occasionally one-on-one. I have always been im- them on: my latest student, Hadi Salmasian, has pressed by the independence of his views—he used these same ideas to take the line of work came to his own conclusions and advanced them further and show that what had seemed perhaps with conviction born of long thought—and by his ad hoc constructions for classical groups could scholarship—he carefully studied relevant papers be seen as a natural part of the representation in or physics and took them in- theory of any semisimple or reductive . to account, sometimes accepting, sometimes not, George’s body may have given up the ghost, but according to what seemed right. I particularly his spirit and his mathematics will be with us for remember a short but highly referenced oral dis- a long time to come. sertation on the Higgs Boson, delivered for my sole benefit. I forget when George took to referring to me as Arthur Jaffe his grandstudent, but a particularly memorable oc- casion when he did so was in introducing me to his Lunch with George advisor, Marshall Stone. The Stone-von Neumann Theorem, which originated as a mathematical Background characterization of the Heisenberg canonical com- mutation relations, was reinterpreted by Mackey I was delighted to see that the program of the as a classification theorem for the unitary repre- 2007 New Orleans AMS meeting listed me cor- sentations of certain nilpotent groups. Both Cal rectly as a student; in fact I have been a student Moore and I have found new interpretations and of practically all my mathemati- applications for it, and now my students use it cal life. George loved interesting and provoking in their work. In fall 2004 I attended a program mathematical conversations, and we had many at the Newton Institute on quantum informa- over lunch, explaining my congenial title. tion theory (QIT). There I learned that QIT had Most of our spurred new interest in Hilbert space geometry. individual meet- One topic that had attracted substantial atten- ings began at tion was mutually unbiased bases. Two bases {uj } the Harvard Fac- and {vj }, 1 ≤ j ≤ dim H, of a finite-dimensional ulty Club. George Hilbert space H, are called mutually unbiased if walked there from the inner product of uj with vk has absolute val- working at home 1 1 2 to meet for our ue  dim H  , independent of j and k. A number of luncheon, and I of- constructions of such bases had been given, and ten watched him some relations to group theory had been found. pass the reading The topic attracted me, and in thinking about it,

I was amazed and delighted to see that George’s room windows. Photo courtesy of Ann Mackey. Generally our con- work on induced representations, systems of im- Mackey in Harvard office. primitivity, and the Heisenberg group combined versations engaged to give a natural and highly effective theory and us so we continued construction of large families of mutually unbi- afterward in one of our offices, which for years ased bases. It seemed quite wonderful that ideas adjoined each other in the mathematics library. that George had introduced to clarify the founda- Some other occasions also provided opportunity tions of quantum mechanics would have such a for conversation: thirty years ago the department satisfying application to this very different aspect met over lunch at the Faculty Club. Frequent- of the subject. I presented my preprint on the sub- ly we also exchanged invitations for dinner at ject to George, but at that time his health was in each other’s home. Both customs had declined decline, and I am afraid he was not able to share significantly in recent years. Another central fix- my pleasure at this unexpected application. ture revolved about the mathematics colloquium, I hadn’t expected the strange-seeming ideas which for years George organized at Harvard. in George’s notes for that reading course to im- George and Lars Alfhors invariably attended the pinge on my research. I had quite different, more dinner, and for many years a party followed in algebraic and geometric, ideas about how to ap- someone’s home. George also made sure that proach . But impinge they each participant paid their exact share of the bill, did. When I was struggling to understand some a role that could not mask the generous side of qualitative properties of unitary representations his character. of classical Lie groups, I found that the ideas from that course were exactly what I needed. And I am Arthur Jaffe is the Landon T. Clay Professor of Mathe- extremely happy not only to have used them (and matics and Theoretical Science at . His to have had them to use!) but also to have passed email address is [email protected].

