by Joel Leonard Wolf
Introduction
In the summer of 2016, not long after I retired from IBM Research, I was invited to a conference at the Fields Institute at the University of Toronto. This conference was to focus on the Baum–Connes conjecture, and would also serve as an 80th birthday party for Paul Baum. Paul was my Ph.D. advisor at Brown University back in the early 1970s. My career since my academic days had taken a very different mathematical turn, but Paul and I had kept in touch from time to time via email. Still, I hadn’t actually seen Paul since the mid 70s, and it took me precisely zero seconds to decide to attend. I have always adored Paul, and this was a chance to catch up, maybe even to learn a little about Baum–Connes.
It’s 494 miles from my home in Katonah (Westchester), New York to the Fields Institute. So I had plenty of drive time to compose my thoughts on the way there, in case I was asked to say a few words. And those thoughts were pretty straightforward:
Paul has always been a magnificent and inventive mathematician, with a keen sense of what problems are important. His mathematics is just plain elegant. And somehow, at 80, he was still going strong, with a world of accomplishments.
Paul was a wonderful Ph.D. advisor, always pointing me in the right direction. He was a great mentor, generous with his thoughts and with his time.
Paul was a character in the best sense of the word. He was always extroverted, gregarious: a joy to be with.
Above all, he was kind to me, and to everybody with whom he came in contact. He was supportive and caring when I needed it most.
The conference was lovely. Paul gave a talk, and it seemed like old times to me. There were many other fine speakers. And there was a banquet in Paul’s honor on the last day of the conference, amazingly enough on his actual 80th birthday. Attendance was huge — everybody wanted to be present. Paul was kind enough to seat me at his table, so we talked all night. After the dinner many mathematicians got up to say a few words about him, and what did they say, pretty much in every case? They said the same things I was going to say. Magnificence and elegance? Check. Mentoring and generosity? Check. A gregarious, fun character? Check. Kindness and supportiveness, particularly to younger mathematicians? Check. They stole my lines! It is unusual for me to be tongue-tied, but I, as the last speaker, could really only echo what had been said before. There was universal admiration and love for the man. What more could you ask for in a mathematician and a person?
Since I have the space here, I will now go into a few more details. I’m going to focus on my time at Brown, emphasizing of course the joys and inevitable agonies of working on my thesis. I’ll give credit to Paul for the joys. I’ll take the blame for the agonies. I look back at my graduate career with genuine fondness. Then I’ll say just a few words about my career after Brown, which began as a Benjamin Peirce Assistant Professor at Harvard University, and then took a pretty significant turn towards combinatorial optimization (particularly scheduling theory and resource allocation problems). I was at IBM’s T. J. Watson Research Center for 32 years, with a 7-year intermission at Bell Labs. This “paper” is not about me, of course. I simply want to explain why I left the gorgeous field of topology, and how Paul still had a major influence on my mathematical thinking and approach. And finally I’d like to give a brief overview of the mathematics in my thesis on the cohomology of homogeneous spaces, again with a nod to Paul. (I wasn’t certain, when I started to write that section, that I could actually bring it off: I have been asked, on more than one occasion, to answer questions about my main results. One Friday, a colleague at IBM Research asked me something very specific. I told him I could not remember the details, but I would try to read the two resulting papers and/or the thesis itself over the weekend and get back to him on Monday. On Monday I had to say I had looked at them and was wondering: Who wrote this stuff? That last word wasn’t the one I actually used. Forty years of working on completely different mathematics can do that to you — or at least it can to me. So it has taken me a while, but I think I’ve now done the section justice. Lots of details have come back to me.)
I’ve written hundreds of papers and patents, but I’ve never written anything quite like this. So buckle up and let’s see how it turns out. You’ll laugh; you’ll cry. I know I had fun writing it. I hope you have fun reading it.
The Brown years — Paul and me
I did my undergraduate work at MIT, graduating in 1968. Cathy, my fiancée, was a year behind me. I came to Brown in part because I wanted to stay in the Boston area, where she was. We married that year, towards the end of my first semester, and we are still married 49 years later. Cathy followed me to Brown the next year, starting a Ph.D. program in experimental psychology. We have always been a team.
