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# Memories of Calvin C. Moore

Calvin Cooper Moore (“Cal” to his friends), had extraordinary talents for both mathematics and administration. His most notable legacy is his crucial contribution as co-founder of the Mathematical Sciences Research Institute (MSRI) at the top of the Berkeley Hills.

Cal was born on November 2, 1936, in New York City. His parents were Ruth Miller Moore and Robert Allen Moore. (His earlier ancestors came from Germany and Switzerland.) His father, an MD specializing in pathology, was an eminent administrator. He served as dean of the School of Medicine of Washington University, St. Louis, from 1946 to 1954, and then was named vice chancellor of the Schools of Health at the University of Pittsburgh. In 1957, he became president of the Downstate Medical Center and dean of the College of Medicine of the State University of New York until 1966 when he retired. So it is not surprising that Cal himself had extraordinary administrative talents. Cal had an older (by 11 years) brother, Richard Allan Moore. Richard served in WWII and in 1953 earned a PhD in mathematics at Washington University in St. Louis. After postdocs, he went to the Carnegie Institute of Technology, now Carnegie Mellon University, and had a distinguished career before dying at age 94. Cal grew up in St. Louis, was a good student, and played tennis and swam.

Cal received a BA in 1958 from Harvard. Already as an undergraduate he began research under the guidance of George Mackey, and so he was able to complete his doctoral dissertation at Harvard in two years, receiving his PhD from Harvard in 1960. The title of his dissertation was “Extensions and Cohomology Theory of Locally Compact Groups.” It grew into the papers

Cal spent one year at the University of Chicago and then accepted an assistant professorship at UC Berkeley in 1961. He spent the 1964–65 academic year at the Institute for Advanced Study, and was promoted to associate professor with tenure in 1965 and almost immediately to professor in 1966, testifying to his excellent research record. Over the next years 13 students received their PhD under Cal’s mentorship.

However, Cal’s administrative talents were soon recognized, and he was appointed as the Dean of Physical Sciences in 1971 (only five years after receiving tenure), for a five-year term. (He had previously been vice chair under John Addison’s chairship of the Mathematics Department.) This was a tumultuous time on the campus and for the department. In 2007, Cal published a very thorough history of the Berkeley mathematics department, *Mathematics at Berkeley*

Cal was first married in the 1960s. After a divorce in the late 1960s, his ex-wife Eva and their eight-year-old daughter Joanne died tragically in 1975. In 1974, Cal married Doris Fredrickson, who was the chair’s secretary in the Mathematics Department (she also made the costumes in the photos in Figure 1).

In spite of all his administrative work, Cal maintained a vigorous research program. While cohomology and extensions of topological groups remained a thread through much of his later research, he made important contributions concerning representations and harmonic analysis for both Lie groups and algebraic groups, ergodic theory for group actions, foliated spaces, and algebras of operators on Hilbert spaces, as well as other related topics. His contributions will be described in more detail in later sections of this Memorial Tribute. Cal was elected to the American Academy of Arts and Sciences in 1973, and he was elected a Fellow of the American Association for the Advancement of Science in 1979.

Many think that Cal’s crowning achievement was his role (together with Shiing-Shen Chern and Isadore Singer) in cofounding MSRI (now SLMath). He had much influence in shaping MSRI, and he was the leader in dealing with the University administration, the NSF, and the national mathematics community. (There is an extensive discussion of MSRI in

In May, 1985, Cal was asked to become assistant vice president for academic affairs at the Office of the President of the University of California (UCOP). At the time he was deputy director of MSRI and was reluctant to leave that position, having done so much to get the Institute up and running. But the opportunity at UCOP was too tempting to pass up. He then became the associate vice president for academic affairs in 1986, and stepped down from that position in 1994.

In 1996, Cal became chair of the Math Department for the first time, serving two three-year terms due to his excellent leadership. (Some of his friends kidded him that he had finally taken on the hardest job at the university. And he said of it, “There is no place to hide.”) Cal later served as the chair of the Audit Committee of MSRI from 2005 to 2017.

Cal retired to emeritus status in 2004, but continued to be very active in departmental affairs as a wise and respected colleague until only a few years ago when his health was declining. Cal died on July 26, 2023. He is survived by his wife Doris Moore and his nephew Peter Moore.

Cal Moore had a long and distinguished career as both a mathematician and an administrator. But I knew him only several years after his retirement.

I was a PhD student at UCLA between 2006 and 2010, studying the interaction of ergodic theory and additive combinatorics. I was drawn more and more into abstract ergodic theory, but was still only vaguely aware of Cal’s contributions to the field during most of that time.

In late 2009, I came upon a problem with a flavour that was new to me. It came down to classifying all those ‘joinings’ between several copies of a measure-preserving system that had certain additional symmetries. I eventually realized that the heart of the problem seemed to require a certain degree-two group cohomology class to vanish. But the class was defined on an arbitrary compact metric Abelian group; it took values in the circle group; and the cocycles and coboundaries had to be Borel measurable. So this was not the classical group cohomology that I had met in passing while learning abstract algebra.

