October 21, 2021
David Wales has pursued big ideas in group theory, algebraic combinatorics, and representation theory over the course of his career. He grew up in Vancouver and developed his interests in mathematics at the University of British Columbia. At Harvard, Wales worked with Richard Brauer on group theory.
In 1967, Wales arrived at Caltech as a research fellow, he joined the faculty in 1968, and has spent his entire career at Caltech. His key collaborators include John Conway and Arjeh Cohen. Wales has served as dean of students and Master of Student Houses.
Since going emeritus in 2008, Wales has remained active in numerous areas of pure mathematics, and he is confident that enough fundamental problems remain that will keep the field vibrant for generations to come.
DAVID ZIERLER: This is David Zierler, Director of the Caltech Heritage Project. It is Thursday, October 21st, 2021. It is my great pleasure to be here with Professor David B. Wales. David, it's great to see you. Thank you for joining me today.
DAVID WALES: It's nice to see you. Thank you for inviting me.
ZIERLER: My pleasure. To start, would you please tell me your title and institutional affiliation?
WALES: David B. Wales. I'm a Professor Emeritus, at Caltech, California Institute of Technology. I have been at Caltech since 1967.
ZIERLER: David, when did you go Emeritus?
ZIERLER: In what ways have you remained connected with Caltech since?
WALES: Until the pandemic, I used to come in every day and work in my math office, doing various things. For a while after I retired, I was quite active in finishing papers that I had started with a Dutch colleague in the Netherlands. I finished those papers around 2014 or 2015. Since then, I've continued working with him. His son started a Dutch company for teaching mathematics on line. They prepare the material, exercises and everything, in Dutch. Then they use a program that translates Dutch to English, but it doesn't do it very well. My friend goes over the materials and makes them a bit more English-friendly. He then sends them to me so I can go over them with a focus on improving the English. For example, sometimes when you say things in Dutch and translate it to English, the adverbs are in the wrong places. He may not recognize that but I do. Sometime he doesn't recognize that a few Dutch words continue into the proof. I can spot those right away. Of course, I don't translate for him; I just tell him it's Dutch. "This is Dutch"—that's what I say. Occasionally, I find mathematical errors in the materials and point those out. So I've been doing that.
ZIERLER: Is this Arjeh Cohen you're talking about?
ZIERLER: Just as a snapshot in time, what are you working on right now?
WALES: Really, that's what I'm doing, although he hasn't sent me any materials for a couple of weeks. I think he said the next will be the end of the calculus part; it may be the end of all the parts [laughs] because it has taken him longer to send it to me. But when he sends it, I'll just do it. It takes about a week, usually, for a chapter of about 50, 60 pages.
ZIERLER: You mentioned that because of the pandemic, of course, you stopped going into the office. Have you been more productive simply by not commuting, not going in, or has the social isolation hindered your research over this past year and a half?
WALES: I don't think it has hindered it. I'm not sure it's called research when you're translating [laughs] not from Dutch to English, but from translated Dutch to English. But one thing I have done is go to more talks than I normally would. It's so easy to turn on my computer at 4:00 and hear a talk—now of course, for a while, there weren't any, but there are getting to be more. There's one today in the Math Department, but they haven't said whether it is going to be on Zoom. I don't know whether I'm going to go in to see it or not, because it is so convenient doing it from home. Of course, if I get tired or don't quite understand what's going on, it doesn't really matter when I'm on Zoom because nobody can see me, whereas in the classroom you have to stay awake, so I try to stay awake.
ZIERLER: With all of these talks going on in Zoom right now, more generally, what's exciting? What's going on in the world of math, circa 2021?
WALES: There are a lot of different areas, and they change. Some are stylish for a while, and then they're not. I think a lot of things are going on. I've done some combinatorics at different times, so there are quite a few things going on there. Of course, there are various other pure mathematical fields, like algebraic number theory, that are going gangbusters with all sorts of things going on
ZIERLER: A broad question—over the course of your career, have computers and computational calculations been important for your research?
WALES: Yes, they have. When I started, of course, there were no computers. I used to do computing by hand. But by the 1990s, everybody had a personal computer. There are mathematical programs that they developed that enable a person to access a lot of things that weren't possible in the old days. One program is Maple. Another program is GAP, which is European. Also, there's Magma, which is another program that does mathematical computing. I haven't used Magma, but I've used the others quite a bit.
Sometimes I've written programs that compute things, and then do it on the computer, whereas if I were to try to do the computations by hand, it would be too hard. You can imagine writing on a big black board, but it would be too small for the things that you could do on a computer. Computers can just do these things on their own. You have to trust them. You have to trust that they're really okay. Some things have to be reprogrammed because there are mistakes, but you can recognize the mistakes and fix them, because you are familiar with the program.
ZIERLER: With over 50 years of affiliation with Caltech, you have a tremendous amount of historical perspective to share. I'd like to ask a few general questions and ones that are specific to math. First, it's always interesting how, at Caltech, there aren't departments like there are at most universities. There are divisions.
WALES: There are divisions, and my division is the Division of Physics, Math, and Astronomy.
ZIERLER: On that basis, because the administrative structure is the Division of Physics, Mathematics, and Astronomy, being housed in that administrative fashion, has that affected your research one way or the other in a way that might be different if you were in a traditional department of mathematics at a different university?
WALES: I don't really think so. Most of the work I've done has been collaborating with other people who are at other universities. That's fine, because of computers, the internet and so forth. Sometimes people in mathematics think that they're kind of left out, because physics and astronomy seem to be driving our Division. For example, there has never been a division chair from Mathematics. They have always been from Physics or Astronomy. Periodically, when we are looking for a new division chair, in my experience, no one really talks about Mathematics or about mathematicians as a possible division chair. I guess that's partly because it has been thought that mathematicians would not understand the issues for physics and astronomy. Over time, there have been different division chairs, and for the most part, they have been very positive and recognize that mathematics needs some nurturing.
The current division chair, Fiona Harrison, has been very good for us, and so have many others. I think in the 1970s and 1980s, the number of mathematicians who were hired on a full-time basis decreased. Of course, the number always decreases when people leave for other places or retire. But ours weren't replaced. They were replaced by people in Physics or Astronomy. At a certain point, President Goldberger—I suppose you know your presidents.
ZIERLER: Murph Goldberger, the one and only.
WALES: The one and only Murph. He recognized what had happened, and made a point to try to get it back up to where it had been. They've done that by adding extra positions in recent years, although it's hard to keep people here because Caltech is a very small place. Other places are more attractive for people who want to work in a big mathematics department.
ZIERLER: What are the advantages of working in a bigger mathematics department, and what explains the desire of scholars like you to stay at a smaller mathematics department like Caltech?
WALES: I just like Caltech. The people who stay are people who like Caltech. The students are extremely good, very good all around, but the very good mathematics students are excellent. You are teaching small classes and have very good students. Not all students really like math [laughs] or like having to take math, but students do have to take math. Over the years, they've decreased the number of math courses that are required of all students.
ZIERLER: What do you think accounts for that? Why the reduced expectations in math courses?
WALES: I think one thing that has happened is that biology has gotten quite a bit bigger than it used to be. I think biologists believe that their students really don't have to know advanced calculus or other kinds of things. I think over time a lot of people in the Engineering Division and other divisions (well, maybe not Chemistry because they're pretty mathematically-oriented) have gotten to feel that way. There used to be two full years of mathematics. Now there's one full year of mathematics, plus different divisions require more courses. I think what has really happened is that mathematics is no longer considered as important for all students as it used to be.
