organization was then only four years old. At this1949Congress I met the founding fathers: Donald Coxeter, Ralph Jeffery, Lloyd Williams, Eric O'Connor, and others. What a fine group of fine mathematicians.

Moser then did a Master's degree at the University of Minnesota, being awarded an M.A. in 1951. For the next two years he was National Research Council Canada fellow, undertaking research at the University of Toronto with Donald Coxeter as his doctoral supervisor [2]:-There I met the founders of the Congress and established a lasting friendship with them and others. I well remember Lloyd Williams, who raised tens of thousands of dollars from private corporations to support the work of the Congress. Also, Ralph Jeffery who started, and guided, the Summer Research Institutes in Kingston where I spent three great summers. I also started a friendship with Father O'Connor who worked tirelessly for the benefit of others. I met Donald Coxeter who played such an important role in my life.

On 2 September 1953, Moser married Beryl Rita Pearlman; they had three children Marla, Lionel, and Paula. In 1955 he was appointed to the University of Saskatchewan as an Instructor, becoming an Assistant Professor there in 1957. This was the year he was awarded his Ph.D. by the University of Toronto for his thesisWilliam Moser's relations with Donald Coxeter were beyond those of a former student: during his whole life he spent many respectful hours visiting Coxeter, particularly later in Coxeter's life. Some of his trips to Toronto have been only to visit Coxeter.

*Abstract Groups And Geometrical Configurations*. In fact this thesis was a major component in the classic text

*Generators and relations for discrete groups*jointly authored by Donald Coxeter and Willy Moser and published by Springer-Verlag in Berlin-Göttingen-Heidelberg in 1957. In the Preface to the book they write:-

See Coxeter-Moser Preface for a complete version of the Preface.When we began to consider the scope of this book, we envisaged a catalogue supplying at least one abstract definition for any finitely-generated group that the reader might propose. But we soon realized that more or less arbitrary restrictions are necessary, because interesting groups are so numerous.

Graham Higman, reviewing the book, explained that it :-

The second publication by Moser... brings together a great deal of information on generators and relations for a wide variety of groups. For instance, it deals with the crystallographic groups, with the fundamental groups of surfaces, with the symmetric and related groups, with the projective linear groups over finite fields, and with groups on real Euclidean space generated by reflections. There is also a systematic method for checking the sufficiency of a set of relations, and considerable attention to graphical representations. The book will be invaluable to anyone who believes, as I do, that progress in Group Theory depends primarily on an intimate knowledge of a large number of special groups.

*On the number of ordinary lines determined by n points*was again of major significance. The paper, which was a joint work with Leroy M Kelly, was published in 1958. G A Dirac, reviewing the paper, writes:-

In fact Dirac was the right person to review this paper for he had made a famous conjecture that there are at leastThis paper is certain to interest a wide class of readers, because of the generality and importance of the problems considered and the elementary nature and great ingenuity of the methods used. Let there be a finite number n(>1)of points in the real Euclidean or the real projective plane, not all on one line, and let every pair of distinct points be joined by a line. The authors consider two most interesting and general questions concerning such configurations of n non-collinear points and the lines determined by them: What is the minimum number of lines determined by the n points, and what is the minimum number of those lines which have the property that each of them passes through exactly two of the n points?

*n*/2 lines with the property that each of them passes through exactly two of the

*n*points. The problem had been around for a long time, for Sylvester had asked about the existence of such lines in a question in the

*Educational Times*in 1893. Moser and Kelly proved in this famous paper that there are at least 3

*n*/7 such lines, and moreover this bound is sharp when

*n*= 7. We note in passing that Moser published a joint paper with Peter Borwein in 1990

*A survey of Sylvester's problem and its generalizations*which looks at developments. Another of Moser's 1958 publications was

*On the relative widths of coverings by convex bodies*, then in the following year he published the paper

*Abstract definitions for the Mathieu groups*

*M*

_{11}and

*M*

_{12}adding to the examples of group presentations appearing in

*Generators and relations for discrete groups*.

In 1959 Moser returned to the University of Manitoba, where he had studied as an undergraduate, being appointed an Associate Professor. He remained there until 1964 when he was appointed as an Associate Professor at McGill University in Montreal. He remained at McGill, being promoted to Professor in 1966. He retired in 1997 when he was made Professor Emeritus. At this time McGill paid him a tribute:-

His contributions to mathematics in Canada, and in particular to the Canadian Mathematical Society, are quite outstanding. He was president of Canadian Mathematical Society from 1973 to 1975. He was Editor-in-Chief of the Canadian Mathematical Bulletin - Bulletin Canadien de Mathématiques - from 1961 to 1970, Chairman of the Publications Committee from 1970 to 1974 and an Associate Editor of the Canadian Journal of Mathematics - Journal Canadien de Mathématiques - from 1981 to 1985. He was honoured for these contributions when he received the Canadian Mathematical Society Distinguished Service Award in 2003. One aspect of his contributions, which we have not yet mentioned, is highlighted in the citation for the award [2]:-During his32years in the Department of Mathematics and Statistics, Moser fashioned and taught his own geometry courses in an original manner, making collections of research problems along the way. In order to promote his specialty, Moser became involved in the teaching of high school mathematics teachers, making films about geometry, organizing mathematical competitions and collecting and disseminating competition problems.

