# George Whitelaw Mackey

### Quick Info

St Louis, Missouri, USA

Belmont, Massachusetts, USA

**George Mackey**was an American mathematician who worked on the representation theory of groups.

### Biography

**George Mackey**'s father was William Sturges Mackey and his mother was Dorothy Frances Allison. The family moved from St Louis to Florida then settled in Houston in 1926. Many mathematicians know that mathematics is the subject for them at a young age but Mackey struggled to find the right topic for him. On the one hand there was pressure from his father to become a businessman. However when he was studying at high school he was inspired by a chemistry book which he was studying and so, as a compromise between different topics that interested him, he entered the Rice Institute in Houston to study for a degree in chemical engineering.

Changes of direction did not come easily to Mackey who described them as a "soul struggle", but he decided to move from the chemical engineering course to study physics. However, he still had not found exactly what he was looking far. As he wrote many years later (see for example [1] and [2]):-

I was disturbed by the way that mathematics was handled in my physics courses and spent a great deal of time trying to redo things in a more precisely defined and rigorously argued manner. ... I wanted somehow to combine the logical precision of mathematics with the (apparently) richer content of physics.After the award of a bachelor's degree from the Rice Institute in 1938 (Rice Institute was founded in 1891 and only became Rice University in 1960), Mackey went to Harvard to undertake postgraduate work. He was one of the first to be awarded a William Lowell Putnam fellowships to undertake research at Harvard. He was awarded a Master's Degree in mathematics from Harvard in 1939, then continued undertaking research there for his doctorate under Marshall Stone's supervision. Although he majored in physics for his bachelor's degree, he moved towards mathematics at Harvard. He later wrote (see for example [1] or [2]):-

In graduate school, I became seduced by the beauties of pure mathematics and put my plans for reforming physics aside.He was awarded his doctorate in 1942 after submitting his thesis

*The Subspaces of the Conjugate of an Abstract Linear Space*.

After the award of his Ph.D. from Harvard, Mackey spent a year teaching at the Illinois Institute of Technology in Chicago. In 1943 he was appointed as an instructor in mathematics at Harvard University and he was to remain on the faculty there for the rest of his career. He served as a civilian in the Operations Research Section of the 8th Air Force in 1944 as his contribution to the war effort. He was promoted to assistant professor at Harvard in 1946, to associate professor in 1948, and to full professor in 1956. Then, in 1969, he was honoured by being named Landon T Clay Professor of Mathematics at Harvard. He retired in 1985, being named Landon T Clay Professor of Mathematics Emeritus at that time.

Early in his career Mackey worked on the duality theory of locally convex spaces publishing papers which include

*On infinite dimensional linear spaces*(1943),

*On convex topological linear spaces*(1943),

*Equivalence of a problem in measure theory to a problem in the theory of vector lattices*(1944), (with Shizuo Kakutani)

*Ring and lattice characterization of complex Hilbert space*(1946),

*On convex topological linear spaces*(1946). His interest in physics continued, however, and he published

*A theorem of Stone and von Neumann*(1949) in which he generalised a theorem about quantum mechanics proved by Stone and von Neumann in 1930. This work was related to

*Imprimitivity for representations of locally compact groups*which Mackey published in the same year. He then produced a series of important papers on group representations including

*On induced representations of groups*(1951),

*Induced representations of locally compact groups*(1952), and

*Symmetric and anti symmetric Kronecker squares and intertwining numbers of induced representations of finite groups*(1953).

In 1955 Mackey gave a course of lectures on the theory of group representations at the University of Chicago in the summer of 1955. A set of mimeographed notes was produced at the time and these later were incorporated as the first half of his book

*The theory of unitary group representations*published in 1976. In the spring of 1960 he gave a lecture course at Harvard University on the mathematical foundations of quantum mechanics. An edited version of these lectures became his famous classic text

*The mathematical foundations of quantum mechanics*published in 1963:-

... the aim of the book is to explain, or at least to illuminate, the essential aspects of classical and quantum mechanics from a point of view more congenial to pure mathematicians than that encountered in physics texts. In a rather unusual way, all physical concepts used are defined in terms of pure mathematics rather than physical connections except, of course, basic notions concerning space and time.In 1967 Mackey published Lectures on the theory of functions of a complex variable which was based on an undergraduate course he gave at Harvard during the academic year 1959-60. He writes in the Introduction:-