ZIERLER: Your answer mentions of course the growth of biology. If you look over the course of your career at Caltech, when you started, physics was probably the largest major among undergraduates.
WALES: Yes, absolutely. And now it's computer science.
ZIERLER: Where does math fit into that in terms of that overall transition from physics being the largest major to computer science, in terms of math, the kinds of interests
WALES: I don't think it has really changed much. There are fewer math courses required for everybody, but there are more math courses that are recommended than there used to be. We have quite a few computer science students in one or two of our introductory classes. Math 6 has a lot of computer science majors. But there are more computer science majors everywhere, I think. One of the things that has happened, of course, is that when students come, they know a lot more about computers than they used to. In the 1980s few knew anything about computers. They were just starting. But now, it has broadened out, and there's a whole field. It helps mathematics in some ways because mathematical problems can come up in computer science and then mathematicians can solve them - - sometimes.
Prime Numbers and Dinner Parties
ZIERLER: Sometimes. [laughs] These conversations are always such a great way, particularly for mathematicians, to talk about their work in a way that would be accessible to non-specialists, to people who are not mathematicians. Maybe at a dinner party or something like that, when you tell people you're a mathematician, inevitably they'll say, "Well, what does that mean?" What answer do you have for them?
WALES: I say that there are a lot of mathematical problems that people work on and write papers about. For example, at dinner parties, my wife says I shouldn't talk about mathematics, but sometimes I do anyways. Problems about prime numbers are always interesting. These are numbers which cannot be factored into products of two or more smaller numbers. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23. It can be shown there are an infinite number of these, but there are certain kinds of primes that exist and finding properties of them, such as whether there are a finite number of certain kind of things, can be considered. Primes are considered the building blocks of all numbers because any number is a product of primes often with primes to a power such as 120=5x3x23.
One kind of primes is twin primes - such as when a number and that number + 2 are both primes. For example, three and five are twin primes. Five and seven are twin primes. Eleven and 13 are twin primes. They've used computers to go as far as they can, and there are always more twin primes. One famous conjecture is, "Are there an infinite number of twin primes?" You can see that you couldn't do it by computer, because computers will just go so far, and as long as they keep coming up with more, you don't know. Maybe someone can prove there's only a finite number. So those are problems that people can understand. Another kind of primes is ones of the form some power of 2 plus one which are called Fermat primes or some power of 2 minus one which are called Mersenne primes. Some of these are 3, 5, 7,17, 31, or 129. Notice some primes are not such as 11 and 101. No one knows if there are an infinite number of these Fermat or Mersenne primes. It seems possible from computer calculations that there may be an infinite number of both of them. Proving any of these would likely lead to new ways of looking at primes.
ZIERLER: For people at the dinner party who might know a little bit about mathematicians and mathematics, the next question might be, "Okay, what kind of mathematician are you?" What are the choices there? What kind of mathematician are you?
WALES: I'm what's called a pure mathematician as opposed to an applied mathematician. The area that I work on is group theory, finite group theory which is part of algebra. A mathematician might do algebra, analysis, topology, mathematical physics, geometry or logic, but I do algebra.
There are many problems in algebra. The one that I've worked on for the most part of my life is finite group theory. One of the main problems that people worked on was finding all finite simple groups. Groups are algebraic constructions which are sets with a multiplication between elements which gives another element of the set. The multiplication has properties much like multiplication of ordinary numbers but with different rules. Certain groups are called simple. In a sense, they are the building blocks of all groups. If you knew all of the simple groups, certain properties of all groups would be known.
Before 1900 there were many finite simple groups known. They include several infinite families. Around the 1950's several more infinite families were discovered. At that time it was thought that perhaps there were no more or only a finite number more. However there were several found between 1960 and about 1980.
Since then, it has been calculated that there are no more. This has been called the classification of finite simple groups. There are the several infinite families and several isolated simple groups. People stopped trying to find more in the 1990's and started investigating properties of the known simple groups. I did the same and started working on other algebras that were related to some of the things that show up with group theory.
Pure Math and the Real World
ZIERLER: The binary between applied mathematics and pure mathematics, is that to suggest that pure mathematics need not have any relation to the so-called real world?
WALES: I wouldn't really say that. What about the numbers? There are the numbers in the real world. That's pure mathematics. Numbers have a relation to the world. For example, you can figure out your net worth using numbers and so forth.
I think applied math is more oriented to more practical problems. The problems that applied mathematicians work on really do come from the real world, whereas the problems in pure mathematics come from other mathematicians, and working on things that are related mathematically to what they have constructed.
ZIERLER: The term "applied mathematics" would suggest obviously that mathematics of course has applications in the real world. But isn't it true to some extent that pure mathematics can be applied to the real world also?
WALES: I would say so, yes.
ZIERLER: In what ways?
WALES: I just mentioned numbers. Another is computer science. Computers really work with strings of zeroes and ones. I can't see the zeroes and ones as I look into my computer right now, but really, they're in there. Mathematics will consider various kinds of things involving the strings. Another area is error-correcting codes. These are codes that correct themselves. If you make a small mistake, the code can correct itself. These are used all the time—for example, in the Mars landers and communicating with them. Many things from mathematics are used in error-correcting codes. They are sort of a cross between applied math and pure math, because they involve many pure math question - for example, constructing them, and finding ways for which you can bound some information that can be obtained in correcting codes
ZIERLER: In your field, as opposed to, for example, physics, where there's theory that guides experimentation and allows for ability to understand what those experiments tell us, how might theory and experiment work in pure mathematics?
WALES: One way I've already mentioned is you can look for twin primes. You can design computer programs that go further and further, looking for twin primes. That's an example of something that you're given—twin primes—and you're thinking they're interesting. But there were other kinds of experiments that you can do with computers that will tell you trends about the rate of growth of certain kinds of mathematical configurations. A lot of times, there will be computer simulations of things that will suggest proving something. Then the computer usually can't prove it, although sometimes it does, but then it gives you an idea of what you should try to prove.
If you see a pattern emerging by doing a lot of calculations, you can see whether you can prove that those were always going to be occurring. That's an example where theory can come from calculations, something you could actually prove.
Fermat's Last Theorem is a theorem where it was thought to be true because of large numbers of computations for which it held. The theorem was not proved, and people tried to prove it for years and years. As they tried to prove it, they developed pure mathematical concepts that people either didn't know about or wouldn't have thought of. In a certain sense, various kinds of number theory, algebraic number theory, and I guess computational number theory, too, have grown out of that. You probably know that Fermat's theorem was proved recently, but it was proved using all sorts of things which we can now study. They developed techniques that other people now use to try to generalize the theory or make it more understandable.
ZIERLER: You mentioned simulations. One of the things that's very exciting potentially is the coming quantum computing revolution.
WALES: Oh, absolutely, yes. That's another field that is very mathematical. Quantum computing machines aren't just theoretical; they have to be built. But there are all sorts of algorithms that—if you could build a quantum computer, you'd be able to do quickly what you can't do now. I think financial institutions could use these things—banks and many other people—but it's conceivable that if a quantum computer really can be built, then they'll be able to solve various problems.