Then I discovered Cal’s thesis

I wrote to Cal himself, describing the issue and asking whether he knew anything about it. He replied quickly and warmly, but in the negative. We kept in touch, however, and over the next year our correspondence turned into a full collaboration. Cal realized that a method for solving my problem could have several other applications, including to a question from his own earlier work: whether measurable and continuous cohomologies coincide when a locally compact group acts on a Fréchet space. The eventual result was a fairly long paper containing several related theorems, included answers to my original question and Cal’s earlier one

Cal brought both immense knowledge and real wisdom to this process. He shared both of these with me freely and without a hint of condescension, despite my general ignorance as a newly graduated PhD. In the process I think he taught me as much about writing papers, and about collaboration in general, as he did about group cohomology. Our time working together was a great pleasure, and offered me real support as I tried to establish a research agenda of my own. I look back on it now as a very special example when working with or mentoring more-junior colleagues.

I met Cal in person only once, in October 2011, while I was attending a workshop at MSRI and spent one day down at Evans Hall. He was still coming in regularly, and had the courtesy to book me an office even for such a short visit, so that I would have “a place to hang my hat” that day. We chatted for quite a while about our joint project, but also more widely about mathematics, and his enthusiasm and sense of mathematical elegance made their strongest impression on me that day. I know that both will be sorely missed.

Tim Austin is a professor of mathematics at the University of Warwick. His email address is Tim.Austin@warwick.ac.uk.

I was a graduate student in Berkeley in 1970–1975. My first real encounter with Cal Moore was in the spring of 1973. I was ready to begin thesis work, but none of the obvious candidates for advisor were available. I finally made an appointment with Cal to ask him for advice on what I should do; he was dean at the time, and it never occurred to me that he himself could be a candidate for advisor. I had had a couple of brief dealings with him previously, so we were slightly acquainted. I had previously found him a little intimidating (an opinion I completely changed later), and, besides, anyone with a crew cut in Berkeley in the 1970s was a little suspect anyway. I explained my situation and asked him what he thought I should do, and he replied, “Well, I’m looking for a student.” Within 15 minutes I had an advisor. My head was swimming when I left his office.

Despite his administrative duties, he did very well by me; he did everything a good advisor should do, giving me help and guidance and making sure I learned what I should know. We had a bit of tension about what topic I should concentrate on: my primary interest was (and still is) operator algebras, and he tried to steer me more toward group representations. I’m happy he did this, since I was exposed to a lot of beautiful mathematics I might not ever have learned otherwise. We finally compromised on studying representations of restricted direct product groups such as matrix groups over rings of adèles (a topic I knew something about from work I had done as an undergraduate) and the and von Neumann algebras they generate, relating to the then-current classification of factors by Araki and Woods and later Connes, and leading to subsequent work I did on infinite tensor products of -algebras -algebras.

He was helpful on a personal level too. For example, one quarter toward the end I found that if I had a certain grad status, I could avoid registering as a student. Other students commonly did this, although it could be regarded as gaming the system. I was hesitant to bring up the possibility with Cal, since I expected that as an administrator he would take a dim view of it; but his reaction when I told him it was something other students were doing was simply to reply rather sadly, “So it’s come to that, has it?” And he went along with it.

I looked forward to discussing mathematics with him during our weekly meetings, and he seemed to enjoy it too as a break from his administrative duties. In fact, I rather naively started to get the feeling that I was becoming his principal remaining connection with the world of mathematics. I was disabused from this notion one day in rather spectacular fashion: I went in for my weekly meeting, we said hi, and he said, “Before I forget, I have something to give you.” He pointed to a huge cardboard box on the floor of his office, and said, “Open the box and take one of each.” I opened the box, and inside were stacks of seven new preprints which had just come back from University printing (this was of course before the days of word processing and the internet). So it seems he was finding time to do a little mathematics after all, besides just talking to me. I started to gain more understanding and appreciation for the extreme level of energy he put into everything.

I gradually also appreciated that he was a more sympathetic person socially than I had initially given him credit for, despite his conservative persona. For example, far from being reactionary, he was a strong advocate for diversity in mathematics. And I was surprised when he argued against establishing the AMS Fellows program on the grounds that it smacked of elitism (I strongly agreed with him on this). But we were still a bit of an odd couple in many ways (which did not decrease my respect for him).

After I finished my degree, I had only sporadic contact with him. My biggest interaction was during the special year for Operator Algebras at MSRI in 1984–85, while the permanent building was under construction and then newly opened. I didn’t actually see too much of Cal, who was deputy director (and really functionally director), but I appreciated how much effort he put into getting MSRI off on a solid footing that year.

One other notable point of contact was in the early 1990s, when he was in the president’s office. My department in Reno came due for a program review. I suggested Cal’s name for the review team, and he agreed to chair the team. While the team was on campus I ran into our dean, who I don’t think knew that I even knew Cal. I asked her how she thought the review was going, and she said pretty well, and added, “The review team is good. And the guy who’s the chair, he’s *really* good.” In his oral report to the department and administration at the end of the visit, Cal said something which has stuck with me (although unfortunately has not always stuck with our administrators): “You can’t have a University without a math department. But more than that, you can’t have a *good* University without a good math department.”

Bruce Blackadar is a professor emeritus of mathematics at the University of Nevada, Reno. His email address is bruceb@unr.edu.

Calvin and I published a paper *Acta Crystallographica*, in 1995. It may be the only paper by Calvin in a non-mathematical journal. I will explain the story behind this paper, which deals with using higher-order correlations in the determination of three-dimensional structures such as DNA. Several Nobel prizes have been awarded for this kind of work where one usually assumes that only correlations of order are available, but one such prize was given in 1985 for work using correlations of order Let me start at the beginning. Back in 1974, when I was considering an offer from Berkeley, I had a chat with the dean at that time: Calvin Moore. Fairly soon our conversation turned to what I was doing, and almost immediately we started working together trying to prove that correlations of order . will do the job.

