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Alfred Hales

Alfred W. Hales

Professor Emeritus of Mathematics at UCLA and

Adjunct Research Staff Member at IDA/CCR, La Jolla

By David Zierler, Director of the Caltech Heritage Project

January 19 and 29, 2024

DAVID ZIERLER: This is David Zierler, Director of the Caltech Heritage Project. It's Friday, January 19th, 2024. It's my great pleasure to be here with Professor Alfred W. Hales. Al, great to be with you. Thank you for joining me today.

ALFRED W. HALES: Thank you for inviting me.

ZIERLER: Al, to start, would you please tell me your current title and institutional affiliation?

HALES: I'll give you two titles. One is Professor Emeritus of Mathematics at UCLA, and the other is Adjunct Research Staff Member at CCR, La Jolla. That's actually IDA/CCR La Jolla. I was director for 11 years of this outfit, but I'm now almost completely retired from it.

[Editor's note: In this text, the phrase "Institute for Defense Analyses" or "IDA" is used in a colloquial manner to refer to the organization legally known as the Center for Communications and Computing (CCC), a federally funded research and development center. Precisely speaking, CCC is only a part of the non-profit corporation IDA. Both CCR La Jolla are CCR Princeton are parts of CCC.]

ZIERLER: Let's start first with UCLA. When did you go emeritus?

HALES: 1992. What actually happened was I had just finished being department chair at UCLA, and two things happened at the same time. IDA1, that I consulted with for many years, started a new branch on the West Coast [here] in La Jolla, and were looking for a director. UCLA offered a very attractive early retirement program at exactly the same time. I thought about it a little bit, talked to my wife a bit, and we decided it was time to do something new and different. I'd always loved the work that I did during summers consulting for IDA. I also always loved UCLA too. It was difficult to make the decision, but we decided to move down here. I took early retirement with benefits, because of UCLA's special offer, and then moved down here, and became director of the new IDA branch down here.

ZIERLER: La Jolla is a lovely place to be.

HALES: Yes, it certainly is [laugh], especially when a large part of the rest of the world is under snow right now.

ZIERLER: That's right. Al, do you remain connected with UCLA math at all?

HALES: Oh yes, quite closely. I'm not sure you're aware of this, but there's a math institute at UCLA called IPAM, Institute for Pure and Applied Mathematics. That started some years ago. For eight years, I think, I was chair of the board of trustees there. That gave me a good excuse to go back up to UCLA, and meet with all my colleagues, and so forth. I keep in touch. I try to keep in touch a lot.

ZIERLER: Now, when you went over to CCR, did you maintain your research agenda as a mathematician, or was this more of an administrative role?

HALES: It was an administrative role, but I continued to do academic research along the lines of what I'd done before—and I'll say more about that later—and also to participate. I did some new research with IDA/CCR also. I certainly managed to keep things going. That's because IDA/CCR is very well set up to treat administrators well, and give them time to do other things too. [laugh]

ZIERLER: Al, some broad questions about UCLA math institutionally. What is the math department known for?

HALES: Actually, it's known for a lot of different things. It had a very strong group in functional analysis when I was there. It also had a strong group in number theory and combinatorics, which is what I contributed to more, and a very strong program in mathematical logic also. I'm sure I'm leaving out a bunch of other things too. There were isolated people who were stars in a bunch of different areas. It was a pretty large department when I was there, 50 or so tenured faculty. They covered quite a few things. I think the department ranked nationally somewhere around 12th when I got there, if you believe these rankings at all. I think they've moved up quite a bit, and now they're close to being in the top five in the country—maybe not quite that but very close. One of the things that made a big difference in raising overall recognition was when UCLA managed to hire Terry Tao, who is one of the greatest mathematicians in the world. That was a big coup. Anyway, UCLA has been doing very well. I think some people even consider that we've passed Berkeley now in mathematics. It always used to be that we were the younger sibling of Berkeley from that point of view. [laugh] But we're pretty much on a par with Berkeley now, which means we're basically in the top four or five or six in the country.

ZIERLER: Al, let's move over to the Institute for Defense Analyses. What is its mission? How far back does it go?

HALES: It started as an informal summer research program in about 1952. The idea was to take the mathematicians that had participated in cryptanalysis during World War II, very effectively, and keep them from forgetting everything they'd learned [laugh] back then, and breed new people in case they were needed again from a national defense point of view. There were a series of summer programs actually at UCLA for five or six years along these lines, bringing in mathematicians to do work during the summer. Then after that, they decided they needed a permanent home, not just to do things during the summer. In the late '50s, probably around '58—I don't know the exact date—they started up an official branch on the Princeton University campus, and that's still going in Princeton. For a long time., it was the only such branch—and, by the way, they're basically an NSA contractor. Everything they do is under contract with the National Security Agency. That's still going. Then a number of years later, they started two more outfits, one in La Jolla—the one that I ended up running for a while—and another one in Bowie, Maryland, near the NSA, that was more computer science-oriented. The Princeton and La Jolla ones tend to emphasize mathematics; whereas the one in Bowie, Maryland has computing right in its name.

ZIERLER: I don't know if this is a crude analogy, but is IDA like JASON2 for mathematicians?

HALES: Yes. That's a very complicated story, and I'm not capable of giving the whole thing to you. But at one time, I believe they were both being run by the same overarching outfit, and then some things dropped out, and changed, and so forth. They're now run completely independently, but they're still very close. For example, JASON takes place in La Jolla, as you may know. I get invited to their summer parties every year [laugh], and there have been JASON members who also consulted for IDA. Originally, JASON was all physicists, but then they started hiring mathematicians, and now other scientists as well. There's an interesting and a very complicated connection between the two outfits.

[Editor's note: The JASON program and IDA's Princeton branch were both administered by the non-profit corporation IDA, but IDA's relationship with JASON ended in the 1970's.]

ZIERLER: Now let's go to your connection with the IDA. How far back does that go? What were the circumstances of you joining?

HALES: My advisor at Caltech was Bob Dilworth. Bob was very involved with the IDA work; had been for many years. There were other people at Caltech too, but particularly Bob, and also Marshall Hall. They told me a lot about this outfit. When I got my PhD, and went off to England on a postdoc, they suggested that I apply for a summer job there to get the security clearance processed, and so forth. That went on while we were in England for a year. By the time I got back, I was at Harvard. I had summers free back there, and the security clearance came through. In the summers of 1964 and 1965, I worked at the Princeton branch. Then after that, we moved back to California to UCLA.

There were a number of years when I didn't participate, although I kept up informally with what was going on. Then in the '70s, I started going back there for the occasional summer, and they started running summer programs occasionally in Monterey, which made it more convenient for Californians. There were four or five—maybe more than that—summers in the '70s and '80s when I was working at least part-time for IDA, either in Princeton or in Monterey, and then a couple in Southern California too. I even ran one in 1988. By that time, they'd decided they needed a permanent home in California, so they started one in 1989. By '92, they'd managed to pull me away from UCLA, so I've been in La Jolla now since 1992.

ZIERLER: Al, of course, without getting into any sensitive details, at a broad level, how can mathematicians contribute to the national defense?

HALES: Basically by the analysis of communications security from both a defensive and offensive point of view.

ZIERLER: Has this been—?

HALES: Now, this endeavor, maybe when it first started back in the early 1900s or even earlier, didn't seem to be that mathematical. But as the years have gone by, it's gotten more and more mathematical now, where it really involves a number of very highly complicated parts of mathematics.

ZIERLER: For you, what have been your areas of interest within the national security framework? Do you get to decide, or is it all assigned to you?

HALES: "Assigned" is one word. "Dangled" is an even better word.

ZIERLER: Because there's such great recent problems?

HALES: NSA likes to dangle several different problems in front of the group, and hope that IDA will latch onto one or more of them, and make breakthroughs.

ZIERLER: Now let's move on to your overall research agenda. For non-specialists, like the cocktail party explanation, they know you're a mathematician, but what would you say if they asked, "What kind of mathematician are you?

HALES: When they ask me that question, I guess I just tell them that I'm interested in algebra and combinatorics, and there are many instances where this is of great interest and application, and that seems to be enough. If I'm really forced to tell what kind of mathematics I do, one thing I could do is to talk about, I guess, one of the most interesting pieces of pure math that I did, which some people think is more applied. Fortunately, it can be described to anyone. It involves the game of tic-tac-toe [laugh], which most people understand. What I tell them is that when you play tic-tac-toe usually you play it on a board that's three by three. It's a two- dimensional board; three on a side. But you can try to play tic-tac-toe in higher dimensions, three or four or five, any number of dimensions, and with any side length also. You could play it on a board that's 100 by 100 by 100, so in three dimensions, 100 on a side. The question is, in those circumstances, who wins? The answer is, if the dimension is very large compared to the side length, then the first player always wins. In the other extreme, it's always a tie. This is a mathematical theorem that I proved with Bob Jewett. Bob was a classmate. He was one year ahead of me at Caltech. We started this when we were both graduate students, but I was still at Caltech, and he'd just moved on to Oregon. Caltech had a role in this since we were both at JPL for the summer. This turned out to be of considerable pure mathematical interest, but you can describe it at a cocktail party—sort of—to people that are willing to accept high dimensions and stuff like that.

ZIERLER: This, of course, is the Hales-Jewett theorem.

HALES: Yes, exactly. [laugh]

ZIERLER: Has it changed over the years, the theorem? Has it been improved upon? Has it been challenged?

HALES: Oh, it is been improved upon and extended in all sorts of directions. I don't claim to completely be able to keep up with it. But one of the ways of extending it is a stronger version where you don't even worry about filling up the whole board. You prove that if you just have one of these huge tic-tac-toe boards, and you put enough X's in, let's say—forget about the O's; just put enough X's in—there'll be a winning path, just by the number you put in. You don't have to fill it up completely. That's one extension. In fact, they even had a math blog, that Terry Tao was very much involved in, to try to find a simple proof. The original proof involved some very complicated ergodic theory and stuff like that. But they found a more simple proof after getting one of these blogs where you get participation from all over the world with people logging in and adding an extra tidbit or two, and the final theorem3 has maybe 100 participants or something like that. But, anyway, it's become, particularly recently, very, very extended in all sorts of directions.

ZIERLER: Al, I wonder if you could explain the relationship between the Hales-Jewett theorem and Ramsey theory.

HALES: It is considered to be a theorem in Ramsey theory. Ramsey's original theorem— I'm trying to think of the easiest way to explain it. It basically is a result that says—how should I put it?—if you have a situation that's complicated enough, there's got to be a pattern. There are always patterns, no matter where you look, and no matter how complicated things are. If there's enough items or parts to it, there will be a hidden pattern somewhere. The pattern in the Hales-Jewett theorem is a winning path, for example. But it might be something along the lines, well, here's a very simple theorem in Ramsey theory. If you have six people, any random collection of six people, you can either find three of them that are mutual friends, or you can find three of them none of which know each other. Somewhere among the sixth, there will be at least either three mutual friends or three mutual strangers.

There's a much stronger example of this. If I hand you a batch of a million people—I'm going to make up the numbers now—then no matter, you can find either 30 of them, all of whom know each other, or 30 of them, none of which know each other. Except that 30 may not be the right thing to go with a million here. But it's the same idea that you have this complicated thing, and you don't think there's any structure to it. But hidden within it, there's something that does have a very specific structure that you might not expect to be there.

[Editor's note: See DHJ Polymath, "A new proof of the density Hales-Jewett theorem", Annals of Mathematics, volume 175 (2012), 1283-1327, DOI:10.4007/annals.2012.175.3.6.]

ZIERLER: We'll describe this more in detail in the chronology but, just generally, I wonder if you can explain, given that you had to prove the theorem, how do mathematicians go about achieving a proof of a theorem? What does that mean?

HALES: Oh, that's a [laugh] difficult question. Almost everyone gets led into that, whether they realize it or not or whether they like it or not by the time they're sophomores in high school because, at least in the old days—and I think it's still true—you study Euclid's plane geometry in high school. Kicking and screaming, you're supposed to come up with proofs of the things or, at least, memorize the proofs and understand them so you can reproduce them on exams.

Everybody gets led into this when they're in high school. They may not follow up on this afterwards, and that's one problem that the whole STEM effort is worrying about right now. Everyone gets led into this, and it's one way to be sure of your results—there are many things in this world that we can't be sure of, particularly in politics right now.

ZIERLER: [laugh]

HALES: But it's nice to be in a subject where you can be absolutely sure that you're right, and no one can quibble. The way to do that is to have a setup where there are axioms that everyone believes in, and then you can prove what you want from the axioms. You learn to do that, supposedly, in 10th grade, if you haven't learned it earlier. You may not follow through on it afterwards. [laugh]

ZIERLER: Of course, scientists have experiments to verify their theories. What is the analog of that for mathematicians?

HALES: Certainly, there's an analog to start with that's very similar to what the scientists do. You will carry out experiments with small models to see if there's some pattern or some result that looks like it might be true. If you find something that seems to be true in all the small cases that you've looked at, then you decide that this really must be something that I'm capable of actually proving. You sit down, and then try to follow the type of thing you might have learned studying plane geometry, and come up with an actual proof. But the background to this is experiments or special cases that will either convince you that you're right - or not - there's a great story about a woman mathematician who was extremely well known, and was being interviewed about her work. The reporters followed her around for a week to see what she was doing. She would write down how she was doing. "On Monday, Tuesday, Wednesday, and Thursday," she said, "tried to prove theorem X. Tried to prove theorem X. Tried to prove theorem X." On Friday, she wrote, "Theorem X false." [laugh]

ZIERLER: [laugh]

HALES: She discovered a counter example, so that was the end of that [laugh], a week's worth wasted. But that was a microcosm of what mathematicians do. They find the pattern that they think always holds, and they try to prove it. If they're lucky, they manage. If they're not lucky, it turns out to be wrong, and they go on to try something else. [laugh]

ZIERLER: Al, you mentioned tic-tac-toe. Does tic-tac-toe in this analogy work as a one- dimensional view? What does it mean to have a high-dimensional generalization of tic-tac-toe?