There's a former undergraduate by the name of Peter Shor who has worked on this a lot, and developed an algorithm called Shor's algorithm. The current algorithms involve factoring very large numbers. Factoring numbers is a mathematical problem. There's some theory that can be developed there. If you could factor very large numbers very quickly, you wouldn't be able to use the kinds of codes that they're using now, because quantum computers would be able to factor numbers quickly. Although I think so far, they can only factor very small integers.
ZIERLER: Have you ever pondered some of the philosophical implications of a career in mathematics?
WALES: I'm not quite sure what you mean by that.
ZIERLER: Like the meaning of infinity, or the meaning of zero?
WALES: Oh, sure, sure. Zero comes up in numbers all the time. For abstract algebra, zeroes are there, so I'm not concerned about zero. I think zero is fine. [laughs] These other abstract systems often do have zeroes, so I know lots of systems that have zeroes, not just the numbers. Infinity is much more complicated to deal with. There are various sizes of infinities which are called cardinals. This might seem surprising as they are all infinite. The smallest is the size of the integers. There's another former undergraduate, Hugh Woodin, who developed various cardinals that are now called Woodin cardinals. If you take all sequences of integers, you get a larger one which has the size of the real numbers. This is a subject in logic often discussed.
ZIERLER: If humanity ever makes contact with an advanced civilization somewhere else, do you think math is truly universal, that their math is the same as ours?
WALES: I do think so, yes. Of course, we'll know [laughs] when we make contact, and not before, probably. Although I think there are people who try and record things, thinking maybe someone else is sending us a message. What would the message be? The primes. You could send the primes. If you were going to send a message out to people, you might just start doing all the primes as they come in order. We think that other people—people is the wrong word, of course—would understand that. Now of course I'd like to send the orders of the smallest simple groups, the smallest of which has order 60. [laughs] But I don't think many people would think that's the first step. I think some people are listening to signals coming in, looking for patterns.
ZIERLER: In the way that physicists are searching for a grand unified theory to unite all the forces, is there a grand unified theory of math that would be required before that?
WALES: I don't really think so. I understand there's a grand unifying theory for physics which would explain everything, but physics is physics, and everything is really gravity and stuff like that. Whereas in mathematics, there are these other concepts that come up that really don't have a unifying property. They're just things that mathematicians think of and work on. There are so many areas, like logic, analysis, algebra, geometry, topology, and they're all different.
ZIERLER: I wonder, then, if math is the language of physics, and physics is searching for the grand unified theory, well, if math doesn't have its own grand unified theory, why would we expect physics to?
WALES: I think people think that physics is more down to Earth than mathematics. Down to Earth means really what you can do every day when you're doing things—moving around, planets colliding. All those things are covered—well, in astronomy—and gravity in physics. There's nothing like that in mathematics that connects logic to analysis to algebra or topology. They all seem to be different areas that mathematicians have built up over the years.
ZIERLER: Given that mathematicians have built these up, how do we know what are simply human constructs, and what are representations of physical reality?
WALES: I think the real numbers are pretty physical. You can draw a line, and on the line are the real numbers. These numbers are more down to Earth and have physical interpretations. There are also imaginary numbers which are in a plane. They might or might not have physical interpretations. They are constructs that people have developed because they have mathematical ideas which relate not necessarily to the real world but to what other mathematicians have developed.
ZIERLER: I'd like to ask some broadly conceived historical questions about your major research areas. Let's start first with group theory. Where did group theory originally develop, and as a graduate student, when you started your career in mathematics, what were some of the big ideas at that point in group theory?
Group Theory in History
WALES: The big idea in group theory when I started in the 60s was showing that either there were no more finite simple groups other than the ones that were known or, if not, what they were. When I started graduate school, that was a big impetus. There were people doing other kinds of things, but many of the people who were doing group theory in mathematics were doing finite simple groups.
Group theory goes back to Abel, who in some sense discovered group theory. Now, it first came about by trying to solve polynomials. Everyone in high school learned to solve a quadratic—ax2 + bx + c = 0. Here a, b and c are called the coefficients. We are assuming the coefficients are integers. An example is x2-2x+1=0. A root is a value for which if putting x as the number gives 0. For example, x=1 is a root of the quadratic above because 1-2times 1+1 =1-2+1=0. In high school you learn a root is minus -b plus or minus the square root of b2-4ac all divided by 2a. In particular, there are at most two roots and they are all simple sums and products of the coefficients and possibly a square root. A cubic is x3 plus a quadratic. The same is true there but you have to also take cube roots. A quartic is x4 plus a cubic and a quintic is x5 plus a quartic.
For any of these equations there is a group which can be defined called the Galois group. Now all the groups involved in the quartics or lower involve no simple groups, but for a quintic there are some which do have a simple group. For this reason, it can be shown there are roots which are not expressions in the coefficients even using fifth roots.
Galois was the first person to show this using the associated group and its properties and the fact that it might be simple. He showed that if the group is simple, you cannot solve it using expressions in the coefficients even using fifth roots or any higher roots. We periodically get people who claim that they've proved that you can solve them this way, but in fact you can't solve them this way. I remember one time a good student came up to me after class and said, "Well, if you can't solve it, where are these numbers because they can be solved? Another said, "Well, if we can't get all the numbers by joining square roots, cube roots and higher roots, what are these numbers?"—well, they're just other numbers that you can't write down just using the coefficients and roots of unity. However, you can solve x5-1=0 because its solution is the fifth root of one, so you can solve some quintics.
Another problem which is simpler and easier to prove is trisecting an angle by ruler and compass alone. By ruler and compass, you can take an angle, and there's a way to draw by compass a couple of arcs like you do in high school, that divides the angle in two. Of course, you can divide it in four by dividing it into two, and dividing those in two. There are various other angles you can get by ruler and compass construction. That means you take a ruler and a compass and by a succession of drawing lines between two points or by putting a compass on a point, drawing an arc, and taking intersections. But you cannot trisect all angles this way. If you could trisect 60 degrees you could take a projection to the x axis and could find a point whose coordinate is the cosine of the angle of 20 degrees. It is known this number satisfies a polynomial of degree three which has no roots in the rational numbers. Also, the intersection of two circles or the intersection of a circle and a line can all be solved by square roots. Now a result on dimensions of vector spaces can be used to show that 3 would have to be a divisor of some power of two which is not true. People still send papers to mathematicians reporting to be able to trisect an angle. What you can do, if you mark your ruler, like in inches for example, then you can trisect an angle. But that's not allowed.
ZIERLER: You mentioned the big ideas in group theory when you were a graduate student. To bring that narrative to the present, in what way have those theories advanced or changed over the years?
WALES: They changed in group theory by finding out more about the simple groups that are known. There are a lot of people that do various kinds of things that take the groups and then analyze them. With the simple groups, there are certain permutation representations. So you can realize any finite group as some group of permutations. I know I haven't said what permutations are. Maybe I won't, unless you want me to?
WALES: Okay! [laughs] Permutations are rearrangements. I'll take 1, 2, 3. You can look at 1, 2, 3 seeing 1 on the left, 2 in the middle, 3 on the right-hand side. You can move those around, like you can do the 3 first, and the 1, and then the 2, or you could just interchange the 3 and the 2, and then it would be 1, 3, 2. Reading from left to right, you can get an order. The numbers come up one after another in a certain order. You rearrange all those orders and that gives you permutations. I think you might be able to see that if someone rearranges something, and then rearranges it again, it's just a rearrangement of the first one. Think of the permutations of 1, 2, 3. If all we're talking about are the numbers 1, 2, 3, maybe even add 4 and 5, and you rearrange once, and then you rearrange again, one is going to be on the left, one is going to be next, and so on. So rearrangements of rearrangements are, again, rearrangements. That's really what a group is.
