September 29, 2021

*Michael Aschbacher grew up in Illinois, Michigan, and California and completed his undergraduate education at Caltech, where he developed his interest in mathematics. At Caltech, he met Marshal Hall and was first exposed to combinatorics and group theory.*

*Aschbacher went to the University of Wisconsin for graduate school and returned to Caltech in 1970 to join the faculty. He has made seminal contributions in pure mathematics, including his classification of finite simple groups.*

*Aschbacher served as Executive Officer of Mathematics from 1991-1994 and was named Shaler Arthur Hanisch Professor of Mathematics in 1996. He is the recipient of numerous prestigious mathematics prizes, he is a fellow of the American Mathematical Society, and he was elected to the National Academy of Sciences in 1990.*

### Interview Transcript

**DAVID ZIERLER:** OK, this is David Zierler, Director of the Caltech Heritage Project. It is Wednesday, September 29, 2021. I am so happy to be here with Professor Michael Aschbacher. Michael, thank you so much for joining me today.

**MICHAEL ASCHBACHER:** OK.

**ZIERLER:** To start, would you please tell me your title and institutional affiliation?

**ASCHBACHER:** I'm Schaler Arthur Hanisch Professor of Mathematics Emeritus.

**ZIERLER:** Is Schaler Arthur Hanisch one person, three people?

**ASCHBACHER:** It's one person.

**ZIERLER:** Can you tell me a little bit about Shaler Arthur Hanisch?

**ASCHBACHER:** Not very much. He's no longer with us, and his family wanted to do something to make his name live on, so they endowed two professorships back in the 90s.

**ZIERLER:** Did Hanisch have any connection to mathematics or to Caltech?

**ASCHBACHER:** Well, he must've had some connection to Caltech because the development got them to commit to fund these two professorships. But I don't have any memory of a deep connection. For example, I don't think he went to school here, but I'm not sure.

**ZIERLER:** What year did you go emeritus?

**ASCHBACHER:** 2013.

**ZIERLER:** And in what ways have you remained connected to Caltech and the research since that time?

**ASCHBACHER:** Well, in every way, except I no longer have to teach or serve on committees.

**ZIERLER:** So it gives you extra bandwidth to just work on the research.

**ASCHBACHER:** Yes.

**ZIERLER:** And just as a snapshot in time, what are you working on right now?

**ASCHBACHER:** Well, we've come into some technical difficulties. I'm working on trying to prove one portion of the classification of the finite simple groups in the category of fusion systems.

**ZIERLER:** Is this a longstanding project? Or new?

**ASCHBACHER:** Well, these terms are relative, but I've been working on it for ten years at least.

**ZIERLER:** Do you still work with graduate students or post-docs?

**ASCHBACHER:** Here, no. I may encounter graduate students or post-docs when I go to a meeting or to somebody else's university. But I haven't had any graduate students or post-docs in a long time.

**ZIERLER:** As a mathematician, in the pandemic, where we all had to socially isolate and not go to work, was that an ideal situation for you, at least from a research perspective?

**ASCHBACHER:** No. It wasn't a terrible burden. But it was nice to travel and to see people. But no, two years, year and a half is a long time to give that up. And also, before COVID, I was used to coming in here to my office and spending a lot of time in the office. With COVID, with nobody around, there didn't seem any reason. I live very near here. I walked to the school. So there didn't seem to be any reason to spend a lot of time in my office. As a matter of fact, I probably wasn't supposed to spend as much time as I did.

**ZIERLER:** And did that conform to your usual work style of researching solo? Or do you generally work with other people, collaborators?

**ASCHBACHER:** I've certainly collaborated, but that's not my normal style. The modern mathematician seems to be very collaborative. And even back in my day, I was less collaborative than most.

**ZIERLER:** So you harken back to maybe even an earlier generation in mathematics.

**ASCHBACHER:** Yes, that's right.

**ZIERLER:** Some overall questions right now that might help translate your specialty to our broader audience. What is the overall field in mathematics that you work in? What would you call that?

**ASCHBACHER:** Well, I work in several, but primarily, I work with finite groups.

### The World of Finite Groups

**ZIERLER:** So if we can break this down in a decision tree, let's say, for example, a graduate student in physics, maybe one of the first binaries that they would have to choose would be experimental or theoretical physics. What would be, in mathematics, that first binary, that first choice that all mathematics graduate students would have to make?