HALES: Think of playing tic-tac-toe on a board that's three by three by three, it's 27—

ZIERLER: A cube?

HALES: —"squares". That's a three-dimensional board now, and you can visualize now putting zeros and X's in these 27 holes or cubicle boxes, let's say, in this thing. By the way, you can actually buy copies of this, or sometimes it's four by four by four. But they used to have in toy stores copies of this. The idea then is that you take turns putting zeros and X's in, and you're trying to construct a straight line, that the line could be this way, or this way, or diagonal, or diagonal that way. There are more directions you can go, and it gets more and more complicated as you have more cells and if you have higher dimensions. If you start trying to visualize four dimensions, then it gets much more difficult. The way to visualize a four-dimensional board would be to think of 3 three-dimensional boards, so here's a board, here's a board, and here's a board. One winning path would be a cell from here, a cell from here, and a cell from here. You're cheating that way, because you can't really draw things in four dimensions. But mathematicians eventually get comfortable thinking in high dimensions. It's actually a convenient way to describe things and improve understanding of things.

ZIERLER: Al, some overall questions about other areas that you've worked on, shift registers. What are shift registers? What's your work there?

HALES: Shift registers have turned out to be very important in computers and in all aspects of communications, and so forth. A shift register, you should think of as—how should I put it? It is just a bunch of little cells. Let's say, there are 20 cells in a row. You have 20 zeros and ones arranged in a sequence in some way. You're going to change this, one cell at a time. But what I mean by that is, to make a change, I'm going to shift everything over like this, one cell. The thing that was here falls off the end. There's going to be a blank over here, because I've shifted. I put something new in at the end. Now I have a new set of 20 things. It overlaps in 19 of them, but one of them is new. I've shifted it once. Now I'm going to shift it again so, again, something will fall off one end. Something new will come in at the other end. There's a formula that tells you what to put in at the new end each time. It's a formula, like, it might be the sum modulo 2 of everything that was there already, or it might be a more complicated formula, like this times this plus this, or something like that. Now think of this sequence of 20 things, and I run it for a million or a billion times. You can ask, what are all the patterns that are going to show up on this? If I send a message, and I use these sequence things to encode the message I'm sending, is someone else going to be able to figure out what the pattern is in this shift register? It's going to depend on how complicated the feedback is.

There've been many people who have devoted almost a whole career to looking at these things. It turns out to be very important and critical in many aspects of secure communications these days. In fact, it's also used every time you open your garage door. When you press the button, a shift register sequence decides whether the button you pressed is supposed to open this door or that door. The sequence that your little handheld thing produces will match what's in the gadget in your garage door, if you're being honest that it's your gadget, and it'll open your door. But it won't open your neighbor's door, because his is showing a different set of sequences. In between garage doors and high-tech cryptanalysis or whatever it is, there was a whole article—which I can tell you more about later—written by Steve Wolfram about this. He estimated the number of times this happens every day, umpteen-trillion times someone in the world uses a shift register. [laugh]

[Editor's note: Hales worked on shift register sequences at JPL with Solomon Golomb, see for example Golomb's book "Shift Register Sequences", 3rd edition, World Scientific, 2017.]

ZIERLER: Oh wow. [laugh] Al, what about Ulm invariants? Who was Ulm? What are Ulm invariants?

HALES: Ulm was a—by the way, you seem to know quite a bit [laugh] about my work already.

ZIERLER: I tried to do my homework!

HALES: Ulm was a, I believe, German mathematician. He was investigating the structure of groups, particularly the infinite abelian groups, to try a way of describing up to isomorphism, what they all were; how many different groups of a given type there were. He came up with something called Ulm invariants, which worked but only in one special case, in the countable case, so infinite but a small countable infinite. Trying to generalize this became an open problem in algebra, in group theory, for a long time. It was approached in several different directions. One approach came from, Paul Hill and some others in the South that involve almost a detour into algebraic topology. At the same time, more or less, I was working with Pete Crawley, although we didn't know we were working on the same problem. Pete was a Caltech colleague of mine, and we had the same thesis advisor, but we were in different locations at the time.

We were coming at it from a slightly different direction, which was a much more simple and effective way of doing it. We all came to conclusions at the same time. This is the Crawley- Hales theorem, the part that Pete Crawley and I were working on. It involved describing an abelian group in terms of the simplicity of its generating set. You might describe it by saying it has generators A, B, and C, and 7A equals B, and 7B equals C plus D plus A or something like that but having an infinite number of generators and relations like this. That was our point of view, but it was all extending work that Ulm had done some years before, quite a few years before.

ZIERLER: Al, you've alluded to it in a few places in your answers, but generally have there been technological advances over the course of your career in communication, in satellites, in computers that have really been relevant for the kind of math that you do?

HALES: Yes. In fact, in many ways, it steered me in at least 50% of what I did, as we'll get to later on. While I was at Caltech, I very fortunately was recommended to JPL for a summer job. I worked two summer jobs, actually, on the Caltech campus. But then I went to JPL in 1958 after my sophomore year. That had an enormous effect on my career. I worked there every summer while I was still at Caltech in undergraduate and graduate school. The kind of math we were doing, they were mostly shift register sequences but other things too, at that stage. When I got to Caltech, they were still a Jet Propulsion Lab thinking about military things. But, all of a sudden, Sputnik happened, and the motivation changed drastically. They still had the same name, but they started worrying about space, and how do we measure how far away Venus is? They used a shift register sequence to do that. You've got to know where Venus is if you're thinking about sending a spaceship. But, anyway, everything they were doing turned out to involve shift register sequences, and the group I was in was doing almost nothing but that, in one way or another. That was just the beginning. After all, space exploration became of great interest and then (computer) communications exploded shortly after that. The world is unrecognizable now, almost, to someone who started in that generation. [laugh]

ZIERLER: Al, do you see your research as contributing to advances in physics?

HALES: Only indirectly. I'm trying to think now. Some of the algorithms that I've worked on have certainly been algorithms that have been useful in physics, and in other parts of science also. In terms of what I've done, I'm trying to think now. I'm not sure about that, let's put it that way. Certainly not, as far as I know, not in explaining dark energy or dark matter—

ZIERLER: [laugh]

HALES: —or the relationship between relativity and quantum mechanics, which are all things I'm interested in. But I don't see any specific application right now.

ZIERLER: Al, as an outside observer, you mentioned algorithms, and it's striking to me, as a historian of science, how algorithmic biology and the life sciences have become. Everyone is talking about the importance of a mathematical perspective in biology, which is a relatively recent development. What's the takeaway for you? Have biologists seen the light? Is that what this is about?

HALES: They've seen the light in several different ways. One is that it's [laugh] very important for them to get mathematicians involved in their work. I should say, by the way, partly just out of general interest and partly from the role of a former administrator, they've been poaching on us.

ZIERLER: [laugh]

HALES: [laugh] There's several outstanding mathematicians who started their lives working in things I'm interested in, and maybe working for IDA Princeton or IDA La Jolla on things that are important. All of a sudden, they decide they can make more money [laugh], but they're working for one of these outfits that's breaking the genetic code or something like that. Several of these people have jumped across to one of the institutes in MIT and so forth. When my daughter graduated—one of my daughters was a mathematician—when she got her PhD from Berkeley, I was asked to give the speech at the retirement then. One of the things I touched on in my speech was what mathematician do to earn a living. I said, "A change from the past is that many mathematicians either think about stocks, genes, or codes."

ZIERLER: [laugh]

HALES: Stocks, they go to work for Jim Simons. Jim Simons was also someone that was poached away from us, by the way.

ZIERLER: That's right.

HALES: Genes, that's the biology you're talking about. Nick Patterson was someone that was poached away from us to work on this. Then codes (cryptanalysis) and so forth. I pointed out these are three fairly new things that the graduating mathematicians from Berkeley that year ought to think about as career roles.

ZIERLER: Al, do you like thinking about the philosophical aspects of math, things like the meaning of infinity, or if math is a human invention, or it exists independent of the human mind?

HALES: Yes, I do like to think about that. Some of the more exotic ones, I only think about when I'm looking around for something different. But some of them are more specific because, in particular, the meaning of infinity, some of my work, like in my thesis work, I was looking at what I was proving was there's no free complete Boolean algebra on an infinite number of generators. But this involves looking at all levels of infinity, not just the first countable level or the size of the real numbers but involves looking at all conceivable cardinal numbers, and showing that in none of these cases could you have a free complete Boolean algebra of that size. I was forced then into something that some people would think of as really exotic but, for me, it wasn't exotic; it was part of my thesis. That aspect of it, I often think about, the size of infinity.

Some of the other aspects, from my point of view, are more exotic, and I don't think about them that often but occasionally.

ZIERLER: Al, if there is a breakthrough in quantum information, and the impact that this could have on encryption, which obviously the NSA is very concerned about, what is your perspective on that? What are the opportunities and risks?

HALES: My interest in quantum research started with my 1982 paper with Ernst Straus which was related to hidden variables. Then the spectacular 1994 result of Caltech alumnus Peter Shor fascinated me from both a number theoretic and computational point of view. And in 2001 our daughter Lisa received her doctorate from Berkeley in quantum computation, following up on one aspect of Shor's work. Ever since then I have followed with great interest all the quantum developments in universities, companies and government agencies around the world, to try to follow possible applications and further advances. Most recently I have noted Caltech's quantum "hub" IQIM described in a recent Caltech alumni magazine. But I am not directly involved with any of this work so I really have nothing to add to what you read in the news media.

ZIERLER: Al, just a snapshot in time. what are you currently working on? What's fun to you these days?

HALES: The thing I've been working hardest on has been actually something you might be interested in. When I went to JPL to work, I was working under Solomon Golomb, who was there for a number of years, and then moved to USC, and became very well known for his work on all sorts of things, including shift register sequences. He passed away in 2016. For the last few years, I've been working with one of his daughters and with another colleague to edit a book in his memory. The book just came out this last summer. It's called The Wisdom of Solomon. I'll send you an email that has a clipping on it.

ZIERLER: Oh wow.

HALES: I think you'll be interested in looking at it, because Caltech, in particular, JPL plays a role in that. The book, among other things, has interviews with many different people, including people who worked under Solomon Golomb at JPL, and Caltech figures in a number of places in a very important way. I think it's something you ought to buy, and certainly your library there ought to have a copy. [laugh]

ZIERLER: OK, good to know. Al, let's go all the way back to the beginning. Let's establish some personal history. Where did you grow up?

HALES: I grew up in San Marino, initially, so very close. While I was growing up in San Marino, my grandparents were living in Pasadena on Arden Road, which, as you know, is very close to campus. There was some unrest in the educational system in San Marino. They were building a new high school or something. At the same time, my parents thought they might have to move in with my grandparents when one of them died. They decided that they should send me to Polytechnic School. Starting in fourth grade, I was at Polytechnic, which, as you know, is adjacent to Caltech and used to be part of Caltech. I was in school at Polytechnic from fourth through ninth grade.

At that time, Polytechnic only went through ninth grade, so I then went on to Flintridge Prep for 10, 11, and 12. But I was at Poly from fourth through ninth grade. A number of my colleagues there—I should say classmates there—were children of Caltech professors. Andy Bacher was Robert Bacher's son. Robert Bacher was on the Atomic Energy Commission, and head of Caltech physics. There was Hallett Smith heading the humanities department at Caltech. He had a daughter at Poly that I dated once or twice. Another person in the humanities department there was—I can't think of her name now—the daughter of someone who taught economics at Caltech, Sweezy, the daughter of Professor Sweezy. I knew indirectly of Caltech in many, many ways through my colleagues or my classmates, let's put it that way. If I can digress slightly—

ZIERLER: Please.

HALES: — and you'll probably find this very interesting. In fifth grade, I got very interested in chess. My father somehow figured out how to encourage me in this. He took me down to the Pasadena Chess Club, and convinced them that they should be nice to me—and they were. For about a year, I would go in one day a week to the Pasadena Chess Club, which was this room filled with, what looked like to me, old men who smoked all the time, so it was dense with smoke. They were very nice to play chess with me, and help teach me the game. There was one man that was particularly nice, who was the strongest player there, and was also head of the Pasadena Chess Club, and his name was Sidney Weinbaum. I don't know if that rings any bells. This was for about a year I did this, and then I got interested in other things. A year later, we opened up the Pasadena Star-News one morning, and it says, "Sidney Weinbaum arrested for treason," or something like that. It was a horrible headline, a horribly shocking headline.

What actually happened was he'd been working on sensitive things at Caltech, and he'd been asked if he had ever been a member of the Communist Party. He said no, because he didn't want to lose his job. But he actually had apparently been somehow connected with it. Anyway, this was during the Communist Scare, so the whole thing blew up into headlines and everything, and he lost his job, and he went to jail. He had been working at Caltech all of his career basically. The Caltech people, including Linus Pauling, jumped in and did their best to save him. It didn't help Linus Pauling's reputation, in a way, because then people started getting upset at him too. But, anyway, many years later, Judy Goodstein did an interview with Sidney Weinbaum after he came out of jail. You have that interview in your library—and it's well worth reading, and it's very interesting. It brings in all sorts of things involving the Communist Scare, and Caltech, and the famous people at Caltech that were on the side of Sidney Weinbaum, and so forth. But, from my point of view, he was a very nice man, and took very good care of me, and helped to teach me chess. [laugh] But it's well worth reading.


HALES: I think there's some other things like that in your libraries too, I'm sure. But that's certainly worth looking at.

ZIERLER: Did chess put you on a mathematical path?

HALES: I was already interested in mathematics, I think, but it's close enough to mathematics so that I was interested. Certainly, in all the places I went, and all the mathematicians I met later on, a certain number of them were always chess enthusiasts. I continued to have some interest in chess, but I never took it up seriously as a devotion, so to speak.