I had taken a wonderful seminar in homotopy theory with Frank Peterson during my junior year at MIT, and I knew then and there that I wanted to be a topologist. I took the usual four courses my first year at Brown, in order to prepare for the qualifying exams: topology, algebra, real analysis and complex analysis. Two of the courses were memorable to me. The topology course was taught by Paul, and I liked him from the start. The real analysis course was taught by Yuji Ito, and the difference in teaching styles could not have been more stark. In those days, children, professors taught using a primitive implement called chalk. Yuji would write complete, perfect sentences on the board, never ever making even the slightest mistake. It was incredible. The resulting notes, taken verbatim, could have formed a book, no editing required. I still have these notes, and also those from two courses on ergodic theory which I took in part to see if he could continue this amazing magic trick. He could. Yuji was close to an automaton. Paul, on the other hand, was bigger on verbally explaining what was important. He would say something like, “Let’s consider a topological space \( X \),” and write a huge \( X \) on the board. And we would, indeed, consider it. With something that big, how could you not? Then he would explain why we should consider it, perhaps describing the Lefschetz fixed-point theorem, proving it cleanly, mostly verbally, but primarily emphasizing its beauty. Just like that \( X \), he gave us the big picture. And it was wonderful, great fun.
By the end of 1970 I had completed my qualifying exams and was ready to start work on a thesis. I knew it would be in topology, and I had several possible advisors from which to choose. Bob MacPherson had arrived at Brown, and I knew he was outstanding. But I really wanted to work with Paul, and thankfully he agreed.
So in the beginning of 1971 we started talking about thesis topics. The very first thing Paul did was insist that I not tackle the cohomology of homogeneous spaces. This was a subject of interest to mathematical luminaries such as Henri Cartan [e1], [e2] and Armand Borel [e3]. It was also the subject of Paul’s Princeton thesis, published as [1]. The natural and desired “theorem” (which I will state precisely later) seemed to have a curse on it, there being several false proofs in the literature. Sadly, Paul had a gap in his own proof, though many excellent results and ideas from his thesis have survived intact.
Accordingly, Paul suggested other topics which would be simpler and more suitable for me. And, of course, the more he pushed me away from the cohomology of homogeneous spaces, the more I wanted to tackle it. I’m stupid that way. Clearly, Paul thought it was important, and I wanted to do something important. By the end of the spring semester Paul had reluctantly agreed, and I began.
The first thing Paul suggested was that I read his full Princeton thesis, in those days available in book form from the University of Michigan microfilms. Cathy and I had planned a one-month vacation during the summer of 1971. It was to be our first trip to Europe, and the only homework Paul gave me for that month was to read the thesis. Cathy had read the book Europe on \$5 a Day but apparently thought it was Europe on \$5 a Month. So in August 1971 we flew (the cheap) Icelandic Airlines to Luxembourg. On the next day we took the train to Calais, then the boat and the train to London. We registered at our (very cheap) bed and breakfast hotel, and I put the thesis on the bed so I could start to read it that evening. And off we went out to see the sights of London. You get one guess as to what I discovered when we returned: the maid had decided to clean up from the previous occupants, and mistakenly thrown out the thesis. I spent several hours looking for it in the trash, to no avail.
So here I was on day two of a one-month vacation, and the dog had already eaten my homework. In those days there was no fix for this: I was toast. I expected Paul to be furious, but when I returned I quickly learned one of his best character traits — kindness and understanding. No problem at all, he said. Just order another copy and start again. He said I was now rested and ready. And indeed I was. Rested, ready and extremely relieved.
And so I started to make some progress. Paul was great at making suggestions, and I followed them up. I remember with great fondness the occasional trips to his home in Providence. He did his work in an attic, and I was honored to be allowed in. (Paul said that his wife and kids were not allowed in, making it that much more of an honor.) In those days Paul and Raoul Bott of Harvard were just finishing up their great and famous work on the singularities of holomorphic foliations [2]. Accordingly, Paul was extraordinarily busy. But I always felt I had full access to him, and he always seemed to find the time to discuss whatever idea I was thinking about. And, of course, to point me in the right direction.