August 2007 Notices of the AMS 833 George also enjoyed lunch at the “long table” ago, the theme “The Mathematical Theory of Ele- in the Faculty Club, where a group of regulars mentary Particles” represented more dream than gathered weekly. Occasionally I joined him there reality. or more recently at the American Academy of I knew George’s excellent book on the math- Arts and Sciences, near the Harvard campus. I ematical foundations of quantum theory, so I could count on meeting George at those places looked forward to meeting him and to discussing without planning in advance. Through these in- the laws of particle physics and quantum field teractions my informal teacher became one of my theory. George was forty-nine, and I was still a best Harvard friends. So it was natural that our student at Princeton. Perhaps the youngest per- conversations ultimately led to pleasant evenings son at the meeting, I arrived in awe among many at 25 Coolidge Hill Road, where Alice and George experts whose work I had come to admire. George were gracious and generous hosts, and on other and I enjoyed a number of interesting interac- occasions to 27 Lancaster Street. tions on that occasion, including our first lunch While the main topic of our luncheons focused together. on mathematics, it was usual that the topic of Our paths crossed again two years later, on- conversation veered to a variety of other subjects, ly weeks before my moving from Stanford to including social questions of the time and even Harvard. That summer we both attended the to novels by David Lodge or Anthony Trollope. “Rochester Conference”, which brought together George seemed to come up with a viewpoint on particle physicists every couple of years. Return- any topic somewhat orthogonal to mine or to ing in 1967 to the University of Rochester where other companions, but one that he defended both the series began, the organizers made an attempt with glee as well as success. to involve some as well. George began as a student of physics and The Rochester hosts prepared the proceedings found ideas in physics central to his mathemat- in style. Not only do they include the lectures, but ics. Yet George could be called a “quantum field they also include transcripts of the extemporane- theory skeptic”. He never worked directly on this ous discussions afterward. Today those informal subject, and he remained unsure whether quan- interchanges remain of interest, providing far tum mechanics could be shown to be compatible better insights into the thinking of the time than with special relativity in the framework of the the prepared lectures that precede them. The Wightman (or any other) axioms for quantum discussion following the lecture by Arthur Wight- fields. man includes comments by George Mackey, Irving When we began to interact, the possibility to Segal, Klaus Hepp, Rudolf Haag, Stanley Mandel- give a mathematical foundation to any complete stam, , C.-N. Yang, and Richard example of a relativistic, nonlinear quantum field Feynman. It is hard to imagine that diverse a appeared far beyond reach. Yet during the first ten years of our acquaintance these mathemat- spectrum of scientists, from mathematicians to ical questions underwent a dramatic transition, physicists, sitting in the same lecture hall—much and the first examples fell into place. George less discussing a lecture among themselves! and I discussed this work many times, reviewing Reading the text with hindsight, I am struck by how models of quantum field theory in two- and how the remarks of Mackey and of Feynman hit three-dimensional Minkowski space-time could be the bull’s-eye. George’s comments from the point achieved. While this problem still remains open in of view of ergodic theory apply to the physical four dimensions, our understanding and intuition picture of the vacuum. Feynman’s attitude about have advanced to the point that suggests one may mathematics has been characterized by “It is a find a positive answer for Yang-Mills theory. Yet theorem that a cannot prove a George remained unsure about whether this cul- nontrivial theorem, as every proved theorem is mination of the program is possible, rightfully trivial,” in Surely You’re Joking, Mr. Feynman. questioning whether a more sophisticated con- Yet in Rochester, Feynman was intent to know cept of space-time would revolutionize our view whether quantum electrodynamics could be (or of physics. had been) put on a solid mathematical footing. Despite this skepticism, George’s deep in- Today we think it unlikely, unlike the situation sights, especially those in ergodic theory, con- for Yang-Mills theory. nected in uncanny ways to the ongoing progress in quantum field theory throughout his lifetime. Harvard George chaired the mathematics department Early Encounters when I arrived at Harvard in 1967, and from I first met George face-to-face during a conference that time we saw each other frequently. We had organized in September 1965 by and our private meetings, and we each represented Roe Goodman at Endicott House. Some 41+ years our departments on the Committee for Applied

834 Notices of the AMS Volume 54, Number 7 Mathematics, yet another opportunity to lunch voting member while still retaining my original together. affiliation with physics. At that point I began During 1968, Jim Glimm and I gave the first to interact with George even more. Following mathematical proof of the existence of the unitary George’s retirement in 1985 as the first occupant group generated by a Hamiltonian for a nonlinear of his named mathematics professorship, I was quantum field in two dimensions. This was a humbled to be appointed as the successor to problem with a long history. George’s old and George’s chair. I knew that these were huge shoes dear friend Irving Segal had studied this question to fill. for years, and he became upset when he learned of its solution. Government At a lunch during April 1969 George asked George often gave advice. While this advice might me my opinion about “the letter”, to which I re- appear at first to be off-the-mark, George could sponded, “What letter?” George was referring to defend its veracity with eloquence. And only after an eight-page letter from Segal addressed to Jim time did the truth of his predictions emerge. One and me but which neither of us had received at topic dominated all others about science policy: the time. The letter claimed to point out, among George distrusted the role of government fund- other things, potential gaps in the logic of our ing. published self-adjointness proof. On finally re- George often expressed interest in the fact that ceiving a copy of the letter from the author, I had a government research grant. I did this in I realized immediately that his points did not order to be able to assist students and to hire represent gaps in logic, but they would require extraordinary persons interested in collaboration. a time-consuming response. I spent consider- George often explained why he believed scientists able effort over the next two weeks to prepare a should avoid taking government research money. careful and detailed answer. His theory was simple: the funder over time will This put George in a difficult position, but his ultimately direct the worker and perhaps play a reaction was typical: George decided to get to role out of proportion. the bottom of the mess. This attitude not only When the government funding of research reflected George’s extreme curiosity but also his evolved in the 1950s, it seemed at first to work tendency to help a friend in need. It meant too reasonably well. It certainly fueled the expan- that George had to invest considerable time and sion of university science in this country during energy to understand the details of a subtle proof the 1960s and the early 1970s. At that time I somewhat outside his main area of expertise. And believed that the government agencies did a rea- for that effort I am extraordinarily grateful. sonable job in shepherding and nurturing science. It took George weeks to wade through the pub- The scheme attempted to identify talented and lished paper and the correspondence. Although productive researchers and to assist those per- he did ask a few technical questions along the sons in whatever directions their research drew way, George loved to work things out himself at them. This support represented a subsidy for the his own pace. Ultimately George announced (over universities. lunch) the result of his efforts: he had told his But over time one saw an evolution in the old friend Segal that in his opinion the published 1970s, much in the way that George had warned. proof of his younger colleagues was correct. This settled the matter in George’s mind once and for Today the universities have became completely all. dependent on government support. On the other We returned to this theme in the summer of hand, the government agencies take the initiative 1970 when George, Alice, and their daughter, to direct and to micro-manage the direction of sci- Ann, spent two long but wonderful months at a ence, funneling money to programs that appear marathon summer school in Les Houches, over- fashionable or “in the national interest”. George looking the French Alps. George (as well as R. Bott warned that such an evolution could undermine and A. Andreotti) were observers for the Battelle the academic independence of the universities, Institute, who sponsored the school. During two as well as their academic excellence and intellec- weeks I gave fifteen hours of lectures on the orig- tual standards. It could have a devastating effect inal work and on later developments—perhaps on American science as a whole. While we have the most taxing course I ever gave. That summer I moved far in the direction of emphasizing pro- got to know the Mackeys well, as the participants grams over discovering and empowering talent, dined together almost every day over those eight one wonders whether one can alter the apparent weeks. asymptotic state. Gradually my research and publications be- came more and more centered in mathematics Personal Matters than physics, and in 1973 the mathematics de- George spoke often about the need to use valu- partment at Harvard invited me to become a full able time as well as possible. And the most im-