We labored on this for about two years, but we could not prove our result in any generality and we came to a friendly divorce. For about 20 years we never talked to each other about this issue. I attended a meeting in Italy where someone gave a private talk claiming to prove that correlations of order were enough. Calvin and I knew that this was not true. When I got back to Berkeley, I talked to Calvin about our old project and we started from scratch. We soon discovered that what we had been trying to prove was also false, but after more work we got the result that correlations of order do the job. By the time that we were doing our second round on this problem, Calvin had moved to the president’s office at the University of California. I would regularly visit him there and we would hide away in some small office and work. I was always amazed at Calvin’s ability to separate neatly two very different kinds of lives.

Now that Calvin is no longer with us I have gone back to our paper where we close by saying that we will try to extend that work in several directions. In particular we talk there about going away from the case of the discrete Fourier transform, relevant in the case of crystallography, to the case of non-abelian groups and their symmetric spaces such as the sphere. Back then we could not think of any interesting applications, but recently I ran into the work of J. Peebles, Nobel Prize in Physics 2019, who back at the time when Calvin and I started on this was looking at correlations of order on the surface of the sphere in connection with the problem of cosmic background radiation. It would be a nice way of honoring Calvin to make further progress in this problem.

F. Alberto Grünbaum is a professor emeritus of mathematics at the University of California, Berkeley. His email address is grunbaum@berkeley.edu.

The short version is: Cal was a superb thesis advisor for me. But it was not only through that that he helped further my career. He took an interest in my progress and aided and influenced me for a long time afterward. I have to confess that when I first encountered Cal, during my first year at Berkeley, I found him a little scary, so he was not the first person I approached about the possibility of supervising my dissertation research. I spent most of my second year talking to another faculty member who seemed much more approachable. But at the end of the year, all I had was a sheaf of five pages of notes sketching various ideas, none substantial enough to develop into a thesis. So, over the summer of 1968, I asked Cal if he would be willing to be my advisor. He not only accepted, but immediately had a project to suggest. As an undergraduate, I had become interested in representation theory, from several courses, but especially from a reading course I had taken with George Mackey (who had been Cal’s thesis advisor). I was especially taken with the orbit method invented by A. A. Kirillov for describing the unitary dual of nilpotent Lie groups. The project that Cal described to me tied in to all of that. It also involved groups and adèles, which introduced me to ideas that were key to some of my later work. The project Cal pointed me to concerned the multiplicities of irreducible representations of a nilpotent group acting on a compact nilmanifold — a manifold -adic where , is a (connected, simply connected) nilpotent Lie group, and is a discrete subgroup such that the quotient space is compact. I was making good progress on the question, and as I worked, I got the feeling that a version of Kirillov’s theory might have strong implications for the representation theory of the discrete subgroups. So I started thinking about that, and was able to show that, given certain restrictions, that still allowed a large selection of examples, strong analogs of Kirillov’s results could be obtained. This was particularly interesting, since these discrete groups are not “Type I,” a technical condition imported from the theory of von Neumann algebras, and such groups were broadly thought to be pathological. These examples showed that even for such “bad” groups, one could say something interesting about their representation theory. It was this work that eventually became my thesis. I was able to finish by the end of 1969. The thesis was not quite complete at the start of the usual period for applying for jobs, so I was a little late with my applications, but I did get an offer from Stony Brook, with its ambitious new chair, James Simons. In September, we moved to Port Jefferson, Long Island, and I started teaching, which I had not had to do in graduate school, making it easier to finish relatively quickly. But over the summer between graduation from Berkeley and going to Stony Brook, thanks to Cal, I was able to attend two high-level conferences. One was in Seattle, at the Battelle Institute, and focused on mathematical physics. Cal was one of the main speakers. He discussed restrictions of representations to subgroups, emphasizing the implications for ergodic theory. This was my introduction to these ideas, which were valuable for later work.