Moser published a fine collection of combinatorics papers jointly with Morton Abramson:Since the beginning of his career in1955, William Moser has been involved with high school/college mathematics in a variety of ways, particularly serving on committees arranging provincial mathematics competitions in Saskatchewan, Manitoba and Quebec. He was a member of the CMS Education Committee when, under the chairmanship of Lloyd Dulmage, it instituted the Canadian Mathematical Olympiad. He edited(some with Ed Barbeau)the booklets containing the problems, solutions and results of the Canadian mathematical Olympiads from1969to1978. During the years1975-1985, he, E Barbeau and M S Klamkin edited five collections of problems(500problems altogether)that the CMS printed and distributed widely. The full collection, corrected and improved, has been published by Mathematics Magazine in its spectrum series as "500Mathematical Challenges" in1995. He has also taught NSF Summer Institutes for High School Teachers(1959-62)and participated in the College Geometry Project(1964-68)at the University of Minnesota, making beautiful films, one about Coxeter.

*A note on combinations*(1966);

*Combinations, successions and the n-kings problem*(1966);

*Permutations without rising or falling w-sequences*(1967);

*Enumeration of combinations with restricted differences and cospan*(1969);

*Generalizations of Terquem's problem*(1969);

*The problem of the second seating and generalizations*(1972);

*Arrays with fixed row and column sums*(1973); and

*Linear and ring arrangements*(1976).

Finally we must say something about Moser's remarkable contributions in publishing surveys of problems in discrete geometry in both books and articles. He edited *Problems in discrete geometry* (1980) which collected together 34 problems, each with references to preceding work. In the following year he edited *Research problems in discrete geometry* which was a collection of 68 problems of combinatorial geometry including distance problems, covering and packing problems, lattice point problems, and visibility problems. In 1985 in the paper *Problems on extremal properties of a finite set of points* he presented updated information of progress on 19 of the problems from the earlier work. In 1989 the First Canadian Conference on Computational Geometry was held in Montreal and Moser presided over the two problem sessions publishing *Problems, problems, problems* in the conference proceedings. This interest in problems in discrete geometry culminated in 2005 with the publication of a 500 page book *Research problems in discrete geometry* published jointly by Moser, Peter Brass and János Pach. The chapter headings give an indication of the content:

Density problems for packings and coverings;

Structural packing and covering problems;

Tiling problems;

Distance problems;

Problems on repeated configurations;

Incidence and arrangement problems;

Problems on points in general position;

Graph drawings and geometric graphs;

Lattice point problems;

Geometric inequalities.

A few years later Moser supervised a second Master's student with the name of Campbell, namely Harvey A Campbell, on a similar topic. In his thesisAs a young Master's student from Scotland attending McGill University I was introduced to the delights of computational group theory by my cigar chain-smoking supervisor William Moser. While working on my dissertation in the summer of1965, I met up with William Moser in Minneapolis where he was heavily involved in the Minnemath project at the University of Minnesota. It gave me an opportunity to visit the Guthrie theatre in Minneapolis, encouraged by Moser who is a great enthusiast for the theatre. At a professional level my career has revolved round the Todd-Coxeter coset enumeration algorithm to which I was first introduced by Moser.

*An extension of coset enumeration*, Harvey Campbell writes:-

May I [EFR] recount a meeting with Willy Moser in 1982 (when he gave me the copy of Harvey Campbell's thesis from which we have just quoted). I attended the 'Finite groups - coming of age' conference in Concordia University in Montreal and visited Moser at McGill. We met on Saturday morning in the Mathematics Department. When I arrived Moser was playing lightning chess with one of his students. It was a game played with such a few seconds on the clocks that it was more a test of reaction time rather than of chess. It was played in great spirit with much hilarity.The author wishes to express deep gratitude to a scholar, Professor W O J Moser, whose ideas and encouragement made this thesis possible.

In March 2003 Moser was interviewed by Siobhan Roberts who was working on her major work on Coxeter *King of Infinite* space. He recounted the following story [3]:-

We note that Moser also told Roberts in this interview that"Donald made many great contributions to mathematics. I made one great contribution," recounted Moser. Moser's opportunity came at the end of Coxeter's1955summer of roving lectures, after his session in Stillwater, at Oklahoma State University. Moser drove down to meet Coxeter and serve as his assistant, taking detailed notes of the well-polished lectures. "At the end of the summer we drove north, to civilisation," said Moser wryly. "We were in my car and Donald asked me if he could drive. It was a new car. Indeed it was the first car I had ever purchased, a green1955Plymouth2-door. I paid $2,000for it and drove it to Oklahoma. But I agreed. I was surprised to see that he was an aggressive driver. At one point he was trying to pass a car while driving up a hill on a2-lane highway. I immediately perceived that this was not a prudent thing to do. He tried to coax the car to go faster but it wouldn't respond. At the last moment I shrieked at him, 'Pull back, pull back'. I was probably his only student to shriek at him. He began to pull back and at that moment a truck came over the hill. He managed to get back in the right lane just in time. I HAD SAVED HIS LIFE! And mine. But saving Coxeter's life was my greatest contribution to mathematics."

*Generators and relations for discrete groups*was, for a time, miscatalogued at the University of Toronto library and mis-shelved in the genealogy section.

We end this biography of Willy Moser by quoting his words from his address to the Canadian Mathematical Society at the time he received its Distinguished Service Award in 2003. Addressing the young mathematicians in the audience he said [2]:-

Be generous and patient as teachers, be active in projects which benefit the mathematical community and, above all, have as long and as happy a mathematical life as I have had, and am still having.

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