When we weren’t talking mathematics Paul and I talked about mathematics. One thing he said was that I shouldn’t be afraid to make mistakes. He claimed that all great mathematicians except Gauss made mistakes — if you weren’t making mistakes from time to time you weren’t trying hard enough. This turned out to be relevant to me, because I was quick to have false Eureka moments. I would think of an idea that almost worked and triumphantly announce my progress — before I had written it down and checked it carefully. Truthfully, I’ve done this throughout my career: It’s solved! Oops, it’s not. I have never inferred from this trait that I was a great mathematician. Unlike Gauss, I just periodically made misteaks. More on this shortly.
Another metamathematical discussion we used to have was the relative importance of the various branches of our field. Paul always claimed that the geometric areas (algebraic and differential topology, algebraic and differential geometry) were the topmost level, and I, of course, agreed. At the other end of the spectrum were the more somehow discrete areas, such as combinatorics. I agreed with this too. More on this shortly as well.
During my entire time at Brown I can remember only one colloquium given by a nonacademic. Roy Adler from the IBM T. J. Watson Research Center was an ergodician. Both he and Yuji were students of Shizuo Kakutani at Yale, and Roy’s talk described his excellent work on toral automorphisms. There was a party that evening at Yuji’s house, and I remember asking Roy how in the world he wound up at IBM Research. He claimed not to really know, but he said that Watson was effectively a world class university in which you didn’t have to teach. I’ll come back to this also.
I have one other aside from my graduate days, one which involved Paul and Walter Strauss, not me. But it’s too funny not to mention. Walter, who worked on differential equations, was also at Brown. And both he and Paul had old cars of the same model and year. Paul reminds me that they were Plymouth Valiants. (They probably weren’t all that valiant.) Naturally, Paul and Walter would compare notes on their cars from time to time, and one day Walter offered Paul four tires. So Paul asked the obvious question: why didn’t Walter want them? Well, it seems that Walter had gone to New York for a week-long meeting at NYU. And he had found, to his (brief) delight, a great parking spot on a street right near the university. Now New York was not in great shape in the early 70s. Even briefly abandoned cars were often stripped by thieves for parts. The city, in an effort to limit this problem, would quickly tow the cars to the pound. Then, if the car was not worth much, they would assume it had been left there to rot, and would compact it. Thus when Walter went back to where he had left his car, it wasn’t there. After finally finding the right agency, he was told the bad news: his car had been compacted two hours earlier. I guess he could have gotten his car back if he had wanted it — basically in a doggie bag. I have never be able to unsee this image. I don’t recall how Walter got back to Providence, but it’s a good bet that a Greyhound bus was involved. Of course, he had no further use for those tires.
And now back to my thesis. By the middle of 1972, following Paul’s suggestions, I had a proof of the main theorem on the cohomology of homogeneous spaces — but for the easy variants of the problem — real and rational coefficients. And by the end of 1972 I had found what seemed like a beautiful, short proof of the more important variants — fields of characteristic 0 or (under a mild restriction) of characteristic \( p \). I was very excited and Paul said he was as well. All that was left was to write it up and celebrate. I had made Paul proud.
I started applying for academic positions on the United States Election Day, in November of 1972 — Nixon won 49 out of 50 states, which should have been an omen. A number of offers came in quite quickly. I can recall Vanderbilt, Penn and British Columbia by early 1973. There may have been others. But the offer that interested me the most was from Harvard, which had a four-year nontenurable position for young mathematicians. I was offered a Benjamin Peirce Assistant Professorship. There is no doubt that Paul was instrumental in getting me this and the other positions. And, of course, he was working closely with Raoul, the Harvard chairman at that time. That surely did not hurt. Now having no chance of tenure wasn’t good — but staying in the Boston area was exactly what I needed: my wife was again one year behind me, and wouldn’t get her Ph.D. at Brown until 1974. Harvard wasn’t too bad a university, of course. That didn’t hurt either.