**ASCHBACHER:** Maybe pure or applied mathematics. Applied mathematics means whoever is on the faculty at the particular institution its applied.

**ZIERLER:** So how do you understand pure mathematics versus applied mathematics? Applied toward what? Let's start there.

**ASCHBACHER:** Well, it could be any semi-real-world problem. For example, when I came here on the faculty, applied mathematics at Caltech meant fluid dynamics. But I don't think it means anything like that anymore.

**ZIERLER:** What does it mean now?

**ASCHBACHER:** I don't know. I've lost connection with the applied group. Besides, there would also be what I would think of as applied mathematicians that are in other groups than the applied mathematics group.

**ZIERLER:** So for you in graduate school, that first binary decision would've been pure mathematics?

**ASCHBACHER:** Yes, but I made that decision as an undergraduate.

**ZIERLER:** And then, if we could further break down pure mathematics, what would be the next step in your mind?

**ASCHBACHER:** Well, maybe broad areas of mathematics. Analysis, algebra, geometry, topology, foundations. There are probably a few more.

**ZIERLER:** And then, from there, as a graduate student, what did you develop as your area of specialty?

**ASCHBACHER:** When I was an undergraduate here, a couple faculty members got me interested in a certain type of combinatorics. And that's why I went to the University of Wisconsin. They had a good combinatorics guy named Bruck, who, in theory, I worked with. And so, I solved whatever problem I set for myself and took it to him, and that was OK. I had to stick around for three years because of the way the rules were written at the University of Wisconsin. And as I was there, the way I approached my thesis problem, I was doing more finite group theory, permutation group theory than the combinatorics. And I got interested in that. My first job was a year at the University of Illinois, where there were two very good finite group theorists. So by then, I decided I would be a finite group theorist. And at that time, that was a very hot area, too, so it was better for a young mathematician to be in.

**ZIERLER:** Now, let me ask some questions about finite groups and historical context. Where do we first encounter the term finite groups? How far back does that go?

**ASCHBACHER:** Oh, 19th century. A hundred and a half years, maybe.

**ZIERLER:** And what are some of the key ideas behind finite groups?

**ASCHBACHER:** Well, first, you have to come to the notion of a group. Galois has this very romantic life, but he's killed at age 20 in a duel. And he had solved this great problem, and he supposedly had written down the mathematics that he had just done before going out to the duel, and he got killed. Not exactly in a duel, but he was stabbed in the stomach and died of intestinal problems a few days later. The work that he'd written down, he gave to a friend. But he had sent copies of it to Cauchy, who was a great French mathematician at the time. And Cauchy lost his papers. So Galois's work was not appreciated until about 30 years after his death. Somehow, maybe Lagrange, or somebody came upon his work and read it carefully enough to understand it. It was very important. Group theory existed in some proto-mathematical form before Galois, but Galois gave the impetus, I think, to group theory beginning. That would be probably in the mid-19th century. Although, the notion of an abstract group wouldn't have come into being until maybe 30 years later.

**ZIERLER:** And where did that happen?

**ASCHBACHER:** In the work of some German mathematicians and French mathematicians.

**ZIERLER:** When does this start in the United States?

**ASCHBACHER:** Early 20th century. You have to know that mathematics in the United States wasn't worth very much until the second quarter of the 20th century, maybe. And even then, not so much.

**ZIERLER:** And what changed?

**ASCHBACHER:** Well, universities in the United States started to get better, mathematicians stayed in the United States rather than going abroad. A big impetus came with the second World War and the support of all sorts of science by the federal government.

**ZIERLER:** What are some of the big questions in finite group theory? What does it seek to answer?

**ASCHBACHER:** Well, the biggest question was to determine all the finite simple groups. And that was achieved, depending on what you think is achieved, either around 1980 or 2005, something like that.