A group has the identity which leaves all elements fixed. For example, 1, 2, 3 just stays as 1, 2, 3. Or you do any of the other rearrangements that I've mentioned. If someone else does another one, then you're still going to end up with a rearrangement with a sense of something on the left, something next over, then something to the right. That's what a group is. There's another property that they have, but that doesn't matter. You can take any number "n" and take all rearrangements of all the numbers 1 through n in increasing order. The set or rearrangement forms a group. It is called the symmetric group of n elements. But you can take subsets of those, for which the multiplication is also in the subset. What that means is you have a fixed set of rearrangements, which is not all of your rearrangements but just some of them, and you can take two of those and rearrange them, and then they stay in your small list. That is called a subgroup of all the permutations. All finite simple groups can be thought of in terms of subgroups of permutations. This means the group of the permutations is a subgroup of the symmetric group for some n. People are looking at those things, and the properties that they have. The simple groups of finite order, and even all finite groups are subgroups of some permutation group.
There's another way of looking at groups, that is matrix groups. You start with all non-singular matrices - that means for a matrix A there is a matrix B for which the product is the identity matrix with zeros everywhere except on the diagonal where all values are 1. You can take subsets of those which are also groups. There are many subgroups found this way. A lot of people are working on things like that. They have meetings, and I go to some of them. Since Covid, they've all been on Zoom or on some variant of Zoom.
ZIERLER: Let's move on to another major area of research for you, algebraic combinatorics. Tell me about that field.
WALES: That's where you take certain algebraic things and try to count some kinds of things, or try to do various things with them. That's a big area, combinatorics, in mathematics. I've done some but only in narrow fields. One thing I did was a paper with Ku where there are certain kinds of permutations that have a certain property called derangements. The simplest thing you might think of is counting them which had been done years ago. With each derangement comes a graph from which there are complex numbers called eigenvalues. We wrote a computer program which computed many tables of them. Some properties about their sizes was evident, and we were able to prove some results about them. Combinatorics itself is really the mathematics of counting things about certain objects. In this case we found properties about sizes. There's a lot of mathematics that's connected with them.
ZIERLER: In what ways has combinatorics changed over the years?
WALES: They've found more things and discovered different things. Group theory changed when all finite groups were discovered, but group theory still goes on where different things are being studied. Similar things have happened with combinatorics. There was no big thing that was discovered that stopped the search for various things, as happened with group theory, but certain things may come into or go out of fashion. I guess that's the way to put it. Good mathematicians discover certain things, and quite often there are connections to other things. If you have a good idea that enables you to find something that people didn't know before, there are probably other ideas in what you found that you can develop or use in other places.
ZIERLER: The term "algebraic combinatorics," does that suggest there are other branches of combinatorics that are not algebraic?
WALES: Yes. Algebraic combinatorics is just a certain way of doing what was combinatorics. It uses algebraic methods to study them. For example, the real combinatorics is just counting. How many permutations are there? How many of various things are there? It's just counting. That's really combinatorics. But at a certain point, maybe around 1970 or so, people started using algebraic techniques to study some of those. That's algebraic combinatorics.
ZIERLER: Finally, one term I'm not familiar with—representation theory. What is representation theory?
WALES: It's more general than this. If it's referring to any groups—not just finite groups—you can realize a group as some set of non-singular matrices. When you study those matrices, that's called a representation. Representation theory, especially in group theory, considers groups, and you look at some matrices that really are the same as the group, then you study those matrices. Maybe they're not the same as the whole group, but they're the same as parts of the group. They are things that you can study. It has been used in number theory and various other areas of mathematics. It is very common in mathematical physics. This always involves taking some construction and finding some matrices that are easier or more concrete, which reflects properties of the construction.
ZIERLER: Among all of these fields of research, for graduate students, even undergraduates starting out right now, what's most exciting? What is the area most ripe for future discovery, would you say?
WALES: I don't really know. I think a lot of people are finding computer science to be exciting and really a lot of that is pretty mathematical. It's a specific kind of mathematics, but I would say that in the next 20 years, things are going to come out of computer science that people will find exciting. Also, number theory. Number theory is very hard, and it takes a lot of studying before you can really get started on the main problems. You might start with simple ideas, but pretty soon they get pretty complex. I think people like that. People think of mathematical physics as a field of mathematics, and it is a field of mathematics. It's more applied than some fields, but there are lots of things in mathematical physics that people get excited about. Another exciting field is quantum computing.
From Vancouver to Harvard
ZIERLER: David, we've covered the big questions in mathematics and teaching in your career. Let's go back all the way back to the beginning for your personal history. Let's go back to Vancouver. Let's start first with your parents. Tell me about them.
WALES: When I was born in 1939, my father was a high school physics and math teacher. Later he became a principal and then an administrator where he basically started a junior college. My mother got a degree in Montreal, at McGill in library science. But when she got back to Vancouver, it was the Depression. It was then considered inappropriate for a married couple to both be working, so she never got a job as a librarian. Today, you would describe her as a stay-at-home mom of three boys – me and my younger brothers, Terry and Keith.
ZIERLER: Were your parents originally from Vancouver?
WALES: No. They both came from Eastern Canada. My father was born in Winnipeg, and my mother was born in Ontario. My mother's father was a minister, and he brought the family out to Vancouver - I guess in the early 1900s. My father's father brought his family to Vancouver from Winnipeg, also sometime in the early 1900s.
ZIERLER: You would have been very young, but what memories might you have from World War II?
WALES: [laughs] Someone the other day, one of my friends, said, "I think someone is pretty old because he remembers World War II." And I had to say, "Well, I remember World War II." I didn't say, "But I'm not very old." What I remember is that people were always talking about the war. When I was born in 1939, the war was in Europe. When I can actually remember the war, it was also in Japan. Our house had three big windows in the back and they had to be blacked out every night, so we had what were called blackout blinds. My mother and father drew the blinds down every night because people were afraid that the Japanese would bomb Vancouver the way they did Pearl Harbor. Vancouver was never bombed although there was talk about Japanese planes or submarines being seen offshore.
Another thing I remember is what I now know was V-E Day. We had a roomer living downstairs in the basement. One day she called up to my family saying "The war is over." Then it had to be explained to me that the war was over in Europe but not in Japan. At that time, we had one of those old Model-T Fords that my mother drove around town. We went shopping, and there were all kinds of people in the street, waving flags and shouting and cheering, and so it was exciting.
Back then, we had a summer cottage which was up the coast away from Vancouver. It was an isolated cottage with no phone, but there was a nearby local store with a phone where we knew fairly well the man running it. He came down one day and said, "They dropped the atomic bomb." My father tried to explain to us what the atomic bomb was. I remember him talking about whether the chain reactions could blow up the whole world. That same man came down a couple of days later to say that I had a new brother—my mother had phoned the store to say that we had a new brother named Keith. Of course, he came down to tell us that. I remember those things as happening at the same time—that the war was over, and we had a new brother. About that same time, my other brother, Terry, crawled into the engine well of our boat, and they couldn't find him. My father was just beside himself looking all over, but then my father found him in the engine well. They were saying, "Well, you almost lost one, but you've got another one." I remember that.