The second conference was focused on representation theory of semisimple Lie groups, especially the work of Harish-Chandra. It was held in Namur, Belgium, giving Lyn and me the opportunity to see this part of the world, enjoy Belgian French fries, and meet several people with whom I developed lasting relationships, perhaps most notably Paul Sally. During these summer conferences, I heard about the new ideas of Langlands that were catching peoples’ attention, and the resulting interest in representations of groups. The Stony Brook Math Department was full of geometers, which was Simons’ area. I seemed to be the only person with a primary interest in representation theory, but there were several people, including Ron Douglas and Bill Helton, who worked on operator theory, which I had also studied quite a bit, so I started talking with them. This resulted in several of my earlier papers being in operator theory — one with Douglas, on Toeplitz operators on the quarter plane, and two with Helton, on integral operators on the line, which uncovered some nice phenomena that helped make the connection between operator theory and -adic that was strongly exploited by Alain Connes. My time at Stony Brook was interrupted by two leaves, the first in 1971–1972 at IAS in Princeton, meeting Harish-Chandra and attending his lectures, and forming strong friendships with Paul Sally and Joe Jenkins. I also met Jacques Tits there, and though we did not interact mathematically, he very kindly arranged for me to be invited to the Sonderforschungsbereich at the University of Bonn, run by Friedrich Hirzebruch, for 1973–74. I had realized that the approaches I had used to study representations of discrete nilpotent groups might also apply to -theory groups, or at least their open compact subgroups. This connection was further strengthened by hearing about Harish-Chandra’s philosophy of cusp forms, and supercuspidal representations, and the possibility that some supercuspidal representations might be induced from open compact subgroups. So during my IAS year, I worked pretty hard on -adic representation theory. This led me to appreciate the central nature of the Heisenberg group and the associated (projective) representation of the symplectic group, which was first strongly brought to the attention of mathematical research by David Shale and Irving Segal, in the context of quantum field theory, and then by André Weil, in the context of automorphic forms, especially Siegel modular forms. Although the main paper that came out of that year is titled “On the character of Weil’s representation,” I have taken to calling this representation the “oscillator representation.” During my Bonn year, I found myself continuing to think about the oscillator representation. Its many connections with mathematics and physics have occupied the bulk of my research since then. In particular, from results about holomorphic representations coming from several authors (Steven Gelbart, Kenneth Gross and Ray Kunze, Masaki Kashiwara and Michèle Vergne) and also from some numerical calculations which Jacques Tits and Günter Harder helped me understand, I was led to formulate conjectures about “dual pairs” (maximal pairs of mutually centralizing subgroups) of the symplectic group, which were the focus of my work for many years. -adic

In between IAS and Bonn, I continued to work on representations. Toward the end of the year, I decided to visit Langlands in New Haven, to find out if/how the results I was finding might connect with his ideas. When I mentioned that my thinking had started with nilpotent groups, and that it was related to some dissertations that Dan Mostow had directed, he took me to visit Dan. I guess I made a good impression, because Dan, who was the chair of the Yale Mathematics Department at that time, started a process that led to an offer to join the department, which I did directly after returning from Bonn. I remained at Yale until I retired in 2016. This lengthy excursion into -adic representation theory and operator theory had substantially delayed the preparation of my thesis results on nilpotent groups. My first published paper was a direct outgrowth of the thesis problem Cal had posed to me, but the related work on representations of nilpotent discrete groups, which formed the actual body of my thesis, was at first rejected for being poorly written, and a suitable revision had languished. Cal had noticed this, and when he visited Yale in 1976, he urged me to get these results into print. I explained the problem I was having balancing doing revisions with new work. At that time, Cal was serving as an editor for the -adic*Pacific Journal*, and he suggested that I should submit the papers there. So I did, with the result that eight of the delayed papers appeared in the December, 1977 issue. While at IAS, I also learned some more about ergodic theory and its connection with representation theory. I wrote up some of the ideas this stimulated in a preprint, which I circulated a little, in particular to Cal. At the 1977 AMS summer conference in Corvallis, Oregon, Cal mentioned that he thought some of the results of my preprint could be generalized, to establish a very general ergodic theorem for flows on homogeneous spaces. I was happy to get a chance to work with Cal, and with his leadership and energy, we soon produced the paper

During the 1980s, Cal was quite busy with administration at Berkeley, and also with starting the Mathematical Sciences Research Institute (now the Simons-Laufer Mathematical Sciences Institute). I benefited from several visits to MSRI. The first was in 1984, before its current beautiful building in the Berkeley Hills was complete. It was a conference celebrating the work of George Mackey. One very nice feature was that it was attended by five generations of mathematicians: Mackey; his advisor, Marshall Stone; Mackey students, represented by Cal; Mackey grandstudents, including me; and a Mackey great-grandstudent, Norman Wildberger, now at UNSW in Australia (see Figure 2).

During this period, Cal became the editor of the Research Announcements (RA) section of the *Bulletin of the AMS*. As his duties at Berkeley grew, he found he was not able to keep that up. In 1988, he asked me to take over for him. Although my editorial skills were limited, I agreed and served as RA editor for three years. This had a significant and unexpected impact on the direction of my career. In the middle year of such a position, one is a member of the Council of the AMS, so I attended the Council meeting at the Winter Meeting in 1989. This was the year that NCTM was publicizing their newly written *Standards for Mathematics Education*. They made a presentation about them to the Council, seeking endorsement from the AMS. I was surprised to learn that mathematics education in the US was so problematic, although I could see some evidence of this in my son’s experiences. I started thinking about the problem, and gradually got more and more involved with mathematics education.

The first result, which still gives me satisfaction, was a concerted effort to improve my own instruction. This took several years, but eventually I did manage to become more effective as a teacher. This was good for my students, for me, and for the Yale Mathematics Department. Another result was my involvement in several national mathematics education activities, including the Mathematical Sciences Education Board, run by the NRC; a committee of the AMS that reviewed the revision of the NCTM Standards in the late 1990s; recruitment to the NRC Study Committee asked to review the research on mathematics education by the Department of Education, resulting in the publication of *Adding It Up* *Notices of the AMS* the book *Knowing and Teaching Elementary Mathematics* by Liping Ma, which had a huge impact on the US mathematics education community. (Eventually, this led to my being recruited by Texas A&M University, to lead a project there to strengthen the mathematics preparation of their elementary teaching majors.) Also, MSRI decided to promote the involvement of mathematicians in mathematics education, and I was a member of their board of advisors, and participated in several mathematics education conferences there during 2000–2010. This gave me several chances to see Cal during that period.

In summary, both for his guidance during my graduate studies, and for his continuing beneficial attention in the half century since, I am enduringly grateful to Cal.