ZIERLER: Were you always advanced in math, even in high school? Was that sort of your area of strength?

HALES: Yes. I think in third grade, I remember being ahead of everybody else in terms of the multiplication tables. But what I really remember is in fourth grade at Polytechnic, I was participating in a play that the whole campus was putting on. It was run mainly by the ninth- graders, but they needed some fourth-graders to act as many soldiers in the play or something. During rehearsals, I was sitting around with some ninth-graders, and a ninth-grade girl decided to teach me the binomial theorem as a fourth grader. She was learning this in ninth grade but, for some reason, she knew that I was interested in mathematics, and I guess she thought that it would help her understand it if she could teach it to a fourth-grader—so she did. She taught me the binomial theorem. [laugh] I never forgot that, and I'm still in touch with her. I think from then on, mathematics has always been, starting in fourth or fifth grade, has been something I knew I wanted to do for a living. I did, tentatively, when I entered Caltech, put down math and physics as possible majors. But after several laboratories, I decided I'd scratch the physics. [laugh] I would take physics courses, but I wouldn't take any physics labs. [laugh] It was mathematics from then on.

ZIERLER: Was it foreordained that you would go to Caltech? Did you apply elsewhere?

HALES: Yes. I guess I actually applied to three places. I applied at Pomona because my half-brother had gone there, and I applied at Stanford, and I applied at Caltech. I didn't actually complete the applications for two of them because by the time I got around to the final aspect of it, it was clear that Caltech was what I wanted. But I did go out, and talk to the head of Pomona. I don't remember exactly. But, anyway, I didn't complete the applications there, but I definitely decided on Caltech. But there was a strange reason that that was the first choice. It had nothing to do with mathematics directly. It had to do with the sport of badminton. By the time I was in 10th or 11th grade, I had become wild about the sport of badminton. I got into that because the daughter of friends of ours in Pasadena, who is now my wife, and is sitting in the room behind me, I'd gone down to watch her play in a badminton tournament. Both I and my brother became devotees of the sport. He became a national champion. My wife became a national champion. I came close but no cigar on that, but I won several tournaments. The point is that the Pasadena Badminton Club was the top club in the country. If I was going to keep up with that sport, I was going to have to stay in or near Pasadena. It certainly helped a lot that Caltech was also the best school in the country [laugh] for my academic interests.

ZIERLER: Al, tell me some of the professors in math, as an undergraduate at Caltech, who were really important for your education.

HALES: I'll give you four names if that's okay. The first was Basil Gordon, who actually had just gotten his PhD at Caltech, and stayed on one extra year. He was a student of Tom Apostol. He was teaching a freshman course, and I happened to sit in on his office hours, and I would still be sitting in them if he hadn't had to go home that day. [laugh] He was so magnetic that I could have listened to him for hours and days. That was the first one. The second person was Tom Apostol himself, whom you may have met. Did you know him?

ZIERLER: Of course.

HALES: I took his math analysis course as a sophomore, and the hard problems were wonderful and taught me a lot. That's number two. Number three was Bob Dilworth, who ended up being my thesis advisor. Then the fourth was Marshall Hall, who came to Caltech in my junior year or something like that, and had a lot to do with my interests also.

ZIERLER: Do you know the circumstances of recruiting Marshall Hall to Caltech?

HALES: Yes. I was almost involved with them, in a way, because in 1958—that's when I started working at JPL during the summer—the man I was working for, Solomon Golomb, that I mentioned to you, got involved. At that point, Marshall Hall had come out to Caltech to be interviewed, I think. Solomon Golomb knew Marshall Hall, and was interested in his mathematics, and met with Marshall Hall when Marshall Hall was out at Caltech for this visit.

After that visit, Solomon Golomb wrote up a summary of his conversation, and he passed it around to all of us at JPL to read about who Marshall Hall was, and why he was important, and why it would be great if he came to Caltech. I was aware completely of all this before Marshall Hall ever came, as a matter of fact. Then when Marshall Hall came the next year, I took all the courses I could from him.

ZIERLER: Al, what were the big and most exciting ideas in mathematics that you remember as an undergrad?

HALES: I think the most exciting idea was a course that I took from Dilworth. It was a graduate course in algebra, but I took it as an undergraduate. It was a three-quarter course, full- year course. He began in a somewhat unusual way by spending almost the whole first quarter on axiomatic set theory, which one doesn't always come across right away in an algebra course. But he thought it was important. I got a quarter of axiomatic set theory, which had a great deal to do with all my work on abelian groups. Then for the rest of the course, I would say, three quarters of the mathematics I did after that, or at least half of it, stemmed from things that I learned or was led into by that course or a subsequent course that Dilworth taught on infinite abelian groups, which led to the Ulm's theorem work. I think that was the most important.

The second most important would be either Marshall Hall's lectures on combinatorics or Sol Golomb's talks on combinatorics that first summer that I was at JPL, Sol Golomb was giving an informal course on combinatorics to everybody who would listened to it, and I was certainly one of the most avid listeners. I got two doses of that, one from Marshall Hall, and one from Sol Golomb. Based on those, later on when I went to Harvard as a Peirce instructor, I taught the first course that had ever been taught in the Harvard math department on combinatorics. I based it on the courses I got from Golomb and Marshall Hall. That was the first time they'd actually had that at Harvard, which is strange because Harvard was very advanced in most all of mathematics. But they were quite a bit behind on combinatorics at the time, I think, because they thought it was too applied a subject, and they didn't want to get involved in it. They were wrong, of course, but now they realize that. [laugh]

ZIERLER: Al, tell me about your work as a summer student at JPL.

HALES: Sol Golomb sat me down, and he had a problem involving shift register sequences. It involved factoring polynomials over the two-element field, and we didn't have computers then, at least, there were no computers that were capable of doing this. What he did, he handed me a pencil and a piece of squared paper, and that was my computer. I was supposed to try to factor all the polynomials of a certain sort over the field of two elements that I could in the length of time that I had to do this. As I said, it was a strange kind of thing to do. Nowadays, you would poke a button on a computer, and it would do the whole thing instantly for you. But that was not the thing then. I said they were particularly interested at that point because they were trying to devise a good way to measure the exact distance to Venus. They were going to do that by bouncing a signal off of Venus, and looking at the signal as it came back, and decide how many bits had passed during that time by looking at the whole sequence.

They needed a sequence that was billions of bits long that they could use for this purpose, and it was going to involve using these polynomials that I was working on. I didn't know how. He didn't explain to me until later why or how he was going to do this, but that's what I was doing. The first thing I did that I was proud of was to discover a theorem. I discovered that a certain polynomial could never divide a three term polynomial. That turned out to be a new theorem that nobody there had noticed before. It made my work much easier, which is why I found it, because I was trying [laugh] to make my work easier. But I got credit for that, and I think, because of that, they were quite happy to keep hiring me back for a number of summers after that. It meant that Sol talked to me a lot more about the work he was doing, because he knew I would probably understand it, and so that was it.

ZIERLER: Being at JPL shortly after Sputnik, did you feel the Cold War? Did JPL feel like a center for national security?

HALES: It did after I'd been there for a little bit. I wasn't aware when I first got there that anything that was being done there was classified in any sense. There was a fence, but no one paid too much attention to it. People came in and out of the fence. But, all of a sudden, they did start worrying a lot about this, and the fence turned out then to be a real fence. All of a sudden, they started checking people when they came in, making sure they were who they said they were, and so forth, and they had the right clearance. But it was interesting because in the building I was in, that Sol was in, there were several foreigners who didn't have the ability to get security

clearances, at the time anyway, so they modified the fence. They built an "inner wart" in the fence surrounding the building so that it would actually be outside the fence! That's where all the people that didn't have a clearance would be sitting. The fact that they had to modify their fence just to take care of this made me all of a sudden realize that there were things going on that I didn't know anything about. In some ways, if I can jump way ahead at this point, after Sol died in 2016, there was some work he'd done at JPL on an Army contract that was never released, but some other people had done it 10 years later on the outside, and gotten all the credit for it. His daughter and others felt that that wasn't fair. Sol should be sharing the credit for that. The question was, could we get his original work released so he would get credit for it? I helped doing this. I wrote a letter to the appropriate Army office or something, and they managed to find the old paper, and they ended up releasing it. In this book that I told you about, we actually have a portion of the paper that was initially written at JPL. But I didn't really realize any of this till later on. I thought they were just doing space work.

ZIERLER: Al, your decision to stay at Caltech as a graduate student, is that all about your interest in working with Bob Dilworth?

HALES: No. There were several things. That was complicated. I did look at several other schools as possibilities. One was Harvard as a possibility, but it was mainly badminton still. But it was also the fact that I'd taken enough graduate courses while I was an undergraduate at Caltech, so that I realized that if I could do a quick thesis, I could get my PhD in only two years. Whereas if I went anywhere else, it would be at least three and maybe four or five years. It would save a substantial amount of time if I stayed at Caltech, and did this. That worked out, it turned out, and it was a very good thing for a bunch of different reasons. One is I did manage to find a good topic to write a thesis on, and so it all worked out. I got my degree in two years.

Also, I got to continue the badminton activity. Furthermore, I got back together with the young woman who is now my wife sometime, I guess, early in my first year of graduate school. We got married at the end of that time. It worked out from all conceivable points of view to have done that. Then I ended up going to Harvard on a postdoc later on, so I didn't miss out on Harvard. It's just that the timing was different.

ZIERLER: Tell me about developing what ultimately would become your dissertation topic at Caltech.

HALES: Dilworth had an earlier student named Dick Pierce, about 10 years older than I was. He'd done extremely well, and I think he'd actually done the same thing I ended up doing. He'd gotten his PhD in two years by staying on. By the way, that was another reason that I did it myself. I had this model to follow because Pierce had done it 10 years earlier. Anyway, Dilworth suggested that I write to Pierce, and ask him to suggest some open problems in Boolean algebras, which is what I was getting interested in. Dick Pierce sent me a letter in which he described the four most important open problems in Boolean algebras. I looked at these, and one of them looked particularly interesting.

ZIERLER: What were the four?

HALES: Oh boy. I think I can only remember two of them now.

ZIERLER: OK, we'll go with that.

HALES: One was the non-existence of a free complete Boolean algebra on an infinite number of generators. That was the one I solved. The other one I can remember was classifying complete Boolean algorithms under direct summation. You can combine two Boolean algebras by taking their direct sum, and getting a larger one. You can take the opposite view. If I give you a complete Boolean algebra, can you decompose it completely into indecomposable pieces?

There's certain technical aspects to this, and that's something that I was always interested in, but that's not what I chose. There were two more that I can't remember now. But, anyway, I managed to solve one of them just barely, by the way, because at the same time I solved it, a student of Tarski's named Haim Gaifman at Berkeley solved it with a completely different technique than the one I used. But we did our work at exactly the same time, so we published "together", as a matter of fact. The timing was good because either one of us could have been completely outdone by the other if the timing had been slightly different. But we did them at the same time. Then it turned out that a couple of years later, Bob Solovay came along. He had another proof, which was much simpler than the one that either I or Haim Gaifman came up with. Now it's always known as the Gaifman-Hales-Solovay theorem, I think.

ZIERLER: Simple is always better?

HALES: No, not always better, because sometimes the more complicated ways lead you in other interesting directions. But it's simpler, at least, in terms of people who are trying to learn it, to use it in other places, anyway. What Solovay actually discovered was that an example other people had been working on for many years, for other reasons, was an example of the kind of thing we were looking for, and he'd managed to prove that. It just added to the knowledge that these other people were working on. But sometimes simpler is better, and sometimes it's not better because the techniques you develop to do the more complicated techniques helps you to do other things as well.

ZIERLER: That's one. What was the other, the one you worked on?

HALES: That's the one I worked on.

ZIERLER: That's the one you worked on?

HALES: Yeah. The other one I said is the one decomposing something into a direct sum.


HALES: The next thing I did was to work on a problem, another problem that Dilworth had mentioned, the one that led to the work on Ulm's theorem. That was the next thing that was going. But at the same time I was doing this, I was working on combinatorics problems connected with things I'd gotten into at JPL. In particular, I was working with Bob Jewett on this particular question that led to the Hales-Jewett theorem. I was doing two things that seemed completely different anyway.

ZIERLER: Hales-Jewett did not compose part of your dissertation? That was separate?

HALES: No, it had nothing to do with the dissertation.


HALES: To tell you the truth, I don't think that either Bob or myself felt that, I mean, we thought it was a nice theorem, but we weren't sure anyone else would be interested, or at least not many people would be interested, partly because places like Harvard, as I just mentioned, didn't think that combinatorics was a very important thing. They thought it was a kind of applied math, and that it wasn't interesting, and so forth. The whole mathematical world wasn't quite ready for this. I don't really think that either Bob or I felt that it was that important at the time, or that it was our most important work. Even in retrospect, I don't think we even thought it was. But the outside world ended up feeling differently [laugh], to our great—

ZIERLER: What's the takeaway? What's the—?

HALES: —to our great benefit. It wasn't till 1971, I think, my wife and I had gone to the University of Washington for a year. I had a sabbatical. We came back from the University of Washington, and bought a house in Pacific Palisades. We'd only been in that house a short amount of time when the phone rang, and I answered it, and it was Gian-Carlo Rota, who was a very famous mathematician from the East Coast. He was calling to tell me that I was a co- recipient of the Pólya Prize in Applied Combinatorics. First of all [laugh], I didn't know about the Pólya Prize. I'd never heard of it before, because it was the first time it was ever offered, and I was just stunned. In fact, there were five of us that shared in that: Bob Jewett and I for the Hales-Jewett theorem, and then three other people that had proved another theorem following on that. Bruce Rothschild and Ron Graham, and I can't think of his name now from Europe had proved another theorem following on then. We five shared in the first Pólya Prize. That's the first time that I realized—that we'd both realized—that the outside world considered this really important, partly because someone had decided to offer prizes in applied combinatorics. Anyway, you just never know.