ZIERLER: [laughs] The fact that your father taught physics and math, did that rub off on you? Do you think that affected you?
WALES: Oh, I think so, yes. I was always good at physics and math, and I think it's partly because of him. He helped me to understand things. He was pretty good at that. When it got up to physics, not all fathers can really help their kids, but he was very good at that, and he helped me with mathematics, too. He didn't teach me, but he made sure I understood fractions and numbers, things like that. I think the fact that he taught physics made me think that was a natural thing that you'd study at college, and so I did.
ZIERLER: Did you go to the school that your father taught at?
WALES: Yes, I did.
ZIERLER: That was a public school or a private school?
WALES: It was a public school. He was vice principal, then. He left to become an administrator some time before I was in the tenth grade. People knew that I was at the same school as my father. In a way I'm not sure that I liked that. There was talk about sending me to a different school where he wasn't the vice principal, but that never happened.
ZIERLER: In math, in middle school and in high school, did the teachers have enough to keep you occupied? Did you read outside of math class?
WALES: Only when I got up to the twelfth grade, senior year. We called it the twelfth grade, then, but now I know it's the senior year. I think by my junior and senior years, I had a teacher who gave me a few extra things. I think people now have calculus in high school, but calculus was not taught in high schools back in those days, especially at my high school. But he gave me a couple introductory books in calculus, which I started to read, but I really just took the courses and did pretty well in them.
ZIERLER: When it was time to start thinking about undergraduate options, did you ever think about going to school farther away than the University of British Columbia?
WALES: No, and I don't think anyone in my high school did, except for one person who went to Bellingham and another person whom I heard later went to Toronto. I know there were people in Vancouver who would have gone other places—but I didn't know them.Today, high school students start thinking very early about what colleges they're going to and where they're going to apply, but we didn't do that in Vancouver in 1957. I never thought of applying anywhere except UBC. Other places would have been in the United States or other provinces, so that seemed too far away to me. Also, my family lived fairly close to the University of British Columbia, and I could live at home. Also, I think I got a scholarship, because I had good grades, to the University of British Columbia.
ZIERLER: Did you know it was math that you wanted to pursue as an undergraduate?
WALES: No, I actually started planning to do engineering physics. The first year, I took engineering physics courses including math and physics. While doing that, I decided I liked physics better and didn't really like the engineering part. So, I started taking honors physics and math courses. Then as I got on, I decided that what I really liked about the physics was the math. By the time I was a senior, I decided to stay in math.
ZIERLER: Was there a particular professor that really turned you on to math?
WALES: I guess there were quite a few, yes. I did pretty well, I guess, so they were nice to me. [laughs] There were some I can still remember. They were very energetic and so forth
ZIERLER: Tell me about your decision to stay on for the master's degree at UBC.
WALES: One reason was that I was getting married, and my wife needed another year to complete her education degree. Also, the easiest thing for me to do was to stay and get my master's at UBC. If you took honor's physics and math at UBC, you took many physics courses but not that many math courses. By staying on for another year, I could take more math courses.
ZIERLER: At some point, your vista must have opened up where a place like Harvard was in reach to apply to.
WALES: Yes, I guess so.
ZIERLER: Where else did you apply for graduate school?
WALES: I remember them very clearly—Harvard, Princeton, Berkeley, and Caltech. I didn't get into Princeton, but I got into all the other ones. I chose Harvard.
ZIERLER: Why Harvard?
WALES: Well, [laughs] that's what my teachers said I should do. I didn't know at the time I was going to be coming to Caltech later. There were a couple of people at Caltech that one of my teachers thought I might like to work with. One of them was Olga Taussky who was a professor in mathematics – well, I guess she wasn't a professor, but that's another story, [laughs] but she was in the Math Department.
ZIERLER: Had you ever been that far east before?
WALES: Yes, I had. In 1960 when I was a third-year at UBC, I got a job at the Defense Research Board in Ottawa, Canada. My brother Terry then had a job working for an insurance company in Montreal, so we drove our car back east together. He lived in Montreal, and I lived in Ottawa. They're just maybe 120 miles apart. We'd drive to the other person's place every weekend and then take the train home. During each weekend we'd drive around and see the sites. Then the next weekend we would switch who drove and who took the train. Anyway, we just traded around. One time my brother, a classmate from Vancouver and I went to New York City for a long weekend. That city was quite a shock for me.
ZIERLER: What were your impressions of Harvard when you first arrived in Cambridge?
WALES: Well, when I arrived there in September of 1962, the other students seemed really good to me. I felt out of place because, even though I had a master's degree, many people seemed to know a lot more mathematics than I did. In fact, some of them did know a lot more than I did, in part because they had been to better schools. I remember one person who talked about all the things he knew, and I had no idea what they were. But by taking more courses, I was fine.
ZIERLER: Who ended up being your advisor at Harvard?
WALES: Richard Brauer.
ZIERLER: What was he known for? What did he do?
WALES: For group theory! [laughs]. Brauer did representations of finite groups. That's where I learned representation theory. That's where I learned it all. I wrote a thesis. It took me five years, which was typical at that time. My thesis was on what are called finite linear groups of degree seven. It was a long thesis, 200 pages, with lots of calculations and diagrams. Years later, with the classifications of finite simple groups, it became straightforward to do this. So my thesis was—wiped out is the wrong way of putting it, but anyway, superseded by the classifications that I mentioned earlier.
ZIERLER: As you were thinking about thesis topics to work on, what were the choices that you had, and why did you pick that one?
WALES: When I first started working, Brauer suggested some reading and some possible problems in group theory. After a while, he suggested a topic that he had just written about involving finite linear groups of degree five. He suggested I work on the next case which was degree seven, because the degrees were prime for his paper. I think that's why I chose that topic. But I was also thinking of doing lattice theory with another mathematician, Garrett Birkhoff. I talked to Birkhoff for a while, but before long, I decided I liked group theory better.
ZIERLER: Is lattice theory at all related to lattice gauge theory in physics?
WALES: I don't think so but I don't know what lattice gauge theory is. Lattice theory is a part of combinatorics. There are sets for which certain of the elements are ordered with some extra properties. For certain pairs of elements, one is less than the other. People who work on lattice theory can draw lines that give pretty pictures. Drawing pictures is always nice because you can't do that so easily with groups. That was one thing I liked about lattice theory. The sets are not groups, [laughs] but they might be, and they can be studied in a similar way.
ZIERLER: This gets back to the questions about what mathematicians do. When you're writing the thesis, how do you know when you're done?
WALES: [laughs] Brauer told me!
WALES: What I was doing was using representation theory to classify all finite groups that have seven-dimensional representation. In the wording that I said before, some groups can be realized as a set of 7 by 7 matrices which is a representation of degree seven. The multiplication in the group is the same as the multiplication of matrices. You asked about representations before. A representation of degree seven is a set of 7x7 non-singular matrices which is that particular group. My thesis was to find all 7x7 ones. I didn't find them all, but I did various things to study them. The main problem was to find simple ones, as the important ones can be simplified to those. It involved a lot of funny numbers and things with diagrams, which I worked on. Later, in the 80s when the finite groups were classified, that approach became trivial, but so did my advisor's paper, too.
ZIERLER: You were in good company.