August 2007 Notices of the AMS 835 portant point was to conserve productive time for special person abound throughout mathematics. work. Like me, George had his best ideas early in But they also can be heard over lunch at the the morning. I was unmarried when our discus- long table in the Faculty Club and at the weekly sions began, and George emphasized to me the luncheons at the American Academy. I am not need to have a very clear understanding with a alone. Everyone misses our fascinating luncheon partner about keeping working time sacrosanct. companion and friend.

David Mumford

To George, My Friend and Teacher* As a mathematician who worked first in algebraic geometry and later on mathematical models of perception, my research did not overlap very much with George’s. But he was, nonetheless, one of the biggest influences on my mathematical career and a very close friend. I met George in the fall of 1954—fifty-three years ago. I was a sopho- more at Harvard and was assigned to Kirkland House, known then as a jock house. In this un- likely place, George was a nonresident tutor, and

Photo courtesy of Arthur Jaffe. we began to meet weekly for lunch. My father had Ushers at the wedding of Arthur Jaffe, died three years earlier, and, my being a confused September 1992, (left to right): , and precocious kid, George became a second fa- Bernard Saint Donat, George Mackey, Arthur ther to me. Not that we talked about life! No, he Jaffe, , Konrad Osterwalder. showed me what a beautiful world mathematics is. We worked through his lecture notes, and I ate them up. He showed me the internal logic George also described at length how he enjoyed and coherence of mathematics. It was his person- his close relationship with Alice and how they en- al version of the Bourbaki vision, one in which joyed many joint private activities, including read- groups played the central role. Topological vector ing novels to each other, entertaining friends and spaces, operator theory, Lie groups, and group relatives, and traveling. He also described how he representations were the core, but it was also the even limited time with daughter Ann. But when lucid sequence of definitions and theorems that he was with Ann, he devoted his total attention to was so enticing—a yellow-brick road to more and her to the exclusion of all else. more amazing places. George floated multiple warnings about mar- This was my first exposure to what higher riage that I undoubtedly should have taken more mathematics is all about. I had other mentors— seriously. But years later when I remarried, George , who radiated the mystery of math- served as an usher on that occasion; he even end- ematics; Grothendieck, who simply flew—but ed up driving the minister to the wedding in the George opened the doors and welcomed me into countryside. Afterwards George shared a surpris- the fold. In those days he led the life of an English ing thought: my wedding was the first wedding don, living in a small apartment with one arm- that he thoroughly enjoyed! In honor of that con- chair and a stereo. Here was another side of the vivial bond, I wore the necktie chosen for me and life of the intellectual: total devotion to your field, the ushers at my wedding at my presentation in which was something I had never encountered so the Special Session for George in New Orleans. intensely in anyone in my family’s circle. When Shortly after George retired, I served as depart- I graduated, my mother came to Cambridge and ment chair. At the beginning of my term I made wanted to meet one of my professors. We had a strong case that the department needed more lunch with George. After that, she said, “This is office space, as several members had no regular what I always thought a Harvard professor would office. Within a year we were able to construct be like, the real thing”. seventeen new offices in contiguous space that Back in the 1960s, government funding of had been used for storage and equipment. But mathematical research was just starting, so of before that happened, I had to ask George if he would move from his large office of many years is University Professor in the Division of to a smaller one next door. As usual, George at Brown University. His email ad- understood and graciously obliged. dress is [email protected]. George’s straightforward analysis of the world *This note is adapted from David Mumford’s address at left one completely disarmed. Memories of this G. Mackey’s memorial.

836 Notices of the AMS Volume 54, Number 7