Roger Howe is a professor emeritus of mathematics at Yale University. His email address is roger.howe@yale.edu.

I arrived in Berkeley as a grad student in 1973, thinking that it would be good to work on aspects of representation theory or operator algebras using somewhat “modern” techniques, and that’s how I ended up as a student of Marc Rieffel. But I also met Cal Moore, who at that time was serving as Dean of Physical Sciences, and so his office wasn’t in Evans (the mathematics building) but next door. In spite of his major administrative job, which greatly limited the amount of time he had available for mathematics, Cal agreed at one point to supervise a reading course for me to study his papers on cohomology of locally compact groups

Now it’s hard to recall what things were like before the internet, but back in the 70’s, there were only three ways to find out what someone else was working on: to hear them or someone else talk about it (informally or in a seminar or conference), to get a preprint in the mail, or to see a new paper after it appeared in print. Some time after my reading course with Cal, I was browsing through the new journals in the library and came across a short paper

I came back to Berkeley in 1984–1985 for a stay at the new Mathematical Sciences Research Institute (MSRI, now renamed SLMath), and that gave me another chance to interact regularly with Cal. At this point he was deputy director of MSRI, but in fact he was the *de facto* executive director, dealing with all sorts of things such as the finishing touches on the new building in the Berkeley hills as well as personnel matters and helping to run the scientific program. What I remember most is how effortless Cal made administration seem, and how easily he could move back and forth between administration and thinking deeply about mathematics. While there are plenty of people who can do one or the other, Cal was very unique in being able to do both, and to switch effortlessly between them. That made him a very unusual mathematician, one who will be sorely missed.

Jonathan Rosenberg is a professor of mathematics at the University of Maryland. His email address is jmr@umd.edu.

I first met Cal Moore in late 1979. I was spending 1979–1980 at UCLA for a special year in operator algebras run by Ed Effros and Masamichi Takesaki. I had some friends at Berkeley and had never been there, and so I drove up from Los Angeles to spend a few days there.

At the time, I was developing a spectral sequence to use to study the of -theory I had already tried explaining the idea to a few of my functional analysis friends and hadn’t gotten far. (At that time there were only a very few functional analysts who were familiar with “high-tech” algebraic topology.) I wanted to meet Cal, so I knocked on his door and introduced myself. When he asked me what I was working on I was hesitant to say. To my delight, Cal knew what spectral sequences and -algebras. were and what they were good for. He understood clearly the statement of the result and the idea of the proof. This was my first experience with an analyst who understood spectral sequences, and I was thrilled. -theory

Several months later I finished writing up the paper and had in mind some other papers that I wanted to write on “Topological Methods for I decided to submit the first in the series, the spectral sequence paper, to Cal for the -algebras.”*Pacific Journal*. I mailed it to him (yes, we used mail in those days) and got back an immediate acceptance. He told me that he didn’t need to send it out to a referee since he already knew the content! Eventually *PJM* published four papers in the series, with Cal as the editor. It all went very smoothly.

Back to UCLA. Effros and Takesaki ran a seminar for the 1979–80 academic year on Alain Connes’ brilliant paper “Sur la théorie non commutative de l’intégration” *proof* of the theorem. By the end of the year the goal had shifted to trying to understand the *statement* of the theorem.

As the academic year ended, we heard that there had been a similar seminar taking place at Berkeley, led by Cal. It occurred to me that perhaps we could team up to write a book which would provide the necessary background to understand the statement of Connes’ theorem and then to prove the theorem in detail. So I contacted Cal. He liked the idea and agreed at once to team up. Later several people told me that Cal was hard to approach, but I certainly didn’t have that experience.

Connes’ theorem is a generalization of the Atiyah-Singer index theorem. One starts with a compact smooth manifold which is foliated, meaning very roughly that locally it looks like and the local extend around the manifold creating s smooth manifolds called leaves. For example, the torus -dimensional can be foliated by copies of the circle. However, if one picks an irrational angle and launches a vector field from some point then its path will be a copy of itself, and can be foliated by copies of winding around an infinite number of times.

Connes considered the situation where each leaf of the foliated manifold had an elliptic operator associated to it, and these operators varied measurably as one moved from leaf to leaf. How is the analytic index even defined? The naïve picture is to

- 1.
Take the Fredholm index of each operator and ,

- 2.
Average these indices to get the analytic index of the operator .

There are two serious problems with this approach. First of all, the leaf may not be compact, and hence and/or may be infinite. Second, a direction transverse to the leaves may not be smooth or continuous or even measurable. Connes solved both of these problems. Very very roughly, to the operator he associates an index *measure*. He also assumes one is given an invariant transverse measure. Pairing one with the other and then integrating over the compact manifold yields a real number, the analytic index of On the topological side, there will be some sort of symbol class associated to each . and then one needs a generalization of the classical Chern-Weil theory of characteristic classes to define the topological index of another real number. (You can see why understanding the ,*statement* of the theorem was a challenge at UCLA!) Connes’ index theorem asserts that these two indices coincide. The proof requires for -theory and a lot of analysis. -algebras

Our plan for the book was to have alternating chapters on topology and analysis, so that experts in one of the areas could skim some chapters and study the others in detail. Cal knew all of the measure theory and global analysis needed. I got to work on the topological side. But—it is not so simple, and of course the analysis and the topology are very significantly interwoven. The fact that the leaves could not be assumed to be compact made for very extensive work, both on the analysis and the topology side. We mostly worked by mail, though Cal did come to Wayne State, where I was based, and I spent a couple of weeks in Berkeley. (I stayed for a while at Cal’s home. This was complicated for me because I keep kosher, but Cal and Doris were extremely hospitable and went out of their way to meet my religious dietary needs.)