**ZIERLER:** There's disagreement on what's been achieved?

**ASCHBACHER:** No, there's no disagreement on what's been achieved. But claims were made that the classification was complete around 1980, and that turned out not to be correct. There was a great strategic mathematician, Daniel Gorenstein, who organized this project and came up with an outline of how one should try to prove this classification theorem. And he parceled out problems to various people, and some things were done independently. But eventually, he decided that a solution to the main problem had been achieved. But really, that meant that in some cases, he had confidence that a person he had assigned a problem to and was working on that problem would finish off the problem. In one case, for a very long, hard problem, the person didn't finish it off. And as a matter of fact, he eventually stopped working on the problem and started working on something else. And the mathematics world became aware of this in, I don't know, 1990 maybe, something like that. And it took a while for people to attack this problem. And between 2000 and 2005, it was finished off, this large problem that had been left open.

**ZIERLER:** Operating in a pure math environment, have you ever seen your research, either by your own volition or by people who follow your research, applied to the so-called real world? Or is there a bright line between the two?

**ASCHBACHER:** Well, it depends what you mean. Depends on what you'd regard as an application.

**ZIERLER:** What do you regard as an application for pure math?

**ASCHBACHER:** Well, this won't necessarily be about group theory, but about error-correcting codes. There are mathematicians who are coding theorists who produce these codes. For example, JPL communicated with one of the early satellites using a famous code, the Golay code. To obtain secure communication, you would want an encryption and decryption scheme. And there are various schemes like that that use very sophisticated mathematics. There's one that uses factorization of integers. It was shown, by an undergraduate here, not while he was an undergraduate, but while he was working at Bell Labs probably, that a quantum computer could solve, in real time, factorizing one of these integers as a product of two primes. And so, this encryption scheme, if there was a quantum computer, would not be worth anything. So those are two applications.

**ZIERLER:** In a scientific context, a theory is either validated or not experimentally or observationally. How are theories in math validated?

**ASCHBACHER:** Well, I don't think that term would be appropriate. You have a theory; you prove things within that theory.

**ZIERLER:** How, then, do you know if a theory is right or not?

**ASCHBACHER:** Again, the term right is probably not appropriate. What I think of as a mathematical theory is a bunch of concepts, theorems, examples. You could ask if a theorem is correct. It would be legitimate to ask that question, just like you can ask whether the classification of finite simple groups is correct. But to say the theory is correct doesn't, it's not really a question to ask.

**ZIERLER:** How do you measure progress in the field?

**ASCHBACHER:** Well, there are problems that people within, or even outside, the field decide are important. If people are able to prove those theorems or make some other kind of progress–the problem may not demand just a theorem, it might demand interesting examples or counterexamples. If you can produce those things, then that's progress.

**ZIERLER:** I see you still work on a blackboard despite having computers all around. It's still important for you to write things out physically.

**ASCHBACHER:** Well, you can't really see from what's up there, but a lot of what I would write down would be little pictures, diagrams, things like that. And you can't do that very well on a computer.

**ZIERLER:** That does beg the question how increasing computational power has been relevant for your research over your career. Or has it not?

**ASCHBACHER:** Well, it depends what you mean by relevant. Maybe one time, I used a computer. But I sort of don't like using a computer because it's a crutch that allows you to computationally solve some problem, whereas if you didn't have that crutch, then you'd have to think, and hopefully, be able to solve the problem because you produced some new points of view, new ideas, so you could have a greater understanding of a problem if you just solved it or some portion of it using a machine. But in finite group theory, for example, with these finite simple groups, there are some infinite families of them.

And then, there are 26 groups that don't seem to fit into any one of the families. They're called sporadic groups. And the sporadic groups were discovered over a period–Mathieu discovered five of them back in the mid-19th century. There were 21 left, and they started to be discovered in 1965. For about ten years, they were discovered at a rate of about two a year. To discover it meant you were working on some problem, and you were trying to show that the answer to the problem is, perhaps, some set of simple groups. And you reach a point where you have something that looks like a simple group. You have some object, some object that may or may not exist, some supposedly simple group, and you start to build up a bunch of self-consistent information about this thing.

And after a certain point in time, if enough information is proven along the lines of, "If this thing exists, then the information about it is so-and-so," and it's all self-consistent, then people start to believe, "Well, there's got to be something here." At that point in time, this thing was said to be discovered. But now, there are two other questions. First, you want to know, "Does it really exist?" So there's the existence question. And then, you'd like to know, "Well, a group satisfying these various properties is unique up to isomorphism." That's uniqueness. So there's the existence in the uniqueness problem. And particularly in the early days, the period from '65 to '75, often an existence and uniqueness were established using a machine. But then, for example, some of these things started to be so big, that they exhausted the power of the computers at that point in time. As a matter of fact, one still can't analyze the largest sporadic group (the Monster) to any degree using a computer. The computer power isn't there even yet.