WALES: [laughs] Yes. There is some representation theory in the classification of finite groups, but largely it's not representation theory but concentrates on subgroups. There's one theorem that really has a lot of representation theory that was proved by Walter Feit and John Thompson, which is that no odd order groups are simple, except one of order two which is simple. That was a hard theorem that had a lot of applications. I was studying parts of that. That's why people study representation theory. Some people thought that approach might actually be used for the classification of finite groups. However, it wasn't, except for that case. That's where you start. You start with a group of even order, and then work on from there. You don't start with an odd-order one. So it was used, only in that sense, but there are other things that are also used, techniques that I used in my thesis.
ZIERLER: In the Math Department at Harvard, were there any women or minorities either on faculty or among the graduate students?
WALES: I don't think there were any women on the faculty. There was one female graduate student in my year. I think she was the only one, but I'm not sure.
ZIERLER: Besides Brauer, who else was on your thesis committee?
WALES: A person by the name of Daniel Gorenstein, who was working at Northeastern University in Boston. He was also famous in the classification of finite simple groups. He outlined a strategy which was used in the proof. In a sense, he took it upon himself to outline how the proof might work. There are different classes of groups that you might want to prove first, and then you'd be able to use that in the other ones. He had a big program and announced it with fanfare. Many people worked on this project, including my Caltech colleague, Michael Aschbacher. Some people said Gorenstein was the quarterback and Aschbacher was the star running back, but there were many others too.
ZIERLER: After you defended, what opportunities were available to you?
WALES: I applied for jobs in 1966-67 when I was working on my thesis. I didn't apply for many jobs, but I did apply to Caltech. Caltech had a famous group theorist, Marshall Hall, who was here at that time. It was a natural place for me to apply. I also applied to Yale, where Walter Feit worked, and the University of British Columbia. After Caltech offered me a job, I told the others that I was going to Caltech.
Coming on Six Decades at Caltech
ZIERLER: Your initial position at Caltech was what? It was a postdoc?
WALES: That's what it would be called now. But back then, they didn't call it that. I had what was called a Bateman in mathematics. It was a two-year position. At that time they were not as common as post-docs are now.
ZIERLER: You were called a Bateman Research Fellow. Do you know who Bateman was?
WALES: Yes, in a way. At that time there was a picture of Bateman in the math building which was then called Sloan. The building is now called Linde Hall in honor of Ronald and Maxine Linde who gave money to refurbish it. It was quite a refurbishment and now looks wonderful. There is still a picture of Bateman in the building as well as one of Ronald and Maxine Linde.
ZIERLER: At what point were you made the offer to join the faculty?
WALES: I came to Caltech in September 1967 as a Bateman. In February 1968, Caltech offered me a faculty position. I think it was because Marshall Hall appreciated my work. During that year he and I had worked on a simple group that he had constructed. Together, we showed it was unique for its order. I had also interacted well with Herb Ryser who did combinatorics. Around the same time, Marshall Hall, his student Richard Lane, and I also constructed a new combinatorial design which at the time was important.
ZIERLER: As you joined the faculty, how had your research changed since your graduate school days?
WALES: In my graduate school days, I was just doing everything I possibly could to finish my thesis with those 7x7 matrices. Once I started on the faculty—well, once I started as a Bateman—I wrote out those results, plus many more. That was a good first year at Caltech because I was working with Marshall Hall, and together we did something that was pretty good. Then I used some of the ideas from that work in other ways. I wrote several papers at that time. Much of my work was done late at night because I had two young children: Jonathan was born in 1966 and Jennifer was born in 1968.
ZIERLER: What was that collaboration with Marshall? What were you working on?
WALES: There's a simple group that was called the Hall-Janko group. It's also called J2. The J stands for Janko who had predicted this group and wrote down what the representations were. There was a thing called the character table connected to the representation, and Janko had written that down. It has a certain centralizer of an involution, which are all the elements which commute with a certain element. It has a subgroup of a certain kind, which Janko used to predict it.
Marshall Hall first constructed J2 in Oxford, as a permutation group. Then he and I worked to show that there was only one group of that order which is 604,800. Interestingly, 604,800 also happens to be the number of seconds in a week. I don't think my work changed when I became an assistant professor. I just did the same kinds of things as I had done as a Bateman. I can't remember exactly when I started working on other things.
ZIERLER: What was the research that you think was most important for you gaining tenure?
WALES: I got tenure at Caltech in the 70s. I think my thesis at Harvard was really important, especially what I learned during the course of it. Also, my work with Marshall Hall and John Conway.John Conway was a very famous mathematician. He was then at Cambridge and later at Princeton. He died around a year ago of Covid. In 1972 John Conway was a visitor at Caltech. As I mentioned, Janko had predicted the J2 group that Marshall Hall constructed as a permutation group. Janko had also predicted another group called J3. After Conway arrived at Caltech, he suggested that he and I should construct J3 as a group of matrices. We knew it had an 18-dimensional representation. In other words, you can write it as 18x18 matrices. After all, there are not that many positions. It's just 182 positions. You just have to get the numbers right, and then do all the multiplications, and you'll see that it's a group but we used other methods.
Before this, John Conway had found several simple groups, but always involving symmetries of a lattice, the Leech lattice. But he had never sat down and really constructed a group using matrices. We constructed J3, bit by bit, small parts by small parts. We constructed it by finding many of its subgroups, some as 5x5 matrices, but they got bigger and bigger, and so we finally wrote all of them and constructed it. That was a big thing, because not everyone knew—we didn't either— how to construct it. There was no group of people who were sitting around constructing groups like that, so we figured out how to do it, and we did it.
After that, Rudvalis sent us—really he sent it to Marshall Hall—his prediction of this other group called Ru, the Rudvalis group. He didn't call it the Rudvalis group, but we do now. John Conway and I spent days and days and days constructing it. We had an icosahedron that we'd move around to get various motions in three dimensions. An icosahedron is a bunch of points and lines, and you can move them around and get the same figure. You can find the icosahedron in three space, so you can rotate it, all the different ways, and it stays an icosahedron. So that means it's a group, in a sense, like I said before. We used that quite heavily, and surprisingly, a number of times, and then we constructed it. That's a good thing to do, constructing finite simple groups. I think that's really why I got tenure.
ZIERLER: Tell me about your sabbatical in the U.K. at Oxford and Cambridge. 1976.
WALES: I had a six months sabbatical in 1976. I was in Cambridge from January through March and then Oxford from April through June. I had intended to work with John Conway at Cambridge, but he was then really busy working on something else, in combinatorics. There were no groups to construct, so he was working with another person, Richard Guy. Conway spent most of his time with him.
I took courses, talked to other mathematicians, and worked on things. Another group theorist, Gordon James, was at Cambridge at that time, and I took a course from him. But mostly I was working on a problem that I solved when I was there.
Of course, Cambridge is a good place to be. There are a lot of good people in many different areas of mathematics. It's a very congenial place. Mathematicians there have an afternoon tea at 4:00 in their lounge upstairs. We actually had that at Caltech for years, but it stopped because of the pandemic. At tea, Cambridge graduate students and faculty would gather around and talk, in a very animated way, about things they were doing. It was especially lively when John Conway was there.
I went to my office in the math building almost every day when I was in Cambridge. This was in part due to my housing situation. I was living in a small room in the basement of house. My landlady said I could only have heat from 6:00 to 10:00 in the morning and again from 6:00 to 10:00 in the evening. It was January, and Cambridge is very cold in January, so [laughs] I would have to go to work to keep warm. I shared an office with another group theorist named James Ward, but we didn't work together.