At some point I got totally confused about what was happening locally on forgetting that I was supposed to be differentiating in the , direction but not in the direction, and so I started writing Eventually we realized that we could phrase the whole theorem in that context. We called the resulting spaces .*foliated spaces* and we allowed to be, for instance, a Cantor set. We knew examples of such spaces (they are sometimes called *laminations*) but I can’t say that we foresaw applications of the Index Theorem for foliated spaces in physics!

We got some very substantial help along the way from colleagues and friends, and we were especially helped by Bob Zimmer, who wound up authoring an appendix of the book and coauthoring another.

The book took several years to produce. Cal had other things to do—for instance building MSRI from scratch. He was pretty busy in the 1980s. He was deputy director of MSRI from 1981–1985 from the beginning and through its 1985 move from Fulton St. to its permanent home at the top of the Berkeley Hills. From 1985–1994 he was assistant and then associate vice president of the University of California system. Eventually the book was published by Springer in the MSRI series in 1988 and titled *Global Analysis on Foliated Spaces*

In 1987, I became an associate dean of the liberal arts college at Wayne State. The college had 29 departments, 400 faculty, and around 2000 employees. I was associate dean in charge of money. I frequently consulted with Cal on how to do my job, and he was invariably helpful. I think that the most valuable advice that he gave me was his wonderful slogan “*Never assume malice when pure incompetence will explain everything*,” which helped me many times during my five years as dean and beyond! I learned Cal’s way of influencing the direction of things without attracting undo attention. He was always modest in my experience—for example, giving S.S. Chern and Is Singer maximum credit for the establishment of MSRI, though as far as I could see, Cal did most of the actual work.

The years passed, we would correspond occasionally, but we worked on different projects. Then around 2000, Cal told me that Cambridge University Press was taking over the publication of some of the MSRI series and wanted to do a second edition of our book. By then our generalization of Connes’ theorem to foliated spaces had found important applications, particularly in mathematical physics with the gap labeling theorem. So we got to work. The book had to be totally retypeset (and I spent many pleasant winter afternoons proofreading the new text on the Haifa beach—the closest to the Technion, where I was on sabbatical). More importantly, we made a few corrections and inserted comments at the end of each chapter devoted to developments since the first edition was published. Most of this work was by email and went very smoothly. The second edition appeared in 2004

Cal was always a pleasure to work with. He never seemed to complain when I asked him to explain yet again some mathematical concept or technique. He was always cordial and welcoming to me. While at Berkeley, he introduced me to luminaries like Chern and Mackey. He loved to go to a particular coffee shop in Berkeley after lunch (I don’t like coffee but learned to like cappuccino). I remember one time going with Cal and Serge Lang, who proceeded to get into a hot argument that somehow mixed politics and math in ways that I did not understand!

I am very grateful that I had the privilege to work with Cal and to learn from him over a period of many years. He was a great mathematician and a great person.

Claude Schochet is a professor emeritus of mathematics at Wayne State University and a visiting professor of mathematics at the Technion. His email address is clsmath@gmail.com.

I met Cal and started working with him over 40 years ago, at the beginning of MSRI. This was long ago, at the end of the 1970s. Back then, I was the National Science Foundation program director with responsibility for overseeing the new math institutes.

It is hard to overstate the significance and importance of the role Cal played in the beginnings of MSRI. Cal and I worked closely together, and from the beginning, I was impressed by his vision and dedication to what MSRI has become.

Self-promotion and publicity seeking were not aspects of Cal’s persona. He saw what needed to be done, what should and could be done, and he proceeded to do it.

Cal was central in the creation and infancy of MSRI. There was no one more so. He was aware of the history and the controversies leading to the establishment of MSRI, including its impact on the Math Section’s budget. His sensitivities to the occasional needs for diplomatic behavior and responses were admirable. It is well known that MSRI was initially housed in an office building in downtown Berkeley. What is perhaps less well-known is that an early NSF site visit team met with Chancellor Heyman and expressed strong concern about the MSRI facility. I recall some moderately contentious discussions with Chancellor Heyman in which the NSF site visitors made clear that the ultimately chosen site would be for the flagship institution for Mathematical Sciences research. I do not know the precise roles that the MSRI leadership had in discussions with the Chancellor’s office regarding the MSRI site. I do remember being notified, by Cal, of the (spectacular) proposed site, near the primate research center, now the Field Station for the Study of Behavior, Ecology and Reproduction, FSSBER. (See Figure 3 for a picture of the groundbreaking ceremony, as well as Figure 4, one of my favorites, showing Cal and me at the site in an unusually playful mode.)