ZIERLER: Al, what's the lesson, what's the takeaway in not appreciating in real time the significance of this work?

HALES: I think it's just that you never know. You should choose problems because you think they're interesting, not because you think they're necessarily important to the outside world. That's an easy thing to say, but if you don't have tenure yet, there are other things you have to [laugh] think about too.

ZIERLER: When did you first meet Bob Jewett? When does he enter the scene?

HALES: I think he entered the scene on a volleyball court. Volleyball was another sport that I found very important. In fact, volleyball was, I think, the way I got into the student houses. There were only four student houses when I got to Caltech and, since I lived locally, I didn't get in them. They didn't have room for everybody. But at freshman camp, somebody realized I was a really good volleyball player. The head of Blacker House recruited me to join Blacker House as soon as they had an opening. I think in January of my freshman year, I got into Blacker House, and I played on the Blacker House volleyball team. Bob, I think, was playing on the Fleming volleyball team, and I think that's where I met him before I met him in a math course. But we overlapped so many ways, both volleyball and math wise, that I can't remember the complete timing at this point.

ZIERLER: Al, in the late '50s and early '60s, were there computers at Caltech that mathematicians used?

HALES: Yes, but they were pretty crude. First of all, they were huge. It's not something you carry around, and certainly not in your pocket, and it doesn't fit on your desk either. But for the timing, though, on all this, I'm trying to think when the first one actually appeared that I was aware of. I guess I was never aware of one appearing on the Caltech campus while I was there. They probably were around, but what I was much more aware of was the fact that there was a synchrotron that was functioning, some nuclear thing that was functioning in a building right next to the math building. That was much more interesting than a computer [laugh] from my point of view. As I said, when I was working at JPL, they may have had computers that would do some things, but they certainly wouldn't do things like try to factor polynomials or anything like that. They could presumably add, subtract, multiply, and divide rather rapidly. I'm trying to remember when my father first got one of those in his office. But they wouldn't do anything sophisticated at all.

ZIERLER: What was Dilworth's style like as an advisor? Did you work closely with him?

HALES: Not that close, partly because I already knew what I wanted to get involved in. He had a huge effect on me from the courses I took from him, which pointed me in the direction.

But, as a matter of fact, he was actually not even at Caltech during my first graduate year, I think. He had a sabbatical, and he was maybe out of the country—certainly out of town—during then. He didn't have that much of an effect in terms of me as I was actually working. I was just working on my own. But he had a huge effect on what problems I was working on, and giving me the background in order to attack them.

[unrelated conversation]

ZIERLER: When did you feel like your thesis was complete enough where you were ready to defend?

HALES: Sometime like February—I'm guessing now—maybe February of my last year, something like that. I don't remember the precise timing. What I do remember is that there was one basic book on Boolean algebras that everybody was using at the time by a Polish mathematician named Roman Sikorski. Sikorski was visiting the United States at that time. In his visit, he first went to Berkeley, and then he came down to Caltech. I remember sitting in a room with him when I first met him, and he asked me what I was working on. I said, "I'm proving the following theorem." He said, "But Haim Gaifman at Berkeley just told me that he was proving that." [laugh] That's when I found out, at that stage, we had both managed to do it, and we both told him within a week of each other about this. I'm guessing it was probably around January or February of 1962 was when I finished.

ZIERLER: What did you see as your contributions, absent the Jewett-Hales theorem, with the dissertation itself? What did you see as your contributions there?

HALES: Oh, to my dissertation?


HALES: It was a very interesting question. But, basically, I wanted a thesis, and this was an interesting question that I found I could answer, so that was my contribution. In a way, it's unfortunate in that it closed off an area of research. It closed it off so thoroughly that there was nothing to follow up in that direction, at least, nothing that I wanted to get involved in in that direction. It was not an avenue for further research; it was just answering a nice question but not something to do next. What I was going to do next ended up being partly Hales-Jewett but then also the work on Ulm's theorem, which came a couple of years later. I was really moving in that direction after that, moving in the infinite abelian groups direction, so away from Boolean algebras.

ZIERLER: Were you looking at postdocs exclusively, or were you also considering faculty opportunities?

HALES: I got one faculty recommendation. I think someone at Caltech, either Marshall Hall or Olga Todd—Olga Todd was another person that influenced me a lot, by the way—had suggested that Cornell might be a good place to go if I wanted to go directly. I thought about Cornell but finally decided not to do it. I guess I applied for a Fulbright and an NSF postdoc. I think the NSF postdoc would've been to a Scandinavian country. Sorry, I mean the Fulbright would've been to a Scandinavian country.

The NSF postdoc was to England, and that's what I got, and that's what I took. I went there to work with Philip Hall—no relation to Marshall Hall but a very close friend, a colleague of his. Marshall had a lot to do with my going there, because Marshall Hall had been there himself sometime earlier, working with Philip Hall, so he thought it would be a good place for me to go. He made a very interesting and telling observation. He said, "If you go there, don't go as a postdoc. They don't like postdocs or they don't recognize postdocs. It's not an official position there." He said, "Go as a research student, that means a graduate student, a research student not working for a degree. That is a thing that they recognize at Cambridge." I said, "Yes, all right, I'll do that if you think that's a good idea."

That was almost the best suggestion I ever received from anyone because, as a research student not working for a degree, I had everything open to me that was available to a graduate student. I went to all sorts of interesting courses and learned a lot. I went to King's College there—that was one of the colleges. But since I did have a Ph.D., a different college, Jesus College knew about me through Dilworth, and hired me to tutor for them as a sort of junior faculty member. In that sense, I was unofficial—and they couldn't pay me because I was getting a postdoc salary.

They gave me High Table dining privileges instead, so I got to dine with all the full professors in that college. It was my first year at Cambridge, and Cambridge really had a screwy rule that if it's your first year there, even as a research student, you're eligible for all athletic teams. I played on the Cambridge badminton team, and that was the first official team of any kind I'd ever played on, because Caltech didn't have one. I played on the Cambridge badminton team, and we beat Oxford that year. [laugh] I had triple opportunities because of Marshall Hall's suggestion as to what to apply for.

ZIERLER: Now, had you wrapped up your work with Bob Jewett at that point, or that was ongoing?

HALES: Yes, we had. We finished and submitted the paper before we went off to England.

ZIERLER: Was the reception immediate, or did it take time to land?

HALES: No. I think somebody had some suggestions, and we had to make some changes and so forth. It took another year before it was accepted, I think.

ZIERLER: What was the research you did during your postdoc in England? What did you focus on?

HALES: I didn't do that much research, to tell you the truth. I took the opportunity to audit all the courses that I hadn't had a chance to take at Caltech by going through too quickly, if you know what I mean. I did my best to take courses in every area I could think of, including Philip Hall's lectures on group theory. I learned a lot of group theory, new group theory from him, and I did a little bit under him, but nothing I ever published. That was not really a year that I got research done; it was really a year that I learned more. I didn't get back to working on the Ulm's theorem stuff till the year after that when I was at Harvard, started at Harvard.

ZIERLER: Was that valuable, treating your postdoc almost like an undergraduate?

HALES: Yes. It was a sort of combined undergraduate and graduate year. It was a very valuable year from many points of view.

ZIERLER: Now, what was your appointment at Harvard?

HALES: I was a Peirce instructor there, Benjamin Peirce instructor. At that time, Harvard and most other universities had very few non-tenured positions. They had full professors. They had associate professors. They had assistant professors who didn't have tenure yet but were likely to get tenure. At Harvard anyway, they had six Peirce instructor positions, which are positions, teaching positions where you're not going to get tenure; it's just temporary. I didn't realize at the time, but that was true at most schools at that time. UCLA was the same way when I went there and so forth. In a way, that was bad but, in a way, it was good because you weren't feeling while you were there that you're competing with anyone directly for anything. You're just there to learn and work. Jumping way ahead, if you look at the faculty at Harvard or at UCLA now, it's totally unbalanced in the other direction. They have far too many non-tenured people, and I think it's not a good way to do things. But that's the way the world has changed.

ZIERLER: Your time at Harvard, this was not a tenure track position?

HALES: Not a tenure track position. I knew when I went there that it was a three-year position. I knew it was a three year position, and that after that, I'd have to go somewhere else.

ZIERLER: Culturally, what was math at Harvard like? What were the big areas of focus?

HALES: Algebraic geometry. Almost everything, at the time, seemed to be algebraic geometry. That was because of Grothendieck, the famous and weird French mathematician, who had visited there not all that long before, and had had a huge effect on the department. I think the department already was somewhat loaded in that direction, and it was even more loaded after this. That was a very exciting subject and a very exciting year. I did my best to learn as much as I could about algebraic geometry, and I still do as much as I can to learn about it; not necessarily that it helps me in the work I'm doing, although sometimes it does, but it's just that it's a very fascinating subject.

ZIERLER: Why? Why is it so captivating?

HALES: I don't know how to describe that, actually. But it uses extremely sophisticated algebraic techniques to give you an insight into something that's a purely visual subject, in a sense, studying geometry, although you're studying geometry of higher dimensions, not just two or three dimensions but four or five or more. But it's the combination I think of sophisticated and complicated and weird algebraic structures that give you an insight into something that you feel you can get your hands on, objects that have some sort of geometric meaning.

ZIERLER: When was it time for you to go on the job market, and what were your prospects?

HALES: I started thinking after two years, and partly because someone from UCLA came through on an interviewing trip, and was interviewing all the Peirce instructors. I was tempted at that point, because we really wanted to move back to Southern California, where our families were. My wife Virginia, as I mentioned, comes from Pasadena. I should mention—I don't think I mentioned it before—she's a granddaughter of one of the Greene brothers, the Gamble House architects, and you must know something about that.

ZIERLER: Of course.

HALES: We had several reasons to go back to Pasadena. We go back not only to visit Caltech but also for my wife to give talks at the Gamble House every now and again.

ZIERLER: Oh wonderful.

HALES: I got an offer from Caltech, sorry, I got an offer from UCLA after two years at Harvard, because I said these interviewers came through. I thought about skipping the third year, and going back. I finally decided not to do that. I decided to stay a third year at Harvard. UCLA very nicely renewed the offer a year later, so I didn't miss out on anything. By the end of my third year, I had two other offers also. One was from the University of Washington. That's where Pierce was, the man I told you that suggested my thesis problem. The other was a weird offer from the Rockefeller Institute. The Rockefeller Institute had no math department at the time, but they thought they ought to have one. They stole Gian-Carlo Rota away from MIT and must have made a huge offer to him, and told him he was supposed to build a math department. He offered me not a regular position but a visiting position at the Rockefeller Institute in New York.

That was very tempting, but after talking it over with my wife and my advisor and so forth, it seemed like I'd had enough postdoc positions, it was time to do something to give a more stable life, so I turned it down. That was a smart thing to do, in retrospect, because after several years at the Rockefeller Institute, Rota had a breakdown, actually. I think he couldn't face the situation he was in, somehow. He missed the real students too much or the academic atmosphere, and he left and went back to MIT, and had a stellar future career, continuing his past career at MIT. He was involved with creating that Pólya Prize that I ended up getting later. [laugh] I turned down Washington, and went to UCLA. But, as I mentioned, we went on sabbatical to Washington at our first chance to see what we might have missed. Anyway, my choice was UCLA, and that turned out to be by far the best choice. UCLA has done much better than Washington has, in my opinion, mathematically because they have more money. The state of Washington does not have an income tax and, because of that, I think the universities there have always been deprived of enough money to grow as fast as they might have. Whereas that has not, at least not until recently, been a problem in California. California had more money, and UCLA has grown enormously, and has improved its stature mathematically greatly. I made the right choice, there was no question about that.

ZIERLER: Were you part of UCLA math's growth? Were you part of a wave of hires in that era?

HALES: Yes, a huge wave of hires, actually. In fact, they overdid it because not only did they have more money, but they decided to go to a four-quarter system to have a full summer quarter also, which meant they'd have to have a lot of extra people to cover the extra courses. That part didn't survive. After a few years, they decided that was not a good thing to do. They abolished the summer quarter and went back. They didn't cut down on the number of people they were hiring but, for a short period of time, they overdid it. But still they did expand a lot.

ZIERLER: How did you and your wife adjust to life on the Westside?

HALES: [laugh] They put in a freeway about that time. That helped. [laugh] We lived in Rancho Park, not too far then from UCLA, for the first few years. I continued to consult at JPL during that time, because that wasn't quite so far, and the driving wasn't too bad. But as the years went on, the driving got worse. When we came back from Washington, we bought a house in Pacific Palisades, which was even further away from Pasadena than Rancho Park was, and it was just too much. I gave up my consulting job at JPL at that point. But we loved living on the Westside, and it was close enough so it was easy to go back on weekends and so forth to see our family in Pasadena. That part all worked out extremely well. By the way, family wise, I should mention that my brother Stan became a mathematician also. He went to Pomona, but he went to Harvard for graduate school, and he was at Harvard while we were there. He was a grad student while I was there as a Peirce instructor. Then after getting his PhD from Harvard, he went back to Pomona, and taught at Pomona for a long time. Eventually he became President of the College of Wooster in Ohio.

ZIERLER: Al, what was your research as an assistant professor? What were the big things that you worked on?

HALES: It started out with me finishing off the Ulm's theorem work.

ZIERLER: What remained to be done?

HALES: I was in the middle of it. I didn't actually publish that work until I'd been at UCLA for two years. I published it just in time for it to be helpful in getting me tenure. Basically, I was publishing about 1968, so I'd been working on that for at least two or three years at Harvard, plus two years at UCLA. It took a long time to come to fruition. By the time I got to be an associate professor with tenure, I'd just finished that, and I was starting to work on other things. Actually, some of what I was working on didn't really yield any wonderful results, at least not for a long time, but it was work. In a way, it was following up on the algebraic geometry I learned at Harvard but in applications to algebraic number theory, something called the Langlands program, which I got interested in during that year at the University of Washington. I worked on a number of things connected with that, which didn't actually, at least at that time, come to fruition. But there were some other things too, combinatorial things too. There was a period when I was not publishing that much, because I was branching out in new directions.