For better or worse I chose to de-emphasize the intuitive geometric aspects of the subject and to present it as a deductive system starting with axioms for the real and complex numbers. While I assumed a fair amount of 'mathematical maturity' on the part of the students I made a point of assuming no previous formal knowledge of mathematics. I defined all terms - even those of calculus and elementary algebra - and proved all theorems except those which I decided to leave to the students as exercises. In principle (but not in practice of course) the notes can be read with understanding by someone with no previous knowledge of mathematics. Of course, in order to avoid excessive pedantry I left many simple arguments to the imagination of the student - especially after the first few chapters.During the academic year 1966-67, Mackey delivered a series of lectures on group representations and their applications at Oxford University in England where he was George Eastman visiting professor. As for some of his earlier courses, mimeographed notes circulated from that time on but eventually became part of a properly published textbook. This happened when

*Unitary group representations in physics, probability, and number theory*was published in 1978. Mackey has written many beautiful survey articles and in 1992 the American Mathematical Society and the London Mathematical Society in their wonderful series 'History of Mathematics' published

*The scope and history of commutative and noncommutative harmonic analysis*by Mackey. This consisted mostly of survey articles written by Mackey as expanded versions of lectures given at conferences between 1977 and 1988. Jorgensen writes in a review:-

It is well known that the author's approach to representation theory has proved very successful in finding the unitary representations, and their decompositions, for the groups which come up frequently both in (pure) mathematics and in physics, as well as in mathematical physics. But the author's theory is surprisingly versatile, with applications in number theory, harmonic analysis, ergodic theory, quantum mechanics, and statistical mechanics, and these applications are worked out in detail.... The reviewer notes that there are also interesting applications to operator algebras. ... The new material in the present book is concentrated in the last 50 pages and it centres around lattice models in statistical mechanics, PDEs in hydrodynamics, Kac-Moody Lie algebras, and the Korteweg-de Vries equation.Clifford Taubes, chairman of Harvard University's mathematics department gave this appreciation of Mackey:-

He was a gentle man, but he was the most intellectually honest person I've ever known. He was a realist. He believed that people shouldn't delude themselves. He wouldn't let people get away with cosy preconceptions. Here, I'm not just talking about mathematics, but aspects of life, your beliefs. You'd get into these conversations about politics and things in everyday life and he'd say, 'Well, why do you believe that?' You really had to back up what you were saying.Many doctoral students at Harvard having Mackey as a thesis advisor went on to have academic careers. Let us mention a few: Andrew Gleason (Ph.D. 1950), Richard Palais (Ph.D. 1956), Calvin Moore (Ph.D. 1960), Edward Effros (Ph.D. 1962), Lawrence Brown (Ph.D. 1968), and Caroline Series (Ph.D. 1976).

Mackey married Alice Willard on 9 December 1960; they had a daughter Ann Sturges Mackey born in 1963. Comments by Alice and Ann in [1] say much about George Mackey. When Ann was born, Alice persuaded Mackey to give up driving because:-

... he had a tendency to go left when right was right.Each morning and evening he set aside time to read and play games with his daughter and would not allow work to intrude. Ann said:-

He was a great believer in quality time and he really embraced it. When he was with me, he was really, truly with me.After his daughter went to bed, he would read aloud to his wife while she did needlepoint - biographies, non-fiction, and the classics of English literature. Alice said:-

When he was not working, he was totally attentive to the person he was with. People would say, "He really listened to me."Among the honours Mackey received for his outstanding contributions to mathematics was election to American Academy of Arts and Sciences (1953), the National Academy of Science (1962), and the American Philosophical Society (1971). A long time member of the American Mathematical Society, he was vice president in 1964-65 and received the Society's Steele Prize in 1975. Rice University gave him a distinguished alumnus award in 1982.

### References (show)

- K Gewertz, Mathematician George W. Mackey, 90 : Obituary,
*Harvard News*(Cambridge, March 2006. - B Marquard, George Mackey, professor devoted to truth, theorems,
*Boston Globe*(Boston, 28 April 2006).

### Additional Resources (show)

Other pages about George Mackey:

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### Honours (show)

Honours awarded to George Mackey

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update July 2007

Last Update July 2007