Cal understood the uniqueness and opportunities of having a semi-independent relationship with the University. It’s my conjecture that he was the main proponent of the idea to present that structure as part of the proposal to NSF. That structure continues through today. I wonder how under appreciated this may be. There were many meetings involving NSF staff, the NSF advisory committee, and the site visit committee (early on, shortly after the 1981 award). As I said, the relationship between MSRI and NSF was without precedent, and was unique: NSF’s dealings were with an academic unit that was not part of the UC-Berkeley administration. Cal and I had an unusual relationship. We were often on different sides of an issue, but there was never any ill feeling or lack of respect in terms of actions and reactions. Cal was engaged, responsive, always supremely ethical. SLMath/MSRI is now in an extraordinary building and on an extraordinary site. But it’s important to be aware that, from the beginning several decades ago, the building quality and attractiveness was already exceptional. Cal was responsible for much of it. It was he who did the initial staff hiring, managed virtually everything, and most importantly set the tone. His title was deputy director, but we at NSF generally thought of him as the one in charge. Indeed, when the second director was to be chosen, many thought that Cal would be named director and were surprised that he was not. My work with MSRI was among the most satisfying periods of my life. It was an honor and a pleasure to have worked with Cal. He will always be remembered.

Alvin Thaler was a program director in both the mathematical and computer science divisions of the National Science Foundation (NSF). His email address is alvinthaler@gmail.com.

## References

[ AM13] - Tim Austin and Calvin C. Moore,
*Continuity properties of measurable group cohomology*, Math. Ann.**356**(2013), no. 3, 885–937. MR3063901,## Show rawAMSref

`\bib{MR3063901}{article}{ author={Austin, Tim}, author={Moore, Calvin~C.}, title={Continuity properties of measurable group cohomology}, date={2013}, issn={0025-5831,1432-1807}, journal={Math. Ann.}, volume={356}, number={3}, pages={885\ndash 937}, url={https://doi.org/10.1007/s00208-012-0868-z}, review={\MR {3063901}}, }`

[ BM00] - M. Bachir Bekka and Matthias Mayer,
*Ergodic theory and topological dynamics of group actions on homogeneous spaces*, London Mathematical Society Lecture Note Series, vol. 269, Cambridge University Press, Cambridge, 2000, https://doi.org/10.1017/CBO9780511758898. MR1781937,## Show rawAMSref

`\bib{MR1781937}{book}{ author={Bekka, M.~Bachir}, author={Mayer, Matthias}, title={Ergodic theory and topological dynamics of group actions on homogeneous spaces}, series={London Mathematical Society Lecture Note Series}, publisher={Cambridge University Press, Cambridge}, date={2000}, volume={269}, isbn={0-521-66030-0}, url={https://doi.org/10.1017/CBO9780511758898}, review={\MR {1781937}}, }`

[ BG74] - I. D. Brown and Y. Guivarc’h,
*Espaces de Poisson des groupes de Lie*, Ann. Sci. École Norm. Sup. (4)**7**(1974), 175–179. MR427538,## Show rawAMSref

`\bib{MR427538}{article}{ author={Brown, I. D.}, author={Guivarc'h, Y.}, title={Espaces de Poisson des groupes de Lie}, journal={Ann. Sci. \'{E}cole Norm. Sup. (4)}, volume={7}, date={1974}, pages={175--179}, issn={0012-9593}, review={\MR {427538}}, }`

[ Con79] - Alain Connes,
*Sur la théorie non commutative de l’intégration*(French), Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978), Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 19–143. MR548112,## Show rawAMSref

`\bib{MR548112}{article}{ author={Connes, Alain}, title={Sur la th\'{e}orie non commutative de l'int\'{e}gration}, language={French}, conference={ title={Alg\`ebres d'op\'{e}rateurs}, address={S\'{e}m., Les Plans-sur-Bex}, date={1978}, }, book={ series={Lecture Notes in Math.}, volume={725}, publisher={Springer, Berlin}, }, date={1979}, pages={19--143}, review={\MR {548112}}, }`

[ Cou01] - National Research Council,
*Adding it up: Helping children learn mathematics*(Jeremy Kilpatrick, Jane Swafford, and Bradford Findell, eds.), The National Academies Press, Washington, DC, 2001, https://nap.nationalacademies.org/catalog/9822/adding-it-up-helping-children-learn-mathematics.,## Show rawAMSref

`\bib{NAP9822}{book}{ author={Council, National~Research}, editor={Kilpatrick, Jeremy}, editor={Swafford, Jane}, editor={Findell, Bradford}, title={Adding it up: Helping children learn mathematics}, publisher={The National Academies Press}, address={Washington, DC}, date={2001}, isbn={978-0-309-21895-5}, url={https://nap.nationalacademies.org/catalog/9822/adding-it-up-helping-children-learn-mathematics}, }`

[ GM95] - F. Alberto Grünbaum and Calvin C. Moore,
*The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets*, Acta Cryst. Sect. A**51**(1995), no. 3, 310–323, DOI 10.1107/S0108767394009827. MR1331196,## Show rawAMSref

`\bib{MR1331196}{article}{ author={Gr\"{u}nbaum, F. Alberto}, author={Moore, Calvin C.}, title={The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets}, journal={Acta Cryst. Sect. A}, volume={51}, date={1995}, number={3}, pages={310--323}, issn={0108-7673}, review={\MR {1331196}}, doi={10.1107/S0108767394009827}, }`

[ HM79] - Roger E. Howe and Calvin C. Moore,
*Asymptotic properties of unitary representations*, J. Functional Analysis**32**(1979), no. 1, 72–96, DOI 10.1016/0022-1236(79)90078-8. MR533220,## Show rawAMSref

