## Representing George W. Mackey (1916–2006)

**September 24, 2006**

George W. Mackey

I believe it was Harvard president A. Lawrence Lowell who, in the early 1930s, introduced the undergraduate tutorial system, a clone of the one in place at Oxford. Under this system, a sophomore, having declared his major, was assigned a faculty tutor in the field. The student and tutor met every two weeks or so for an hour; they discussed suggested readings, and sometimes the student was given a paper to write. The tutorial sessions were independent of the courses and did not "count" toward a degree.

My sophomore year at Harvard began in the fall of 1940. Having switched my major from astronomy to mathematics, I was assigned George Mackey---fresh from Texas, as my tutor. When he met with me in the faculty room in Sever Hall for my sessions, he had just earned a master's degree from Harvard, having previously received a bachelor's degree from Rice. I believe I was one of the first---possibly *the* first---of George's undergraduate tutees. I don't know who others might have been (neither undergraduate tutees nor PhDs in math being guaranteed to stay in the math business).

What did George put me onto? Group theory. I read in R.D. Carmichael's recently published introduction to the subject and reported back to George on what I had read---faithfully but half-heartedly. My brain seems not to be hardwired for questions of this kind. I was glad to learn of the broad landscape of group theory but not about each blade of grass. (In later years I was gratified to read that the great Jacques Hadamard harbored the same opinion about groups.) My mathematical heart lay elsewhere. Our relationship was not made in heaven, and my experiences with George as an instructor ended in the spring of 1941. I was given over as tutee to David V. Widder, the Laplace transform man.

In my undergraduate days, I hung out with some of the math majors, but also with a sophisticated group of literary aspirants, and in my newly acquired snobbism, I regarded George as an unworldly genius from Texas, lacking in social and intellectual skills. I got it wrong. I have it from John Wermer, George's second PhD student (the first was Andrew Gleason) that in addition to his utter devotion to mathematics, George interested himself in all manner of ideas and spoke forthrightly from honest conviction, if not always from knowledge. I was told he once gave a copy of a book on politics as a dinner present to his hostess, remarking that it had brought him new insights that were worth sharing. George married (a bit late, I thought) and is survived by his wife, Alice, one daughter, and two grandchildren.

George went on to get his PhD under Marshall Stone in 1942, with a thesis titled The Subspaces of the Conjugate of an Abstract Linear Space. Later, he sought to bridge mathematics and physics with *Mathematical Foundations of Quantum Mechanics* (1963), *Mathematical Problems of Relativistic Physics* (1967), *Induced Representations of Groups and Quantum Mechanics* (1968), *Theory of Unitary Group Representations* (1976).

As an undergraduate, Mackey had majored in physics but found himself increasingly interested in mathematics. As he wrote later in life in an autobiographical sketch, "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."

One may reasonably conjecture that Mackey had read and was influenced by Eugene Wigner's seminal 1939 paper on the representations of the inhomogeneous Lorentz group and was following in Wigner's footsteps in his desire to build a bridge between group representations and physics. Mackey elaborated his motivation in an introduction to an address he gave in 1999 at Ohio State University:

"Number theory and physics may be linked in a very significant way through their use of a common tool. This tool is a synthesis of Fourier analysis, the representation theory of finite groups and the Hilbert spectral theorem. Namely, the unitary representation theory of locally compact groups. There are indications that one can hope to embed a rather large part of both number theory and quantum physics in a suitably developed expansion of this theory."

It's a well-documented phenomenon that theoretical physicists tend to give mathematicians a hard time. They have a different take on the stuff. Over the years, physicists seem to prefer to create their own mathematics. I have in mind such people as Paul A.M. Dirac, Oliver Heaviside, Richard Feynman (dare I forget Newton and Hamilton?). Mathematicians, for their part, tend to thread the whole world through the eye of their own particular needle. The straight line, along with all other phenomena, could, according to George, be best understood through group representations.

Over the years I used to meet George from time to time, in Cambridge or at national math meetings. He had by no means forgotten his quondam student (a phrase used by the ultraconservative Roger Merriman of his former student Franklin D. Roosevelt, with a strong emphasis on the "dam"). From something he once said to me while I was working at the National Bureau of Standards, I got the strong impression that George considered me a traitor: (1) for accepting a non-academic job and (2) for leaving group theory behind in favor of approximation theory and numerical methods.

Time, so they say, heals all wounds and softens all opinions. Some years ago, the publisher Klaus Peters (AK Peters Ltd.) wanted to give my book *The Education of a Mathematician* a bit of a local sendoff. He arranged for me to give a talk in the Natick (Massachusetts) Public Library, to which he sent cordial invitations to mathematicians in the Boston area. George came, bringing his wife Alice. I was tickled. After my talk, he came up, shook my hand, and said something that can be translated as "All is now forgiven."

Henry Sloane Coffin, of the Union Theological Seminary, speaking of children, once quipped to my father-in-law, "After the age of seven, all you can do is to love them." Something similar is true of students---well, some of them at least. And as for George's grandstudents? At least two have gone in applied directions: Jill Mesirov works in computational biology and Joel Spencer in probability.

George Mackey was a mathematicians' mathematician. The incandescence of his mathematical insights endures, as do the ideals he set for his colleagues and students.---*Philip J. Davis, Brown University*