ZIERLER: What were those new directions?

HALES: As I said, one was following up on the Langlands program. It's hard to remember. There were a number of papers at that time, maybe 10 or 12 papers, but they were all in different areas. There were two papers on lattice theory that I was working on with Kirby Baker. Kirby Baker had been a grad student at Harvard when I was teaching there. When he got his PhD, he went to Caltech as a postdoc. Then after being at Caltech as a postdoc, he went to UCLA, and so we collaborated on several papers in lattice theory. I also collaborated with another colleague on several papers in topology, yet another completely different area of mathematics, and also on several papers in combinatorics with people at UCLA. I can't remember what else now. I didn't really get back into serious group theory until my next sabbatical, which was in '77–'78.

We went back to England then, as a matter of fact. By that time, we had three children, so it was a rather different year. But, anyway, I applied to several different places, I applied to the University of London, to the University of Edinburgh, and to the University of Warwick. It turned out the University of Warwick was having a special year on group theory, so that seemed like the right place to go, and that's where I went. We spent a year at the University of Warwick. We lived in Kenilworth, and that's where the kids went to school then. It was a wonderful year, mostly because I met an Indian mathematician who was doing the same thing. He was on sabbatical from India for this special year at Warwick. He was Inder Bir Passi. That's P-A-S-S-I. We hit it off together really well. We wrote one paper together that year, another paper together a year afterwards, and then a couple of years after that, he visited me in Los Angeles. Since then, we've written nine papers together all on one particular area in group rings, which we managed to more or less finish off completely after a lot of work in a lot of different papers. That led to a wonderful collaboration. I visited him in India also in the midst of all this, so that was interesting. But in the middle of all this, about that time—maybe I mentioned to you—IDA started having summer programs on the West Coast, so I started spending more and more summers on IDA work.

ZIERLER: Was that your first contact with the IDA, when they opened up the summer programs on the West Coast?

HALES: No. Oh yeah, that's right. I skipped something. Back when I was at Harvard, for those three years I was telling you about, I had free summers. By that time, I had a clearance from IDA because I'd been recommended. I went to the Princeton branch of IDA in the summers of '64 and '65. I didn't get back into another program with them until '76. That was also at Princeton. But after that time, I was spending almost every other summer with IDA. I got very much into some of their problems. That was all going on starting in '76 through to the late '80s.

ZIERLER: What were the circumstances of the IDA expanding to the West Coast?

HALES: Too many people who had done really good work for IDA in Princeton either came from California originally or ended up in California in academic jobs, and they were reluctant to spend summers in Princeton. It was too complicated, and the weather's not so good in Princeton, and so forth and so on. IDA was realizing they were missing out on a lot of talent by not finding a way of keeping these people involved, and an awful lot of these people were Caltech people. [laugh] There was myself. There was David Cantor. There was Howard Rumsey. There was Lloyd Welch. Anyway, there were a whole bunch of other people who had been at Caltech or had been on the West Coast or at least ended up there. IDA wanted some way to keep them involved, and the only way they could figure to do that really effectively was to open up a branch on the West Coast. They were very reluctant to do this initially because it was too far away from them to visit easily, and there was the computer problem.

You were asking about computers. By that time, computers were very important, and there was no effective way, at least not initially, to transfer information between computers one coast to the other. There was no internet and so forth. This was particularly if the stuff you're doing on your computer is sensitive you're not going to be sending it cross-country. There was great reluctance to set up something so far away, because it was hard to see how they were going to manage it—but they did it, and it ended up working. But it wasn't easy. As I said, I'm sure if I'd take a little bit more time, I could give you a list of a number of other such Caltech people. Unfortunately, most of these Caltech people have passed on already. Lloyd Welch was another person. Lloyd passed away on December 28th.


HALES: He was a spectacular mathematician, and he had gotten his degree at Caltech, and had worked at JPL briefly. Then he moved cross-country to Princeton, and worked for the Princeton branch of IDA. Did spectacular work there. But his wife came from California, and didn't want to stay on the East Coast, so he moved back to California, and went to USC, and spent the rest of his career at USC. But, fortunately, he was able to come down and consult for our outfit in La Jolla, so he was able to keep doing spectacular work, which he did. But Caltech were very involved in all this, at first at a distance.

ZIERLER: Al, was all of the work for IDA, did it all take place within a secure environment?

HALES: Yes, although in some cases the work done was eventually published, That happened with some of my work, actually. There's at least one paper I wrote with Don Newhart of NSA which was published. There were others as well.

ZIERLER: Now, when you were part of the group that won the Pólya Prize in 1971, was that the first year the Pólya Prize was given?


ZIERLER: What was the significance between who George Pólya was and what you were being recognized for?

HALES: He was a very well-known mathematician, and some of his best work was in combinatorics. In that sense, he was an ideal person to name that prize after. I had had some contact with him already, interestingly enough, because when he was at Stanford, he ran an exam for high school students like the Putnam exam for college students. But this is an exam for mathematically talented high school students. It was certainly administered on the West Coast, and maybe it was national too. I'm not quite sure. Anyway, I took that exam one year, and I did well enough to tie for first place with two other people. But in the process of doing that, I looked at some old exams, and I found one problem I couldn't do. Finally, I wrote him a note. I was being rather presumptuous, I thought, but I wrote him a note anyway, and he was nice enough to respond. He warned me that he didn't always respond to notes like this, but he would respond to this one. He gave me a very nice solution to it, so I was very, very happy. I had no idea at that time, of course, that the prize was going to [laugh] appear in his name, and I would end up getting it. But it was nice to have that early contact anyway.

ZIERLER: Was this what finally let it sink in for you, the significance of the theorem?

HALES: Yes, I think so. What it was was the fact that not only did Bob and I share in this prize, but the other people shared in it too. Those three people had their result build on our result. I saw that our result had important consequences. But that was just the beginning. Things exploded after that, and all sorts of other people got interested. There ended up being this mathematical blog thing, where you had people all over the world jumping in to try to do things. But I first started realizing when we got the prize that somebody was paying attention. [laugh]

ZIERLER: Al, when did the Institute for Pure and Applied Mathematics at UCLA get started, and were you part of its founding?

HALES: I was not part of its founding. By that time, I had moved on, I think. But when they were working on that, they called me back to be interviewed at one point, because they wanted to investigate what interaction there might be between IPAM and the outfit in La Jolla that I was running. I came back up to UCLA, and I sat through it. Whoever was doing the interviewing, and deciding on all this, asked me all these questions. I explained that our work was sensitive but sometimes things would spill over to other things too, and there certainly were potential for interaction. But then some time went by, and there were some other people that were involved as heads of board of trustees. But then they asked me to be head finally. Oh boy, I don't remember the precise timing now. About 2000—oh wait, what was the time being—2010 to 2018, I think, something like that, I was chair of the board of trustees there. During that period of time, I was very, very involved. But before that, it had been at a distance.

ZIERLER: It's right there in the name, Pure And Applied Mathematics. Was one of the ideas for the institute to get pure mathematicians and applied mathematicians to talk to each other more?

HALES: Yes, definitely, that was the idea, the main idea. It's been quite successful, I think, from that point of view.

ZIERLER: What have been some of the fruits of that founding idea?

HALES: I think it's compressed sensing was the thing that was most spectacular. Oh boy, you're testing my memory here. There was a summer program on compressed sensing that had a spectacular outcome. There was a statistician at Stanford—whose name escapes me now—who was involved in that, and helped build on it afterwards. But that's the thing that struck me the most. Since then, there have been advances in lots of other directions too, but that's the one that was the biggest, I think.

ZIERLER: What is compressed sensing, and why would that make sense as a meeting of the minds between the pure and applied mathematicians?

HALES: I'm afraid I've forgotten the details at this point.

ZIERLER: [laugh]

ZIERLER: I'm not going to be able to do it. If I thought about it for a while, I could probably reconstruct it mentally, but I don't remember the details. Just a minute, I'm trying to see if I can come up with some buzzwords. I think it's a way of taking a situation where you receive a very large complicated data set which you would like to store efficiently, and using the hidden structure (patterns) in the data to help on this. You have to figure out some way to find the patterns—this is now sounding like what we were talking about before—in it that will help you. But I can't remember the details now.

ZIERLER: Al, when you were asked to become chair of the department, was your approach to that more like a sense of service, that it was your time, or did you really look forward to having administrative leadership and responsibilities?

HALES: I think it was more the first. I had been very involved with administration from a lower point of view, in particular, with recruiting. I was always on the recruiting committee, so I was very concerned about the hiring there, and so forth. I was acting chair for one year when someone else had to take a leave. That was about 10 years before I was really officially the chair. When the time came for me to be the chair, I was aware there was a problem, because the person who was chair before me had not finished up his term. He quit in a conflict with the dean. I was aware of the fact that there was a troublesome dean that I would have to report to if I took the job. But I did feel it was my turn to do it, and I felt like maybe I could do something, so I said yes maybe somewhat reluctantly.

That was about the time that they were creating the new IDA branch in La Jolla, so I was aware that something was going on down there too. But I didn't know exactly what was going to happen. The dean turned out to be very very difficult.. I won't give his name. But I spent basically three years fighting with the dean. He was very good in some ways. He was very good at getting money for the departments that he was in charge of. But he felt that, since he was so good at that, he had the right to tell them what to do, and they had to listen to him. He had no feeling for mathematics at all. He was an applied geologist.

Basically, it was just a whole series of fights. During that period of time, at one point, I went over his head to the provost, and he (the dean) resigned because of that. Then the provost talked him into un- resigning. Then a year later, again because of the conflict, I resigned, and the provost talked me into un-resigning. You can see the conflict was going on all the time. But in the midst of all this, I think we actually made some progress, but it wasn't pleasant progress in any sense, and it was a relief to get out of that. As soon as I got out of it, the La Jolla thing came up. But I should say it was a breath of fresh air to go to La Jolla. The management that I had to cope with in La Jolla, you might've thought was worse but it was infinitely better. I did have to report to two different people. I had to report to the head of IDA, which is a separate organization, but I also had to report to someone at the National Security Agency because that's where the money and the problems came from. [laugh] But they were both wonderful. I had no trouble with any of them, so it was like a breath of fresh air going down there.

ZIERLER: What were the key issues facing the department during your time as chair?

HALES: We were trying to expand a certain amount, and the dean felt that he had the right to tell us who we could hire. He was trying to force affirmative action on us in too ham-fisted a way, so to speak. He thought that he had the right to judge on what people should be hired, and we felt completely different. Part of it was understandable. Deans always want you to hire cheaply, namely, assistant professors. Whereas we had found at UCLA that that was not the right way to do it. The right way was to take people who were like I had been at Harvard, and other people who had postdocs at prestigious other schools, particularly eastern schools, and hire them either as tenured or as high-level assistant professors. Whereas the dean wanted you to hire everybody as a low-level assistant professor because they were cheaper then. I think that was the source of most of the arguments. I still think we were right, because these people that we were hiring needed to have the experience from the eastern schools in order to be suitable for what we were trying to do. But, anyway, there were other arguments too. The department was completely behind me. There was no conflict in the department at all, so I shouldn't complain about that.

That part was wonderful. But it's just that dealing with this particular figure in the administration was not pleasant. There are a lot of interesting stories about that, which I'll tell you on some other occasion. [laugh]

ZIERLER: Beyond those battles, how would you rate your tenure as chair? Was it a successful term?

HALES: Oh yes, it was successful. We hired some really good people. Unfortunately, they didn't all stay. In fact, we hired some really, really strong female mathematicians during that time. Unfortunately, they all got bought away from us by, let's see, one by Princeton, one by NYU, the Courant Institute, and one by Microsoft, I think, something like that. They were wonderful people. But we also hired a bunch of good people who stayed too, so I can't complain about that. But that's the trouble, of course, if you're aiming for the top. Very fortunately, after I left, UCLA managed to hire Terry Tao, and that was spectacular.

HALES: David, you were about to ask a question.

ZIERLER: Yeah. What were the initial conversations that ultimately led to your decision to relocate to La Jolla, to direct CCR?

HALES: To be perfectly honest and I think not very modest, I think that the job was partly created for me. It was created for all these wonderful mathematicians that I was mentioning to you who were on the West Coast. But I think when they thought about doing it, one of the things they thought of first was, who do we get to run it? I think they all thought of me first because, by that time, I was the UCLA department chair. That's what they needed, somebody who not only knew how the business worked, which I did, but had a view of hiring, in particular West Coast hiring, that could help the new outfit hire good people. I was just the right person, except the timing wasn't for me.

They wanted me to come down right away (1989), and that would've been after my first year as chair, and I couldn't do that. I said no. Furthermore, we had one child still in high school at that time in Pacific Palisades, or in Los Angeles anyway. It was not a good time to move, either from UCLA's point of view or from the point of view of the family. I stalled, and I said, "Look, I'd be very interested in doing it, but you're going to have to wait until I finish my term as chair." I went through this several years in a row. They would keep nagging at me, and they would keep trying to find other people to act as chair until I could get there. What they ended up doing was just taking the person who really helped to create it, Mel Sweet, who was my friend and mathematical collaborator, he stayed on, and actually eventually they made him the director for one year until I came down. Then when I came down, he became associate director, and I took the director spot.

ZIERLER: Is this to say, if they created the job for you, was there not a director previously at CCR?

HALES: There was a director of the Princeton branch of CCR. They wanted a director in this new branch. The director in the new branch had to be someone who knew what they were doing from the point of view of the kind of work they did, but also someone who had administrative experience of some sort, and was willing to move to California or was already in California. You put all of those conditions on, you know, it narrowed things down to me and maybe just a couple of other people.

ZIERLER: Al, I'm curious, the early 1990s for CCR, the Cold War had ended. The Soviet Union had collapsed. What did that mean, both from a budgetary perspective, and an existential question for what national security and math looked like?