And so I began the thesis write-up in earnest. I knew it wouldn’t take long: the proof was so beautifully short. And then, poof! As I typed, the proof of the key part of the theorem evaporated before my horrified eyes. I had found one very subtle, but terrible mistake. The curse had struck again! To quote Verdi’s Rigoletto, la maledizione! I was mortified. Would I still have a job? Would I even graduate? I wondered briefly whether I should even point out the mistake. Nobody else, not even Paul had noticed it. But I only wondered very briefly. I knew I had to fess up, whatever the cost. And so I did.
And that brought out Paul’s absolute best. Well, maybe not his first comment, which was existential in nature. He pointed out that if he hadn’t had a gap in his thesis, I wouldn’t have a thesis at all. (I admit this was amusing, though of scant comfort.) But then he reminded me of what he had told me before: that all great mathematicians except Gauss made mistakes, and now I had qualified. I had to remind him that I wasn’t exactly feeling like a great mathematician at that moment, let alone a Gauss. He said that I would fix the problem and all would be well again. That was more comforting. He said that I would laugh about this little crisis one day. (Note to Paul: not yet.) He told me that I had enough to graduate anyway, and after calling Raoul he told me that my position at Harvard was safe. He never reminded me that he had warned me of the curse. Instead, his confidence gave me the courage to continue. And once I had calmed down enough to work, he gave me an excellent mathematical idea. Specifically, he suggested a slightly different tack at the proof, one which made use of so-called strongly homotopy multiplicative (SHM) maps. And he pointed me to Jim Stasheff, who was at Temple during those days. Jim was an SHM expert, and I made a pilgrimage to Philadelphia. One incredibly brutal month later I had salvaged my theorem using the SHM concept, and this time the write-up process went well. The proof was longer than it had been, but I did have a proof, a thesis, a job and a Ph.D. Also, a future. Thank you, Paul.
The years after Brown
As I have noted already, I’m going to keep this section brief. I mainly want to point out how Paul influenced my career heavily in terms of philosophy, if not in mathematical substance. Said differently, I hope Paul did not waste his time by being my advisor, at least not completely.
At Harvard I finally got to teach advanced courses, not just the calculus sections I had been given at Brown. I taught topology at both the undergraduate and graduate levels, and I had great fun doing it. My favorite course was modeled after Milnor’s great book about Smale’s work on the Poincaré conjecture [e6]. These courses were quite well received, and that brought me joy. On the research side I was less productive. I know the horror of the incredible vanishing thesis had taken its toll on me. But there was also a really long recovery from a bout of pneumonia, the very sudden death of my father, my mother’s horrible illness, and (at last something good!) the birth of our first daughter. The Harvard years were certainly my least mathematically creative.
My wife graduated in 1974, taking a one-year job at Wellesley and then becoming a postdoc at MIT. We had an early case of the two-body problem, and in those days universities had antinepotism rules — quite the reverse of the situation today. We were determined to find positions together when my Peirce ended in 1977. One of the graduate students at Harvard was the brilliant Don Coppersmith, who had won the Putnam exam all four years as an undergraduate at MIT. And we became good friends. He had been going to IBM Research every summer, and he told us that we would love it. I remembered Roy Adler. Don said the same thing that Roy had said: it was a university without teaching responsibilities, filled with wonderful mathematicians and other scientists. (At that time the research staff could even take sabbaticals at universities.) Cathy and I decided to try it. So in the summer of 1976 we went to Watson as visiting faculty. And we did indeed enjoy it. In 1977 we each had a few academic opportunities, but never in the same place. We had to ultimately decide between (a) taking positions with half the security, half the salaries and half of us, or (b) leaving academics. We went with heresy, plan (b). And more or less, we never looked back.