In April I went to Oxford where I worked at the Mathematics Institute with the many group theorists there. It was a friendly atmosphere with active seminars. In particular I remember Michael Collins, Graham Higman, Peter Neuman, and Geoffrey Robinson from that time.
ZIERLER: It was a good experience for you?
WALES: Yes, very good. It was an excellent opportunity to interact with mathematicians working in many different areas.
ZIERLER: Back in Pasadena, on the administrative side, you served first as associate dean of students and then dean of students, from 1976 and then until 1984. Tell me about your work in that role.
WALES: I had always loved teaching mathematics to students. Starting in 1976, I got an opportunity to work with students in a different way and to learn about student affairs. At the time, Ray Owen, a biologist, was Vice President of Student Affairs and Dean of Students. He asked me to be Associate Dean of Students, and I did that until 1980. In 1980, Jim Morgan, an environmental engineer, became Vice President of Student Affairs and I became Dean of Students. The title of Associate Dean was changed to Assistant Dean and designated a non-faculty position. Chris Wood was in that position. I was Dean of Students from 1980-1984.
Our office was in Parsons-Gates, and we referred to ourselves as the Dean's office. We handled things typical for that kind of office – granting leaves of absences and various kinds of extensions and exceptions and dealing with a variety of crises that were confidential. Although things were sometimes stressful (especially the calls during the middle of the night), I really had fun with the people who I was working with and enjoyed the contact with students.
ZIERLER: Back on the research side, in the mid and late 1970s, you worked with W. Cary Huffman quite a bit on linear groups. Tell me about that collaboration.
WALES: Cary Huffman was a graduate student of mine. I gave him various problems that he solved. They involved representations of groups as matrices with eigenvalues. I won't go into what an eigenvalue of a matrix is.
ZIERLER: Your work with Phil Hanlon on Brauer centralizer algebras, is that the Brauer who was your thesis advisor?
WALES: Yes, it is. Brauer discovered certain algebras in which he lost interest at some point. Phil Hanlon and I were interested in these algebras and called them Brauer centralizer algebras to distinguish them from another Brauer algebra that involves number theory. We enjoyed working on them for several years and later worked further with Bill Doron who was a post-doc at Caltech and former student of Phil Hanlon.
ZIERLER: When did you first meet Arjeh Cohen and realize that you'd have such a formative, long-lasting collaboration?
WALES: I met Arjeh Cohen at a meeting in Arcata, California in 1986. He had written a thesis which was similar to my thesis, only he had done it a different way. From the beginning it was clear that we were interested in some of the same things, and we spent years working on them.
ZIERLER: Back to the administrative side, in the 1990s, you served as master of student houses. What did that work entail?
WALES: It's a job that no longer exists at Caltech.
ZIERLER: 1991 to 1997.
WALES: I started as Master of Student Houses in the fall of 1991 and continued doing that job until October of 1997. It was a fun job – completely different from what I was used to doing. During this time, I continued teaching math courses but had a reduced load.
For years prior to when I started, the Master of Student Houses was responsible for room assignments in the seven undergraduate houses that existed at that time and for various issues that arose relating to the student houses. Caltech faculty Sunney Chan, Chris Brennen, Bob Oliver and Louis Wilde served in that position before me. A large six bedroom house on Holliston, called Steele House, was designated as the residence of the Master of Student Houses. In 1991, the Vice President of Student Affairs, Gary Lorden, decided to turn these responsibilities over to non-faculty professionals and created the position of Director of Residence Life. Kim West was hired to fill this new position. Gary appointed me to act as Master of Student Houses. My main responsibility was to host social gatherings of students and faculty at Steele House.
When my wife and I and our two Abysinian cats moved into Steele House, it had recently been painted and the kitchen had been completely remodeled. A Caltech staff member, Stan Borodinsky, was in charge of managing the house. Together with the help of Stan and my wife, I outfitted Steele House with the things that were needed for me to host social gatherings there. I also hired a chef, Patty Krause, to make the food for these gatherings. Students were hired to help with serving and clean up. I generally hosted the social gatherings by myself because my wife was a lawyer working very long hours in downtown LA.
Steele House was a wonderful place for hosting students. Up to around 24 people could be comfortably seated in the dining room and adjoining sun porch. We had a pool table put in the library. The biggest attraction was the real pipe organ in Steele House. There was a large room at the side of the house that held the 3,000 pipes. The organ was later removed when Steele House was converted to other uses.
I had two types of social gatherings at Steele House. There were the dinners – about 8 a term – where I would invite students and faculty for dinner. Over time, I tried many different approaches to whom I would invite with the goal that all freshman would come at some time or another. Sometimes the dinners worked well with lively conversations, and sometimes I had trouble getting anyone to talk. But the food was always good! I also had dessert parties where I invited all students from the houses. Patty made huge quantities of cakes, pies, and cookies that disappeared very quickly. I used to do informal counts at the dessert parties and concluded that there were typically about 150 students over the course of the evening.
As I mentioned, room assignments in the houses and issues arising at the houses were handled by the office of the Director of Residence Life which included Kim West, Sue Friedman and Sue Chiarchiaro. I attended the sessions where they trained or met with the resident assistants who were graduate students hired to live in the student houses. This exposed me to a whole new aspect of student life. This was a nice group of people, and I enjoyed working with them and hosting social gatherings for them at Steele House. Kim West and I participated in the Caltech committee that worked on the design of Avery, the dorm that was constructed while I was living in Steele House. I was able to watch this project from our bedroom window, starting with the night that they moved the big old house that was on the site and ending with the day that Avery opened for the first time for students.
ZIERLER: After this time, you enjoyed two more sabbaticals, first back at Cambridge in 1997 and then in 1998, you were in the Netherlands.
WALES: Yes, I was on sabbatical in Cambridge for October – December of 1997. From January to September 1998, I worked with Arjeh Cohen at TUE Eindhoven. My wife and I lived in central Amsterdam on a canal. I traveled by train to Eindhoven four days a week and the other day I worked with Arjeh at his home in Zeist which is near Utrecht. Arjeh and I got lots of work done during this time. Also, my wife and I took advantage of the weekends and various vacation periods to travel extensively in Western Europe. 1998 was an exciting year in Europe because the soccer World Cup was in Paris. Arjeh, his colleagues at TUE, and I enjoyed analyzing the games.
After my sabbatical, I spent various periods of time working with Arjeh in Eindhoven, and Arjeh spent time here in Pasadena working with me. Some of his visits coincided with World Cup games that we were able to watch together at my house.
ZIERLER: I'm curious about your article with William Doran, "The Partition Algebra Revisited." What needed to be revisited?
WALES: I think it was a matter of looking at this algebra more carefully. We didn't discover the partition algebra. The partition algebra had been discovered earlier and had some properties. But we did more work on it. Bill and I worked on this when he was at Caltech as a Taussky-Todd Instructor in Mathematics. This work was connected to my work with Phil Hanlon on the Brauer algebras. There had been papers on the partition algebra, and people knew how big it was. But really it had many more properties than had appeared at first, making it more similar to the Brauer algebras. It's not the same, as it's wilder in some sense, but in other ways it's easier to deal with. You deal with it by looking at the representations. We spent time working on the representations in much more detail than had been done before.