HALES: Let's see. One thing it meant was that as soon as I took over my job, there were links with the Princeton branch that were unexpected, and it had to do with what was going on in the rest of the world that involved NSA's mission and so forth. Then our West Coast branch got drawn into it, just because there were people on the West Coast that would contribute to the national effort. But it was too hard for them to get back to Princeton to do this work. It was much easier for them to come down to La Jolla to do it. We had several people that worked through us on things that had started on the East Coast already, which was interesting. But financially, there was a period of time when our existence was in question, I think, or we'd started already, and somebody on the East Coast started wondering whether they really could afford this or not.

Maybe with the fall of the Soviet Union and so forth, things weren't going to be as complicated. Of course, they were drastically wrong. Things got more complicated. But they thought it might be less complicated, and they didn't need a West Coast branch. There was a brief effort, which failed to close us down, and it didn't work. We survived that. There's been no financial problem since then at all.

ZIERLER: What were the big priorities for you in directing CCR?

HALES: Growing. We started with maybe four or five people that had experience already, including a number of the Caltech people [laugh] I mentioned to you already. My job was to start hiring people as quickly as I could.

ZIERLER: These are full-time positions? You're competing with academic departments?

HALES: Yes, either competing with academic departments or taking people who don't have tenure jobs yet, and getting them before they even get sucked into this.

ZIERLER: What's the elevator pitch?

HALES: It's tricky. Let me think now or try to organize my thoughts here. As I said, a few of them were people that were already down in this area and it was very easy to get. David Cantor left UCLA the year before I did, and came to work at the place. Howard Rumsey had a position at JPL. He retired from JPL, actually, but he unretired, and came to work for our place. That was another Caltech person. There were a bunch of people like that. But what I really was trying to find was young people, and so I was looking at new PhDs and assistant professors at other locations who might have an interest in being on the West Coast. One thing I did was to look at the Putnam exam. The Putnam exam is a national exam for college students, talented college students in mathematics. They put you for six hours in front of extremely difficult problems, so difficult that half of the participants don't get anything right at all, hardly. But a number of people that were already involved with IDA had been people who'd been successful with Putnam results, and had done very well on that. It was pretty clear that that was one thing to do, so that's one thing I did do. I looked at Putnam results for the last several years, and hired people away for that.

There was one young woman named Julie Kerr, who had done spectacularly well in the Putnam exam, better than any woman had ever done before, I believe. We hired her after she had—let's see, how did that work? She got her PhD from Michigan. We offered her a position that was really interesting, actually. She said she wanted to do it, but she asked for permission to spend her first couple of years at NSA, not with us. She had personal reasons for that, and NSA was quite happy to do that. It was good for us because it meant she had experience with our problems before she ever came to us.

Anyway, she came to us, and did spectacularly well. That was one of many people of this sort that I used the Putnam exams to narrow down on. But there were assistant professors at other schools. I did several tours up and down the West Coast, particularly going to Washington and Oregon and Berkeley and Stanford and UCLA and Caltech and UC San Diego, meeting with all the assistant professors at these places to find out what their long- term plans were. Many of them thought maybe they were going to get tenure where they were, but many of them either didn't think they would or didn't want to, and that was a good source of people, and it worked. By the time I left, 5 had grown to 20 or 25. Then my successor and his successor built it even further, so it's quite a bit larger now. I think it's full. The building won't hold any more. [laugh]

ZIERLER: Al, what was your elevator pitch for recruitment? How would you get the best mathematicians to come to CCR?

HALES: The best way to do it is to have them come for the summer. It's partly mathematical and partly money. When you offer them something that sounds intriguing, has interesting mathematics connected with it, you get to meet some other important mathematicians, and you get extra money in addition to your academic salary, that's a very effective recruiting way. That way, we got to look at them to see which ones were good at our kind of work. After all, that's the way most of us had gotten involved initially too. We'd gotten involved with the Princeton branch through summer programs. That worked very well, and it still works very well. I didn't explain this to you. The outfit in La Jolla and the outfit in Princeton double in size every summer. All the offices are single-person offices. During the summer, they all become two- person offices. Most of those extra people are academics. Some of them are from NSA, but that's to bring us the problems we're supposed to be working on. But that's a wonderful recruiting thing, and it gives these people a chance to see what we're like.

ZIERLER: Would you visit NSA? Would it be important for you to be on site at Fort Meade?

HALES: When I was director, I took one trip a month, at least. The idea was to visit NSA and the other sites. I would take three trips to Princeton, and three trips to Bowie, Maryland— that was the outfit, the more computing oriented outfit that I mentioned to you—and three trips to NSA. Then there were three months where all these people would come to La Jolla also. There was a trip every month that involved people from all three sites, plus NSA. I was at NSA at least three or four times a year, and more than that actually, because there were other reasons for visiting. I would say I visited NSA maybe six times a year, and I visited these other sites also, so it involves a lot of traveling.

ZIERLER: What was the dynamic between La Jolla and Princeton? Were they co-equal?

HALES: We were supposedly co-equal. Of course, that wasn't true at the beginning. They were four or five or six times as big as we were. That meant we were competing in our hiring with them. That was tricky, and a couple of places it really got tricky because the people at IDA headquarters were supposed to be managing this kind of thing, because they wanted to encourage us to grow. But they didn't want to do things to upset people in Princeton either. They had to try to manage things and, for the most part, things went reasonably well.

I can remember only two or three times where Princeton wanted to hire someone away from us, let's put it that way. I had to try to step in, and stop them from doing it or, at least, manage it. But, except for that, things were fine, because we all knew each other already mathematically, because we'd all been involved with the Princeton branch before La Jolla was created. We were arguing with friends when we were arguing [laugh], let's put it that way, and it worked. There were problems—well, I don't want to go into it—there are problems with salaries. It sometimes helps, if you're trying to increase salaries for young people to keep them from leaving, to move raise money from the more highly paid people because they have enough money already. But that's hard to do if you don't have that many highly paid people, namely, you're small and brand new. There's some tricky aspects to all this, but it all worked out.

ZIERLER: Al, during your time directing CCR, how did you manage to stay connected to academic research, and how did your time directing CCR influence your academic research?

HALES: Actually, I was very fortunate. I told you I'd met this Indian mathematician, Inder Bir Passi, on that sabbatical I had in England. The work with him came to a real head just as I was moving to La Jolla, and somehow we managed to carry out by correspondence a very successful succession of mathematical papers, during the time I was director. It just worked somehow. I could turn my mind off CCR things, and onto other things. He came to visit several times, and eventually I went to visit him once too, but that was after I stopped being director. But it worked. In some ways, that was my most successful mathematical collaboration, not in terms of the importance of the results necessarily, although they were important, but all my other collaborations had been one-shot deals—one theorem with Bob Jewett, one theorem with Pete Crawley, one theorem indirectly with Haim Gaifman and Bob Solovay—but not sustained work. But with Passi, it was sustained work, so there were nine papers. There were more papers than I did with anyone else. I don't know. I was just lucky, I guess.

ZIERLER: Al, what was 9/11 like for you personally, and what was its impact on CCR and the IDA in general?

HALES: It certainly raised the visibility of what we were doing, and added a lot of pressure. But, in a way, that was good. It helped us hiring in some cases. There were some people that very specifically had worked for NSA, had then gone off to do other things, and when 9/11 hit, they came back and asked, "Can't we renew our clearances, and get back in, so we could contribute to this?" There were some spectacular instances of this, so that was good.

On a completely different subject, since you bring it up, I mentioned my badminton career, that my wife and I were both enthusiastic badminton players, as was my brother. The world's greatest badminton player was an American, Dave Freeman. He died in September of 2001, and he came from Pasadena. His name was Dave Freeman. The Memorial services were in Pasadena, and my brother was supposed to give a talk at the memorial service about Dave Freeman's badminton career. The service was on 9/12 or 9/13, maybe, something like that. Anyway, it was two or three days after 9/11, and my brother, who was in the Midwest at the time, couldn't come to give the talk he'd written because all flights were closed down for a week. There was no travel. Stan emailed his talk to me, and I drove up to Pasadena, and I gave the memorial talk for Dave Freeman, just because of 9/11. I wouldn't have been [laugh] doing that otherwise.

HALES: On another subject, our son-in-law was in mathematical finance. He was supposed to be at a meeting at the top of the World Trade Center building in New York on 9/11, and his plane was late, so he survived.

ZIERLER: Oh my, wow.

HALES: The person that took his place at the meeting did not, so 9/11 had several influences [laugh] in that sense.

ZIERLER: Al, what was the timing? What were your decisions when you thought it was time to step down from the directorship at CCR?

HALES: When I took the job, I asked them what they expected because after having been chairman already at UCLA, I didn't feel I should spend the rest of my life as an administrator. They said, what I was told was, "We hope you'll do it somewhere between four and seven years." I did it for four plus seven years. I did it for 11 years, actually, because they didn't want me to leave, and it was getting hard to find people, partly because of 9/11. It made life even more complicated. But, anyway, I did it for 11 years, and I decided that was enough, and so that was it. It was also a problem of finding the right person to do it. That took several years also. But we did, we definitely found the right person.

ZIERLER: Were you proud of what you had built?

HALES: Yes, proud, and happy to be able to go back occasionally [laugh] once a week, and see what they're up to.

ZIERLER: Is there an emeritus designation at CCR? Could you remain connected?

HALES: No. You either have a clearance or you don't have a clearance. If you have a clearance, and they're willing to take you—but there is no official emeritus distinction. What I am now is I technically retired, and I'm brought back as an adjunct employee, a part-time employee. That's the same designation that we use for our summer employees. They're part-time; not full-time. They're cleared. Once I drop my clearance then I won't be able to go back.

ZIERLER: What did that mean for you in terms of picking up the academic research?

HALES: I'm not going to be around much longer probably anyway. We don't live forever, unfortunately, and I'm not doing much mathematics these days. As I said, what I've been doing recently has been working on this book about Sol Golomb. That's done now. I've written one or two recent papers. The most recent one solves a problem that Sol Golomb came up with many years ago. I'm doing a couple of things like that. I told you that Lloyd Welch has passed away. I'm running a special session on him at a local ITA meeting here in February in his honor.

There's an information theory meeting that happens every February here that I'm doing something in connection with. But other than that, I'm easing out of everything.

ZIERLER: What is so important about Golomb's work that inspired you to take such an in- depth look?

HALES: There's several things I haven't told you because they just didn't come up. But I learned how to be an effective administrator by working under him at JPL. I found that really became useful when I became first department chair, and then when I took over at CCR. Running CCR was very much like the way Sol ran the piece of JPL that he was involved with, and the mathematics is somewhat similar too, and some of the people are similar, or at least were similar, that we first hired. I felt that I owed him a tremendous amount for the effect he'd had on my career.

ZIERLER: Al, for the last part of our talk, a few retrospective questions about your career, and then we'll end looking into the future. Of course, what brings us together is Caltech. Have you remained connected with Caltech, and what does your time at Caltech mean for all that you went on to do?

HALES: It had an enormous influence on me, obviously, because I did both undergraduate and graduate work there, and the JPL work, so I owe it a tremendous amount, and the fact that we can conveniently go back to Pasadena regularly—at least when we were younger, and we didn't mind the drive so much—to see family, to do things with the Gamble House, and to do things at Caltech. It's been convenient to be able to do that. I took a lot of notes here ahead of time. There are all sorts of things involving Caltech that maybe I should tell you on some—what time is it, by the way? It's pretty late. I think we should continue this another time because I've got a lot of other incidents to tell you about that connect with Caltech. Were you involved with the book that I was sent a little while ago, the oral history project book I got?

ZIERLER: No, I was not. This must be from the Alumni Association. HALES: Yeah, it's from the Alum. You weren't involved with that? ZIERLER: No.

HALES: There's some wonderful stories that are partly connected with that, and partly connected with JPL. Is that all right with you if we take time out, and let's continue this on another occasion?

ZIERLER: That sounds perfect. I'll end here. [End of Recording]

ZIERLER: This is David Zierler, Director of the Caltech Heritage Project. It's Monday, January 29th, 2024. It's my great pleasure to be back once again with Professor Alfred W. Hales. Al, once again, it's great to be with you. Thank you so much for joining.

HALES: Oh, you're very welcome. Nice to be here.

ZIERLER: Al, we finished last time where I realized very quickly that there were many other Caltech stories that you had and you wanted to share. That's what I'm here for, so we're going to focus on that today. We left off last time, you were telling me a little bit about your interactions with Richard Feynman. Let's start there.

HALES: This would've been in my sophomore year. I was in the sophomore honors physics section. One of the quarters, Feynman took over. I had the great privilege of having him as a teacher for one quarter that year, which was marvelous and an unforgettable experience.

Then I believe it was the very year after that, he had a second year of teaching undergraduates. He taught one quarter of the upper-division electromagnetism course, maybe even two quarters but at least one quarter of it, and I had him for that also. I had him for two different quarters in two different years and, as I said, an unforgettable experience. That's probably the high point of everything at Caltech, actually. [laugh]

ZIERLER: Al, given that you were able to experience Feynman himself, do you think that made you ultimately a better mathematician?

HALES: Oh yes, definitely.

ZIERLER: How so?

HALES: I think because it gave me a much better insight into the interaction between physics and mathematics. Feynman had this remarkable insight into everything. For example, it was the connection between the field generated by a dipole in physics, and the derivative of a certain function. That was an insight that—who knows—I might have had it on my own, but when he pointed that out, it was startling to me and everybody else. But it was the interaction between the two subjects, and the way he had a view of this, also, and a little bit it's the business that he had a certain irreverence toward everything that was very interesting to see, I guess, because he was so smart. [laugh] I don't know. It's hard to describe, but it was certainly unforgettable.