While nobody at IBM Research forced me, it was clear my career would do better with a change in specialties. And so I drifted quite quickly into combinatorial optimization, actually still quite geometrical in nature — though at the lower end of the mathematical totem pole in the discussions Paul and I used to have. It amuses me these days to see the close relationship between topology and quantum computing — but this is now, and that was then. My wife moved from experimental psychology (language acquisition) to speech recognition and then to human computer interaction. We prospered, both at Watson and during our short intermission at Bell Labs. We always were a team — no more two-body problem worries.
I tried hard in my career to focus on theoretical problems I thought were important, taking my cue from Paul. I can claim a few successes. In [e13] we produced the first polynomial algorithm for a natural problem in what is now known as moldable parallel makespan scheduling. In [e15] we created the first polynomial-time algorithm for moldable parallel response time scheduling. In [e12] we designed the second polynomial-time algorithm for what are known as class-constrained resource allocation problems. (I say second because we were just beaten to the punch by Federgruen and Groenevelt, a fact we discovered while finishing the proofs of our own paper. Sigh.) I am proud of several papers which found elegant optimization techniques for computer science problems which were important to IBM. Two of these are [e14] and [e16]. A final paper of mine has both properties: mathematically a first and a scheduling result which is of value to IBM. This paper [e17] effectively subsumes and generalizes (to malleable parallel scheduling and a wide range of performance metrics) a great deal of my scheduling theory work.
I was a Distinguished Member of Technical Staff at Bell Labs and a Principal Research Staff Member at Watson. I am an IEEE Fellow. I have over 12,000 citations. So I’ve done ok. Is it topology, Paul? No, I went to the dark side. I went rogue. Mea culpa. But all mathematics can be beautiful if you look for it. And I’ve had my fun. No complaints.
I’ve tried in my way to emulate Paul in other ways as well. I have discovered and mentored many fine young mathematicians. I’ve tried hard to nurture them and get them through their occasional but inevitable rough patches. I’ve spent a career with many friends and no real enemies. And I’m pretty sure I was gregarious and fun enough.
Thanks, Paul.
The cohomology of homogeneous spaces
The main theorem of [e10] has a conceptually easy proof, though the details are complicated and reasonably lengthy. We will restrict ourselves to an overview here. For a complete proof see [e10]. The theorem is as follows.
- \( K \) is a field of characteristic 0, or
- \( K \) is a field of characteristic \( p \) and both \( H_{*}(G;Z) \) and \( H_{*}(H;Z) \) have no \( p \)-torsion.
Then, regarding \( H^{*}(BH;K) \) as a left \( H^{*}(BG;K) \)-module via the map \( f^{*} \) and \( K \) as a right \( H^{*}(BG;K) \)-module via augmentation, there exists a module isomorphism \[ H^{*}(G/H;K) \approx \mathrm{tor}_{H^{*}(BG;K)}(K,H^{*}(BH;K)). \]
Let \( T \) be the maximal torus of \( H \). By a result of Paul’s [1] it is sufficient to prove the theorem in the special case \( G/T \). So, restricting ourselves to \( T \), the proof is based on the following diagram: \[ \begin{CD} C^{*}(BT;K) @<{{{f^{\#}} }}<{{}}< C^{*}(BG;K) \\ @V{\alpha}VV @AA{\theta_1}A\\ H^{*}(BT;K) @<{{}}<{{{f^{*}} }}< H^{*}(BG;K) \end{CD} \]
The left-hand side of the diagram is easy. By May [e8] there is a multiplicative homology isomorphism \[ \alpha: C^{*}(BT;K) \rightarrow H^{*}(BT;K) \] which annihilates \( \cup_1 \) products. The right-hand side is more problematic. Now \( H^{*}(BG;K) \) is a polynomial algebra, but \( C^{*}(BG;K) \) is not necessarily graded commutative. Fortunately, \( C^{*}(BG;K) \) is homotopy commutative (via \( \cup_1 \) products) in a very strong way. We take advantage of this to construct an additive homology isomorphism \[ \theta_1: H^{*}(BG;K) \rightarrow C^{*}(BG;K) \] which is the first term in a strongly homotopy multiplicative (SHM) sequence \( (\theta_1, \theta_2, \ldots) \) whose higher terms are written in terms of \( \cup \) and \( \cup_1 \) products.