Bill and I could not call our paper the "partition algebra" because people already knew what that was. The title of a paper often refers to the topic of the paper. For example, Phil Hanlon and I called the Brauer centralizer algebra that because it was related to some centralizer. With the partition algebra, there was no natural thing that it was related to, so we just called it "revisited." People do that periodically to indicate that they have discovered something new which was true for us.
ZIERLER: Yet more administrative service for you—from 2002 to 2007, you were secretary of the faculty.
WALES: You read my CV!
ZIERLER: [laughs] Tell me about that position.
WALES: The faculty has three officers who are chair, vice chair, and secretary. At that time the secretary had resigned for health reasons. Marianne Fraser, the chair, asked me to replace him which I did and continued for a few years. The main job of the secretary is to go to faculty board meetings and faculty meetings and then write up and circulate the minutes. I had a staff assistant who took notes and did a preliminary draft of the minutes. We worked together to finalize the draft and then circulated the minutes.
The secretary is also in charge of running the elections for the faculty officers and for the members of several faculty committees. This part of the job was easy, and my assistant helped me. The first election while I was secretary had paper ballots. We had to count the votes by hand which was quite a job. After that election, voting was done online so that the computer did all the work. As secretary of the faculty, I was also a member of the Steering Committee of the faculty board. That committee meets and decides on the agenda for faculty board meetings and faculty meetings.
ZIERLER: What were some of the key issues facing Caltech that were brought up during these meetings?
WALES: I guess some of the recurring matters were various financial things, concerns of students, new developments, and reports from the president, provost and various vice presidents. Also of interest to me was the fact that the faculty board approved all courses that were designed by the divisions. The courses were first approved by the core curriculum committee who then referred them to the faculty board for approval.
ZIERLER: During the second-to-last year of your tenure as secretary, Caltech chose a new president. There was a change in administration from David Baltimore to Jean-Lou Chameau. I'm curious if you had any particular insight into how Caltech selects potential presidents and how that process of selection plays out.
WALES: No, I am afraid that I have no particular insight about that. But I can tell you, it has nothing to do with the faculty board. [laughs]. All I know is that on a certain day, an announcement is made, and they say, "This is going to be the president, and the president is going to start in the fall."
ZIERLER: Tell me about your decision to go emeritus in 2008.
WALES: At that time and since then, Caltech has offered various financial incentives for faculty to retire at or before a certain age. I found these financial incentives to be attractive. Also, after 41 years of teaching at Caltech and many years of serving on various committees and doing various administrative jobs, I was ready to move on. I was eager to do more challenging international travel for longer periods of time and at times of the year when before I always had to be teaching. Since the early 1970s, I had been hiking regularly in the San Gabriels and Sierras, and I was looking forward to spending more time in the mountains. For example, after I retired, I was able to join friends for three different treks in Nepal.
ZIERLER: And of course you could still work on the math.
WALES: I still have an office in the math building. I still do some work on math but don't work nearly as hard as I did in the past. For many years after I retired, Arjeh and I continued with our work.
ZIERLER: One more specific question on the research. We talked about eigenvalues. I'd like to know about "eigenvalues of the derangement graph"—what is a derangement graph?
WALES: It's permutations that have no fixed points and you can form a graph using them.
ZIERLER: Is there anything particularly deranged about it?
WALES: No, no, no. They're just called derangements. In the example of the permutations I gave you, the rearrangements of 1,2,3, a derangement is one that rearranges everything. In a derangement, everything moves. The wording is, there are no fixed points. Nothing gets fixed. Everything gets moved. For the permutations of three points only, the two which cycle through the three numbers are derangements. For an example, one goes to two goes to three goes to one and the reverse are the only ones. There are many more for higher n. There is a formula that tells you the number, and there are certain properties that it has. What we worked on is certain properties that the eigenvalues have. Do you want to know what an eigenvalue is?
WALES: Do you know what a matrix is?
ZIERLER: I've heard of it, yes.
WALES: Okay, so a square matrix with size n by n is an n by n grid with n rows and n columns. and then you have numbers in all the entries. Maybe it's 5x5, there's numbers in all the entries. For n=2 the first row could have the numbers 3, 4 and the numbers 1, -1 in the second row. For each matrix there is a determinant —do you know what the determinant is?
WALES: There's a thing called the determinant. With a matrix, there's a determinant. It actually uses all the permutations of n. There's a determinant that goes with the matrix eigenvalue. You take the matrix and subtract x times the identity matrix. This is the matrix with all entries zero except on the diagonal where the entries are all zero. The diagonal entries are the ones in i-th row and i-th column. So you subtract x times the identity matrix from the matrix. Take that determinant, which will be a polynomial in x. A root of that is called an eigenvalue. There are a finite number of such roots which are called the eigenvalues of a graph defined in terms of the derangements. For a permutation there is always a matrix associated with it and we do that only for the derangements,
ZIERLER: Now that we've worked up to the present, for the last part of our talk, I'd like to ask a few broadly retrospective questions about your work and your research.
WALES: I think I'm pretty happy with it.
ZIERLER: Of all the things that you've worked on, what has been most intellectually satisfying, that gave you a sense of completion?
WALES: I guess it's one of my papers, with Arjeh Cohen. We spent many years working on it.
ZIERLER: Which paper are you referring to?
WALES: It is "Finite Subgroups of F4(C) and E6(C)".
WALES: That's it. That's the paper.
ZIERLER: This is 1997.
WALES: That's right. That's the one. We were working on that for a long time, off and on since 1990. It's not the only thing we did, but we worked on that for a long, long time. In order to do it, we had to construct several of the groups. That's the one that has been the most satisfying.
ZIERLER: Conversely, the one that's least satisfying, problems that you've worked on that no matter what you do, you always seem to hit a wall?
WALES: I try not to do that. In group theory there are always different things that can be tried. Getting back to satisfying things I have worked on there is the construction with John Conway of the Rudvalis group. It is one of the simple groups. Also the work with Arjeh Cohen showing linear group of finite Artin type are linear. This means they can be written down as matrices. This is always true for the finite groups on which I have mostly worked but these are not finite.
ZIERLER: When you say you avoid those kinds of problems that end up when you hit a wall, how do you know? How do you know when to change course?
WALES: It takes a while. You might have thought we would have given up on the E6 and F4 paper, but we just kept working on it, bit by bit. It seemed like it was impossible at the beginning, and even as we were getting along, we kept finding small things that we still had to do, and more things that we had to do, and more things that we had to do. That's the one I've worked on the most. When you try all the things you can, however, it is time to give up.
ZIERLER: Do you think for undergraduates and graduate students today, there's still fundamental work to be done in the future?
WALES: In mathematics? Oh, certainly. There's no question about it. But it's harder to find. You have to work harder and look further. But still there are lots of areas and lots of problems that people can work on and be quite satisfied with what they're doing and be pleased with what they produce.
ZIERLER: Finally, last question for you, however long you want to remain active, what are the things that you want to accomplish that you haven't done yet?
WALES: I guess you mean active in mathematics. I think I'd like to learn more – not for the purpose of research but because I enjoy the area. I would also like to revisit some areas that I really like but can't remember that well.
ZIERLER: There's always room to learn.
WALES: Yes. So that's what it is. I think when I finish with this work with Arjeh, that's what I'll do, go back and work on things that I like and catch my attention.
ZIERLER: That's great. It has been a great pleasure spending this time with you. I'm so glad we were able to connect and you were able to share your insights with me. Thank you so much. WALES: It has been fun. Thank you.