Every once in a while, I run into classmates, and we reminisce about what things were like. The classmates who were involved in that, that's the first thing that comes up, certainly. As I said, we were lucky because it was the year after I graduated that he started doing the routine Feynman lectures in physics. What he had me for, those were two warm-up sessions for him. One thing that was interesting, it became pretty clear that—I don't want to sound bad for every—but I think he thought that we, being in the honors section of sophomore physics, might be a whole class full of kids that were as smart as he was. After the first exam he gave [laugh], which destroyed all of us [laugh]—

ZIERLER: [laugh]

HALES: —it was clear to him that we weren't as smart as he was [laugh], and the subsequent exams were more reasonable [laugh], which was an interesting experience too.

ZIERLER: Was Feynman uniquely inclined toward mathematical thinking in his teaching of physics, would you say?

HALES: I think so but only to a certain extent, because he realized that mathematicians didn't necessarily understand things the way he did, because he had the unique physical insight at the same time. It's a little hard to pin this down. But the fact that he had equally good insights into both subjects gave him a unique viewpoint on both subjects.

ZIERLER: Al, was the field of mathematical physics available to you? Was that something that bridged the line for you, to some degree?

HALES: To some degree, but it didn't make me feel that I wanted to go into mathematical physics. Mathematics was where my interests were and where I belonged. But the fact that I had a wonderful set of physics lectures—not just his but also Jon Mathews and other people like that—gave me an insight into my subject that not all my colleagues in mathematics had. I'd say it was a unique experience. Another thing I wanted to talk about, which I only got into very briefly in our earlier discussions, was my mathematical lineage. Dilworth was my advisor, Morgan Ward was his advisor, and Eric Temple Bell was Morgan Ward's advisor. I'm going back now.

Mathematically, Bell was my great-grandfather, and I already knew all about Bell in a certain sense because I think I had already read every science fiction story he'd written under the pseudonym of John Taine. I knew a lot about at least his science fiction predilection, so to speak. I was fortunate enough to actually meet Eric Temple Bell once. He died, I believe, at the end of my sophomore year.

But during that sophomore year, I was involved with and maybe even head of the math club. Anyway, I helped get him signed up to give a talk in the math club. As I say, it was just, I think, only about a year before he passed away. It was a treat to meet him and to hear him lecture, although he was pretty old and curmudgeonly, and not functioning too well. But, nevertheless, it was a great experience to at least listen to him talk. I still have all of his science fiction books. [laugh] It was after that that someone—Constance Reid, I think it was—wrote a biography of Eric Temple Bell, which was quite interesting because he had roots in Scotland, I think it was, at least in Europe somewhere, that nobody knew anything about till Constance Reid wrote this biography. That would give an interesting viewpoint on him too. Anyway, I did get to meet all of the people in my lineage, Bell, and then I actually had Morgan Ward and Dilworth as teachers, and Dilworth was my advisor. I feel fortunate to have met that much of my lineage anyway. (Incidentally, Morgan Ward was a distant cousin of my wife.)

ZIERLER: Al, what a great opportunity to ask a question that is unique to mathematicians. In science, generation after generation, they have the benefit of more advanced technology, faster accelerators, more powerful microscopes, what have you. They build on discovery based on those technological advances. In pure math, where that technology might not be as relevant, how do you measure progress?The question is, how, in the sciences where technology makes progress very easy to see and to perceive, how do you measure that progress, even in your own lineage, from one generation to the next?

HALES: There is progress, but it's different in a sense, almost the reverse of what you're describing in the sciences, because in mathematics, to a very, very large extent, it's never the case that we find somebody's been wrong in the past or that their theory has been proved to be inadequate or something like that. It's almost always the case that we build on the past. We don't replace things from the past with new, better theories. We build on this so that basically all the papers that my lineage wrote were correct, true, and interesting, and what I and others have done is build on these rather than replace them. Whereas in physics, it's more the case that you discover that the physicists of the 1800s, or whatever it was, they had insights, but their insights were inadequate to really describe what was happening. The physicists are still struggling with this. They still haven't managed to come to terms with the conflict between relativity and quantum theory. This is still going on. Now dark energy and dark matter may prove that. Anyway, it's a different kind of thing. But the mathematicians are building on things that are almost always—what was found before was correct, and we just extend them, and make them stronger.

ZIERLER: Was Caltech math unique to the extent that you had a perception of peer programs? What were the unique flavors of mathematics at Caltech?

HALES: Of course, I didn't have anything other than Caltech to build on at that stage. In retrospect though, I would say that, at that point in time, Caltech had insights and work in group theory and combinatorics that were the coming thing, in a way. But they weren't the kind of thing that Harvard and MIT, for example, or Harvard and Princeton anyway, were strong in. The Harvard and Princeton work was more classical, and concentrating—as we talked before—on things like algebraic geometry and so forth, which turned out to be very important but in a different direction than what Caltech was working on. I got an insight into one piece of modern mathematics, the part that Caltech was strong in, but I was missing really a good background in things like algebraic geometry, which I tried to get in my postdoctoral years.

I think, as I mentioned to you, in my first year after my PhD, I was at Cambridge, England, and then the three years after that, I was at Harvard. I partly spent time there not just extending my own research but trying to learn the subjects that Caltech had not been strong on. But what I did have was wonderful background in things like combinatorics. As I mentioned to you, I got to teach the first combinatorics course that had ever been taught in the Harvard math department. I was helping to fill in holes in Harvard's background, so to speak, but I was learning a lot of things that had been holes in my Caltech background.

ZIERLER: Culturally, was it uncommon or common to stay on for graduate school in math at Caltech?

HALES: Uncommon, and some people tried to talk me out of that. But I did it partly because I saw that I could jump-start my career by finishing in only two years, because I'd taken a lot of graduate courses as an undergraduate. What I did was uncommon and, in general, I think I would recommend to people not to do that. But in my case, it worked out very well. Also, partly, it worked out very well for me mathematically, but it also worked out very well for aspects of my personal life, which I may have mentioned before. I was going to say a little bit more about other aspects of my background.

ZIERLER: Please.

HALES: I don't know. I've got a wonderful story, which has nothing to do with my mathematical career, but it's something interesting about it. I was in Blacker House, not for my very first semester or first quarter, I mean. But eventually by the end of my freshman year, I was living on campus in Blacker, and I did that all four years. One of the highlights of that was—I guess it was in my junior year—my grandparents, who lived near campus on Arden Road, had both passed away by that time. The house on Arden Road was vacant, and my parents were going to move into it, but they hadn't gotten around to doing that yet. During that year, some of the Caltech students, including people I knew pretty well, had stolen the Occidental Tiger from Occidental. This is a big, huge sort of papier-mâché or plastic, or something like that, replica of a tiger that was their mascot. They had stolen the Occidental Tiger, and so we had that on campus. The question was, how are we going to keep the Oxy students from coming on campus and stealing it back? The solution, which I guess I partly suggested was, in the middle of the night, we took this tiger off campus, around to my grandparents' house, and hid it in their garage.

ZIERLER: [laugh]

HALES: We managed to do that in such a way—

ZIERLER: Oh, because you're local, this is perfect.

HALES: It worked out perfectly, because it was only about a half a block from campus, but no one would've known, no one from Occidental would've known that there was any chance that it would be there. It sat there for a week or two, I think. Finally, I believe what happened was that the Caltech president—which, by that time, was DuBridge, not Millikan—and the Occidental president—and I don't know who was president of Oxy—got together and negotiated a treaty, so to speak [laugh]. We had to give the tiger back to Occidental at that stage [laugh] to keep everybody happy. But it was ours for a week or two weeks or something like that, living in my grandparents' garage. [laugh]

ZIERLER: Al, in those days, of course, Caltech was an all-male school.

HALES: Right.

ZIERLER: Was Occidental, was that like where you would go for mixers? Was it coed?

HALES: That was one of the places we'd go for mixers. A lot of the mixers, though, were actually on campus. What happened was we would bring young women from either Occidental or from some of the local high schools onto campus for the various proms and things like that. That was interesting to some extent to me. But, on the other hand, some of these girls that were being brought from local high schools or local colleges were girls that I already knew because, after all, I'd gone to high school in Pasadena. I didn't find these girls, I mean, they were girls I knew already, so I didn't find them as interesting as some of my classmates who came from out of state found them. But there were mixers at Oxy and Pomona and at—I can't remember. There was a Catholic girls' school, St. Francis, I think. I can't remember now. Yes, there were mixers like that. But, as I said, they sometimes involved girls I already knew and, in some cases, girls I'd already dated. [laugh]

ZIERLER: Al, what other social aspects of your education did you want to discuss?

HALES: The tiger was one of the things I thought was most interesting. There was something else though along those lines. Now, what was the other one? Just a minute. I'm looking at a list I had here of various things. This is a story that involved two Caltech students. But it really involves JPL more than it does Caltech. I guess I told you already that I was working during summers at JPL.


HALES: One of the days when I went into work, we discovered that one of my JPL colleagues—who was also a Caltech grad student at the time, I think—he was late getting in, and we couldn't quite figure out why he hadn't shown up for work yet. By the time he got to work, we realized why he was late, because someone else at JPL had heard him discussed on the radio. On the way to work, this other fellow had heard a story about a Caltech student finding a snake in his toilet. When this other colleague did get to work, he explained all this. He said, in the middle of the night, he got up to use the toilet. He was married by that time. He got up to use the toilet, and he looked in it, and there was the head of a—I think it was an anaconda—

ZIERLER: Oh boy.

HALES: —snake. What's the other huge snake?

ZIERLER: A python maybe.

HALES: It might have been. I can't remember now whether it was a python or an anaconda. Anyway, of course, they called the fire department and the police department or something like that, and had discovered that the snake had been in a next-door apartment, and had been in the bathtub there but had escaped from the bathtub, and gone down the toilet and through the pipes, and at least part of him had come up in the toilet in the adjacent apartment. But then the question is, how do you get him out, because he's partly in these two different things, and when you try to do anything, he tightens all his muscles up, and you can't move.

Anyway, it was a massive effort to get him out. One of the interesting things about this was that later, I met actually the two Caltech students who'd been [laugh] in the other apartment, and who owned the snake. [laugh] I got to know both ends of the snake, so to speak. [laugh]

ZIERLER: Oh boy.

HALES: In fact, that story, part of that story is written up in the book I told you about, the oral history thing recently done by the Alumni.

ZIERLER: That's right.

HALES: If you look up Bob Tausworthe, that's T-A-U-S-W-O-R-T-H-E, he's the one that found the snake in his toilet. He describes that in some detail in his write-up in the oral history there, which is interesting. Oh no, I've got it wrong. It's the other end of the snake. It's Michael Krieger, K-R-I-E-G-E-R. It's his description in there. He's the one that owned the snake. That's the one that was described. That's certainly a very memorable experience, and it involves Caltech students at both ends, so to speak. [laugh]

ZIERLER: Al, I wonder if one of the survival functions of the pranks was as a release, a pressure release valve from the demands of the curriculum.

HALES: Yes. But I would say it was equally a relief valve for the—how should I put it— emotions or whatever of not having girls on campus. [laugh]

ZIERLER: [laugh]

HALES: I think both things played a role, let's put it that way. Anyway, that was an interesting aspect of campus life. I wonder what it would be like nowadays to be going through Caltech when they do have a coed atmosphere. I guess I'll never know.

ZIERLER: [laugh] I think the girls can participate in pranks just as well as the boys can.


ZIERLER: [laugh]

HALES: Maybe they found other things to occupy themselves, rather than doing the pranks as much. I wonder. I don't know if the pranks are as numerous.

ZIERLER: They're not. I don't think they're as legendary as they used to be—

HALES: Oh, I see.

ZIERLER: —yeah, definitely. Al, I'm curious, to foreshadow to your work with IDA, when you were a student, either as an undergrad or a grad student or both, were you aware of Caltech connections to the national security world, like the Manhattan Project, for example? Were those the kinds of things that gave you a window into those connections to national security?

HALES: I was certainly aware of the connections with the atomic energy aspect of things because, as I mentioned to you I guess, Robert Bacher, who was at Caltech and was the father of one of my junior high school classmates, he was on the Atomic Energy Commission at the time. I was aware of those connections. Also, there was a synchrotron, which is like a cyclotron, on campus at the time too. I don't think I was aware at all of the connections with code-breaking in the National Security Agency at the time. If I was, I was just barely aware of it, when I found that some of my JPL colleagues, particularly Lloyd Welch, left JPL, and went back east to work for IDA. I don't think I became aware of that. I didn't become aware of it until I was in graduate school anyway, and just barely aware of it, because my advisor at that point suggested that I might want to work for IDA during the summers, and would I be interested in him recommending me for a security clearance? I think I learned about it when I was in graduate school at Caltech but not when I was an undergraduate.

ZIERLER: Al, the other questions that I wanted to ask, before I get there, I wanted to know if there was anything else that you wanted to touch on.

HALES: Let's see, what else do I have on here of my list? When I was in Blacker House, I lived in the triple. That was an interesting experience. There were three of us all occupying one room, one rather large room that was on the sleeping porch upstairs. What we did was to move all our beds onto the sleeping porch, so we had a triple room that was completely vacant of beds, and that really gave us a lot of room for desks and things like that. That was a nice living experience, and I don't know if that's still the case. It was interesting. I always wondered, and I think my parents wondered too, how I could ever sleep on a porch overlooking California Boulevard. Didn't the traffic noise keep me awake? The answer is no [laugh], it never did. I don't think it kept any of us awake. We could sleep through it. The other interesting experience I had was when I started graduate school there, my advisor said, "You shouldn't just move back home with your parents, who are now living on Arden Road. You really should get the experience of living by yourself." I made a feeble effort to find some other place to live, not with my parents. I had an opportunity to live in the house that was occupied by George Beadle, who was at that time on the Caltech campus, a professor. In fact, I'd had him for a biology course. He had an extra room in his house, and I had a chance to occupy that room. I went over to take a look at the room, and thought about it a bit. Finally, I basically chickened out, and decided I wasn't that interested in that. It wasn't that great a room, and so forth, so I lived at home. But the interesting thing about that, there were two interesting things about that. One is it was probably a good thing I didn't choose to live there because, in the middle of that year, Beadle was stolen away by the University of Chicago to be president. I had no idea what he did at that stage, whoever was occupying the room, but he wouldn't have been there for that.