(Briefly, suppose \( A \) and \( B \) are differential graded algebras over \( K \), and \( (f_1, f_2,\ldots ) \) is a sequence of \( K \)-module homomorphisms such that \[ f_n : A \otimes \cdots (n) \cdots \otimes A \rightarrow B \] for each positive integer \( n \). Suppose \( f_n \) has degree \( 1-n \) for each \( n \), and \[ \displaylines{ df_n(a_1 \otimes \cdots \otimes a_n ) - \sum_{i=1}^{n} \sigma(i-1)f_n(a_1 \otimes \cdots \otimes da_i \otimes \cdots \otimes a_n ) \hfill\cr\hfill = \sum_{i=1}^{n-1} \sigma(i) \bigl[ f_{n-1} (a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n ) \hfill\cr\hfill {}- f_i (a_1 \otimes \cdots \otimes a_i) f_{n-i} (a_{i+1} \otimes \cdots \otimes a_n)\bigr]. } \] Such a sequence is said to be SHM. See [e5] and [e9] for more details.)
The diagram above is, in fact, commutative. Furthermore, the effect of the higher-order SHM maps is nullified by \( \alpha \).
By the results of Eilenberg and Moore [e4], [e7] there is an algebra isomorphism \[ H^{*}(G/H;K) \approx \mathrm{Tor}_{C^{*} (BG;K} (K, C^{*} (BH;K)), \] which we shall apply to \( H=T \). Note that this \( \mathrm{Tor} \) is more general (and vastly less computable) than \( \mathrm{tor} \), our desired torsion product. To prove Theorem 1 we will actually need to create a yet more general and less computable torsion product, namely \( \mathrm{TOR} \). It is described in terms of SHM sequences, and we will not go into those details here. Suffice it to say that it provides an isomorphism \[ \mathrm{Tor}_{C^{*} (BG;K)} (K,H^{*}(BH;K)) \approx \mathrm{TOR}_{H^{*}(BG;K)} (K,H^{*}(BH;K)), \] which we will again apply to \( H=T \).
Waving our hands a bit more (to avoid many pages of exposition), the key calculation shows that the composite SHM sequence \[ (\alpha f^{\#}, 0, \ldots) \circ (\theta_1, \theta_2, \ldots) \] is actually and magically equal to the strictly multiplicative sequence \[ (f^{*}, 0, \ldots), \] meaning that the most complex \( \mathrm{TOR} \) collapses in one fell swoop to \( \mathrm{tor} \). The resulting sequence of module isomorphisms \[ \eqalign{ H^{*}(G/T) &\approx \mathrm{Tor}_{C^{*} (BG;K} (K, C^{*} (BT;K) \cr &\approx \mathrm{Tor}_{C^{*} (BG;K} (K,H{*}(BT;K)) \cr &\approx \mathrm{TOR}_{H^{*}(BG;K)} (K,H^{*}(BT;K)) \cr &= \mathrm{tor}_{H^{*}(BG;K)} (K,H^{*}(BT;K)) } \] proves the theorem. For the full details, see [e10].
The main theorem in [e11] is a simpler and less important result, namely:
We omit the details, but they are available in [e11].
Last words
So here’s to Paul Baum, the newest member of Celebratio Mathematica. What a great human being. And what a magnificent career he has had — so far, at age 82. I am 12 years younger than Paul, and yet I have retired and Paul is still at the top of his game. In 2017 he was the Gauss Professor at the University of Göttingen. How lovely and appropriate is that?
At the end of the 2016 banquet at the Fields Institute, Paul paraphrased Churchill. He said he didn’t think it was the beginning of the end. He said it might be the end of the beginning. I don’t think it’s even that. I’ll paraphrase Jerome Kern in Showboat: like Old Man River, Paul just keeps rolling along. Be afraid, unsolved topological problems. Be very afraid.