`\bib{MR533220}{article}{ author={Howe, Roger E.}, author={Moore, Calvin C.}, title={Asymptotic properties of unitary representations}, journal={J. Functional Analysis}, volume={32}, date={1979}, number={1}, pages={72--96}, issn={0022-1236}, review={\MR {533220}}, doi={10.1016/0022-1236(79)90078-8}, }`

[ Moo64] - Calvin C. Moore,
*Extensions and low dimensional cohomology theory of locally compact groups. I, II*, Trans. Amer. Math. Soc.**113**(1964), 40–63; ibid. 113 (1964), 64–86, DOI 10.2307/1994090. MR171880,## Show rawAMSref

`\bib{MR171880}{article}{ author={Moore, Calvin C.}, title={Extensions and low dimensional cohomology theory of locally compact groups. I, II}, journal={Trans. Amer. Math. Soc.}, volume={113}, date={1964}, pages={40--63; ibid. 113 (1964), 64--86}, issn={0002-9947}, review={\MR {171880}}, doi={10.2307/1994090}, }`

[ Moo76a] - Calvin C. Moore,
*Group extensions and cohomology for locally compact groups. III*, Trans. Amer. Math. Soc.**221**(1976), no. 1, 1–33, DOI 10.2307/1997540. MR414775,## Show rawAMSref

`\bib{MR0414775}{article}{ author={Moore, Calvin C.}, title={Group extensions and cohomology for locally compact groups. III}, journal={Trans. Amer. Math. Soc.}, volume={221}, date={1976}, number={1}, pages={1--33}, issn={0002-9947}, review={\MR {414775}}, doi={10.2307/1997540}, }`

[ Moo76b] - Calvin C. Moore,
*Group extensions and cohomology for locally compact groups. IV*, Trans. Amer. Math. Soc.**221**(1976), no. 1, 35–58. MR414776,## Show rawAMSref

`\bib{MR0414776}{article}{ author={Moore, Calvin~C.}, title={Group extensions and cohomology for locally compact groups. {IV}}, date={1976}, issn={0002-9947,1088-6850}, journal={Trans. Amer. Math. Soc.}, volume={221}, number={1}, pages={35\ndash 58}, url={https://doi.org/10.2307/1997541}, review={\MR {414776}}, }`

[ Moo07] - Calvin C. Moore,
*Mathematics at Berkeley: a history*, A K Peters, Ltd., Wellesley, MA, 2007, DOI 10.1201/b10579. MR2289685,## Show rawAMSref

`\bib{MR2289685}{book}{ author={Moore, Calvin C.}, title={Mathematics at Berkeley: a history}, publisher={A K Peters, Ltd., Wellesley, MA}, date={2007}, pages={xviii+341}, isbn={978-1-56881-302-8}, isbn={1-56881-302-3}, review={\MR {2289685}}, doi={10.1201/b10579}, }`

[ MR75] - Calvin C. Moore and Jonathan Rosenberg,
*Comments on a paper of I. D. Brown and Y. Guivarc’h: “Espaces de Poisson des groupes de Lie” (Ann. Sci. École Norm. Sup. (4)*, Ann. Sci. École Norm. Sup. (4)**7**(1974), 175–179 (1975))**8**(1975), no. 3, 379–381. MR427539,## Show rawAMSref

`\bib{MR427539}{article}{ author={Moore, Calvin C.}, author={Rosenberg, Jonathan}, title={Comments on a paper of I. D. Brown and Y. Guivarc'h: ``Espaces de Poisson des groupes de Lie'' (Ann. Sci. \'{E}cole Norm. Sup. (4) {\bf 7} (1974), 175--179 (1975))}, journal={Ann. Sci. \'{E}cole Norm. Sup. (4)}, volume={8}, date={1975}, number={3}, pages={379--381}, issn={0012-9593}, review={\MR {427539}}, }`

[ MS88] - Calvin C. Moore and Claude Schochet,
*Global analysis on foliated spaces*, Mathematical Sciences Research Institute Publications, vol. 9, Springer-Verlag, New York, 1988. With appendices by S. Hurder, Moore, Schochet and Robert J. Zimmer, DOI 10.1007/978-1-4613-9592-8. MR918974,## Show rawAMSref

`\bib{MR918974}{book}{ author={Moore, Calvin C.}, author={Schochet, Claude}, title={Global analysis on foliated spaces}, series={Mathematical Sciences Research Institute Publications}, volume={9}, note={With appendices by S. Hurder, Moore, Schochet and Robert J. Zimmer}, publisher={Springer-Verlag, New York}, date={1988}, pages={vi+337}, isbn={0-387-96664-1}, review={\MR {918974}}, doi={10.1007/978-1-4613-9592-8}, }`

[ MS06] - Calvin C. Moore and Claude L. Schochet,
*Global analysis on foliated spaces*, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 9, Cambridge University Press, New York, 2006. MR2202625,## Show rawAMSref

`\bib{MR2202625}{book}{ author={Moore, Calvin C.}, author={Schochet, Claude L.}, title={Global analysis on foliated spaces}, series={Mathematical Sciences Research Institute Publications}, volume={9}, edition={2}, publisher={Cambridge University Press, New York}, date={2006}, pages={xiv+293}, isbn={978-0-521-61305-7}, isbn={0-521-61305-1}, review={\MR {2202625}}, }`

## Credits

Figure 1 is courtesy of John Addison.

Figure 2 is courtesy of Ann Mackey.

Figure 3 is courtesy of Celebratio Mathematica.

Figure 4 is courtesy of the Dolph Briscoe Center.