That's one interesting—the other interesting thing is that the woman I married, Virginia Greene, it was her grandfather who had designed that house. He'd designed it for the Bandini family. This was just north of San Pasqual now. Caltech didn't own that area yet. It was a house that had not been occupied right after Bandini by Beadle. It was occupied before by a famous biologist— whose name I forget now—that was responsible for bringing George Beadle to campus. But, anyway, there's an interesting thing about the house. It was designed by the grandfather of the woman I eventually married, one of the Greene brothers, so that was another interesting thing. Anyway, that's enough of what I jotted down here as things to talk about. It's your turn now.

ZIERLER: Al, I'm curious, just growing up as a neighbor to Caltech, just as a boy, where did Caltech loom in your imagination?

HALES: The fact there was a synchrotron there certainly was something I was aware of. Another thing I was aware of was Feynman, because Feynman was very often in the news. He was a very interesting character in many ways. One of the things he was famous for was playing bongo drums in one of the bars that was on North Lake. He would get written up in the Star- News occasionally. That was something about Caltech I was aware of. The other person I think I already mentioned to you is Sidney Weinbaum, in the last talk. That made a big splash in the local newspapers when he was arrested for supposed communist sympathies. That, of course, also brought Caltech into it because Pauling was involved in the thing, and Caltech made a big impression on everybody locally through the Star-News, what the Star-News wrote about. But beyond that, I don't think I was that aware of things. One other interesting aspect—this is getting down now to when I was actually applying for graduate school, and trying to make this decision—I had a copy of the Caltech catalog. I and my parents were looking through the list of courses involved, particularly courses in mathematics.

They came across this course called "combinatorial analysis" which was considered a huge mouthful at the time, certainly by my parents and by me too. What in the world are they talking about here? I think they might have been used to things like calculus and algebras but not combinatorial analysis. Neither of my parents nor I really knew what was going on [laugh], at least not initially when I was in high school and reading the catalog. [laugh] But that ended up being my subject, and I ended up teaching a course on it at Harvard eventually. That was shortened to just combinatorics. But, anyway, there were instances like that. But, basically, I was a naïve high schooler, and I wasn't that aware of other things that much.

ZIERLER: Al, the road not taken, have you ever wondered if you would've more or less done the same kind of mathematics, had you gone to a different school?

HALES: Yes, I still wonder that because it turns out that the work I ended up being best known for was either work that I'd been gotten into by Dilworth, my advisor, or by Marshall Hall, who I also got to know well, and Sol Golomb. It was the people that I met at Caltech and JPL that seemed to spark the work that I ended up being best known for. I'm guessing that things would've been very different if I had gone to Harvard, for example. I might have ended up going into algebraic geometry, for instance, or something like that. I've always wondered that. That's one reason I've always been very interested in those subjects, and tried to sit in on seminars and so forth. But I never ended up doing any actual published research in these things. Even though I'd like to think that I was very independent, I wasn't really that independent. I was being molded by the people that I was exposed to.

ZIERLER: Al, I don't know if you pay much attention to academic rankings but, if so, what has been Caltech math's rankings over the years?

HALES: Of course, when I think of these rankings, I think of—I can't remember who it was that used to publish these things once a year. There was one particular outfit that used to publish rankings that everybody looked at hard. In the one I remember paying close attention to—not when I was at Caltech but when I went to UCLA—UCLA was ranked down around 12th or something like that. We always were trying to improve that ranking. I think, by now, whatever the corresponding ranking is, I don't know if the same outfit still does it, but it's probably higher, maybe sixth or seventh or something, fifth, sixth or seventh now. What I don't remember is what Caltech's ranking was at the time I was a student there. I wasn't even aware of the ranking, as a matter of fact. I'm guessing that Caltech ranked above UCLA.

My impression was at the time, I guess, at least by the time I got to UCLA, was that the average ranking of the individuals in the Caltech math department was above that of the average ranking at UCLA. But, on the other hand, the UCLA department was much larger, so it covered a broader set of areas of mathematics. But I'm not sure my impression was correct. It's always molded by the fact that you tend to think that the place you're at is better than other places, or you'd like to think that. But, anyway, I don't know what the ranking is now, and my impression is there are many more rankings these days. Many different outfits have their own pet rankings. There was that one scandal recently. Was it somebody at Columbia, now? I don't remember who it was who discovered that Columbia had been fudging on—and I hate to say this, because it might not have been Columbia, but I think it was something like that. He discovered that Columbia had fudged the rankings that they prepared themselves by exaggerating things so that it indicated you can't trust anybody's rankings anymore. [laugh]

ZIERLER: Al, what about the perennial issue at Caltech of size? Caltech institutionally is small. The math department is small. Whatever the rankings are, how does it stay competitive with departments that are so much bigger, like at UCLA, for example?

HALES: It's not easy, but they do. My impression is that they do stay even, at least, in the areas they specialize in. But they do suffer from not having as broad a coverage of things. One thing I'm not sure of right now is the actual size of these departments. I know at UCLA, the department, if you look at it, is much larger than it was when I was there. But they've done that by hiring a large collection of assistant professors, who don't have tenure, and don't have prospects of tenure, and will have to go away eventually. That was not the case when I was at UCLA. Almost all the teaching was done by either tenured or tenure track people. But now they have a much broader collection by bringing in—I don't want to say cannon fodder. But these are people who don't have long-term prospects of staying there. I don't know if Caltech has done something like that or not. It may be that the Caltech department is also much bigger than it used to be by having a lot of non-tenured people around. But I haven't looked at the catalog recently, so I don't know. Are you aware of that?

ZIERLER: Have you stayed connected with Caltech over the years?

HALES: Yes, but only through certain individuals, pretty much. There's some people I have known quite well there, and some of them have moved over to UCLA. For example, I have not seen a Caltech catalog recently and, I'm ashamed to admit, but I haven't been in the building there since it was extensively remodeled. The last couple of times I was actually on campus and had time to walk down and look, to start with, the building was locked up. Often we'd go back and stay at the Athenaeum on weekends. Maybe I mentioned to you, as a former Caltech student, I have Athenaeum privileges there, and I don't need to pay dues at the Athenaeum. I only pay isolated small dues if I stay there, but I don't need to pay monthly dues.

But when I'm back on campus, it's usually a weekend, and the building's locked. It used to be that nobody locked anything up at all, and when I was there on weekends, I could go walk into the building and walk around. But since then, for many different reasons, I think, the building has been extensively remodeled, and I haven't seen the remodeling on the inside, and it's no longer named the Sloan Laboratory. It has the name of a new donor on it. It's locked up on weekends, so that's too bad. But I think that's true on all campuses. The UCLA buildings are also locked up on weekends now. [laugh]

ZIERLER: Al, we'll move to some retrospective questions about your career. First on the pure math side, what do you see as the most significant aspect of your work, either as it's judged by your peers or by how it's changed math generally?

HALES: I guess the thing that has changed math the most is the work I did with Bob Jewett on the Hales-Jewett theorem because, even though we didn't think it was that important a result necessarily when we did it, other people felt differently. It's been built on, and it's become quite widely known now. I personally always thought that my work with Pete Crawley was the most important thing I ever did. But that, in a sense, although I think it was very important work, that sort of finished off a particular area, and it hasn't been built on in the same sense recently that the combinatorial work with Bob Jewett has. That's also true with my thesis work, which was on Boolean algebras. It finished off an area, but it hasn't expanded that much in other directions. It's still true that people work on it and build on it, but it hasn't gotten the national attention. But things like that may change.

I realize after seeing what happened with my work with Bob that you're at the mercy of what people consider important and what other people consider important. Then I guess the other aspect of my work, I should talk about the work later in life with Inder Bir Passi on Jordan decomposition in group rings. That also, interestingly enough, built on something I learned from Bob Dilworth in my algebra course at Caltech. That work has gotten a lot of attention but, again, mostly just among algebraists. Again, I would have to say that the work with Jewett is, right now, the work that has attracted the most attention. But I wouldn't be surprised if things flipped over completely in another generation or partial generation, and something different happened. I just don't know.

HALES: There are a number of other papers that I've written too with a variety of other people, and who knows whether those will eventually be considered more important?

ZIERLER: Is there a correlation you can establish between the perceived significance of a given math project, and how much fun you've had doing it?

HALES: There's certainly a correlation between the fun I had doing it, and how I regard it afterwards. But I don't think there's necessarily a correlation between that and the way the outside world views things. Also, I guess this maybe a ticklish thing. There's work I've done that's sensitive, and may someday be deemed publishable. Who knows how that's going to be viewed? Some things I did at CCR did get published eventually, so I know that this can happen. But whether it will happen or not, I don't know, and whether other people will view it as particularly interesting, one has to wait and see.

ZIERLER: For all of your work in pure math, where do we see its applications out in the so- called real world?

HALES: That's an interesting question. Certainly some, I guess, applications in the real world, I guess what I would have to point to is work which could have been classified but wasn't classified because I did it at JPL working under Sol Golomb. But the work on shift register sequences with him, which appeared in part in his book with that title, has probably been the most influential in the world of electrical engineering and communications, in particular, because this has turned out to, you know, with computers and with digital communications and the internet and so forth, and the fact that shift register sequences are used by millions of people every day, even things as rudimentary as your garage door opener but also in secure communications, that probably has had the biggest influence on the outside world. The shift register sequence work is used in the outside world for everybody's communications now, basically. When you send an email to someone, the fact that it has any security to it at all is partly due to what was going on at JPL. That probably has the biggest influence.

ZIERLER: Now, on the classified side, what does it feel like to have contributed to our national defense? What satisfaction do you take in that?

HALES: I do, I take great satisfaction in that, actually. But, of course, I will never actually know specifically how this has been used because most of the work we did was passed on to NSA and was used by them in various ways that they didn't necessarily need to tell us about. I'm aware of some things, but I'm not aware necessarily of other things. But the fact that communications that I send now to other people are presumably not read by anyone other than the people I'm sending them to is due in part to the work I did at CCR and in part to my work with Sol Golomb at JPL. It's due to the efficiency of the shift register sequences. It's helping everybody, including myself, nowadays.

ZIERLER: I suppose getting all of the renewals to your security clearance is the best feedback you could ask for.

HALES: That's true. [laugh]

ZIERLER: [laugh]

HALES: Although I must admit, at this stage, I'm only working [laugh] a half-day a week or something like that, so I'm not making much of a headway, but I did. There was a conjecture that Sol Golomb and Lloyd Welch made in one aspect of the work he did at JPL that was an open conjecture for many years. It was just settled in a paper that I did in collaboration with Richard Arratia at USC, and Rod Canfield at the University of Georgia. There were three authors, and it was just published in the journal Combinatorics, Probability, and Computing last summer. That's the last paper. I'm not sure I'll ever write another one. But that's the last paper I've written that's appeared, and it's nice to know that we settled the conjecture that Sol and Lloyd Welch made back in the '50s. It's nice know that we still managed to do something even though none of us are young anymore. [laugh]

ZIERLER: Al, I have two final questions, both looking to the future. In your capacity, if you ever work with younger scholars or for the people that are starting their careers in mathematics who will listen to this interview, what advice do you have to give, both in terms of the most exciting areas to pursue in pure math, and how that might be applied for those mathematicians who would be interested in following in your footsteps, and working in support of the American national defense?

HALES: I think, one thing, I think it's healthy for any mathematician to take some time off from the main research program that they tend to be following, to do something different just to get a broader view of mathematics and the world. This could be by working for one of these CCR summer projects, or it could be working for something that's not necessarily sensitive but involves applying other aspects of mathematics. I think it's good for your overall viewpoint to see other people working on other things, and to interact with other people, and to get a slightly broader viewpoint.

You can't spend too much time doing that because you have to keep your nose to the grindstone a little bit to make progress in your main research program. But it's good to get a refresher every once in a while, and that can be done by working. As I mentioned, mathematicians can get jobs with stocks, genes, and codes. You can work for an outfit that applies mathematics to the stock market or to finances in general, you can work for somebody that's applying mathematics to the study of the genome, or you can work for someone that's looking at codes. Any one of those three, those are my favorite ones, but there are many others too. Any of that's going to bring you to meet interesting other people and to learning interesting new things.

ZIERLER: Finally, Al, we've discussed your academic lineage going all the way back to your great-grandfather. What about the timeline going in the other direction, by your students, and even the students of your students? Where do we see this legacy today, and what might that tell us about where math is headed in the future?

HALES: That's interesting. I only had two formal students at UCLA. One of them went into mathematical anthropology, in the department of anthropology. He stayed at UCLA, as a matter of fact, and spent his career analyzing the kinship relationships among African tribes, the mathematical aspects of that. The other one went into the study of semigroup theory, so a branch of algebra. I'm not sure if he had students or not. The lineage, the formal lineage I have is not very large, but I did a lot of work with students—how should I put it—postdoctoral work with students who I worked with both at UCLA and also at CCR. I think, both in combinatorics and algebra, I have a number of people that I can point to in those directions, actually. I don't have a list in front of me right now, but I say that would mainly be in the areas of algebra and combinatorics, and some of these are at CCR, and some of these are either at UCLA or other places, falling into group theory and so forth. Some of them are in India because of my working with Inder Bir Passi over there.

ZIERLER: Of course, algebra and combinatorics, these remain thriving vital fields. There's still plenty of work to do.

HALES: There's still plenty of work, yes, and in some sense, they're thriving even more than they have, particularly combinatorics. The interaction between combinatorics and algebra, I think, is one of the hottest areas in mathematics right now.

ZIERLER: Al, this has been a wonderful discussion. Again, I'm so glad we reconnected to get more of your perspective on Caltech. I want to thank you so much for spending this time with me.