**ZIERLER:** What about simulation? Is computer simulation important for your research?

**ASCHBACHER:** For my research, the computer isn't important. But for some people, sure. I'm not sure simulation would be the word, but there are group theoretical packages. And for lots of people, the first thing they think about doing is input into one of the computer packages and see what slides out, so they can get some idea about what seems to be true. Then, use that to generate conjectures, maybe even beginnings of ideas about how to approach a problem.

**ZIERLER:** Another question about progress, coming from the vantage point from where I'm more comfortable in the sciences, and that is, progress doesn't necessarily mean understanding nature. It just means understanding it at a deeper level of complexity. And you can keep going, and there's probably no end to that progression. In mathematics, is that also the same, that when you achieve a breakthrough, it doesn't necessarily mean that you understand fundamentally the problem, it just raises new questions that couldn't be raised before?

**ASCHBACHER:** Well, probably what you do is understand some problems in the larger domain that you're studying. Say, finite group theory. But there are always other questions to ask. So this classification of finite simple groups, we have that. But then, you use that to go on and solve problems, often from other areas of mathematics. But to do that, just knowing the simple groups isn't enough. You have to know lots of facts about them. And you always want to know more facts. Maybe you think, "Well, I know everything that's worth knowing." But then, some problem comes along, and you see that there are extra things that you need to know to solve that problem.

**ZIERLER:** So in essence, there's no end. Hundreds of years from now, mathematicians will still be plugging away?

**ASCHBACHER:** Oh, sure. But they'll be plugging away at different things than we're plugging away at. As a matter of fact, I got my PhD in '69, so I've been working 50 years or so, and mathematics has changed a lot in those 50 years. I can barely recognize it.

**ZIERLER:** This is perhaps as much a philosophical as a mathematical question, but as I'm sure you probably know, many physicists are critical of string theory because it's not testable, it's simply mathematics.

**ASCHBACHER:** Not physically verifiable.

**ZIERLER:** That's right. So in mathematics itself, how do you know that what you're working on is tethered to reality and is not just conceptual? Or is that not the case, and it's not a concern on top of that?

**ASCHBACHER:** I'd say it's not a concern. If it is connected to the real world in some way that the real world finds interesting, that's nice because that makes your work more important.

**ZIERLER:** Socially useful, you mean?

**ASCHBACHER:** No. If someone is judging a piece of mathematics, at least the way I would judge it, you ask can you do something with it? Can you prove things in other areas of mathematics, or perhaps even better, can you use this theory in a significant way to solve some real-world problems?

**ZIERLER:** That's always a good thing.

**ASCHBACHER:** Yeah, sure.

**ZIERLER:** But that doesn't mean that you'll shy away from research if you can't draw those connections, either.

**ASCHBACHER:** That's right.

### Mathematics and Reality

**ZIERLER:** How do you tether yourself, then? How do you know, if it's not connected to physical reality, that you're working on stuff that is testable, that's falsifiable?

**ASCHBACHER:** Well, it could very well be tethered to some other areas of mathematics. So just as it's nice to have your theory say something in the real world, it's nice to have it say something in the mathematical world. Maybe in some specialty in mathematics that, on the surface, has no connection to your specialty. And if you're able to solve a problem that is of interest in the mathematical world, probably meaning that the professors at good mathematics departments appreciate it, [laugh] then that's giving some prestige to the area of mathematics that you're working in.

**ZIERLER:** A cosmologist would say that the laws of physics are universal, and a cosmo-chemist would say that the periodic table is universal. Is mathematics universal as well? On some faraway planet, it's the same math?

**ASCHBACHER:** Well, this is related to a philosophical problem in mathematics. Do mathematical objects really exist? Or are they discovered, created by the mathematicians that came upon them? And I think you'd find that almost all (particularly good) mathematicians would answer that these notions really exist.

**ZIERLER:** As opposed to being…

**ASCHBACHER:** Abstract somehow. But that they really exist. So your mathematician on this other planet would be led to the same notions.

**ZIERLER:** Meaning that it's more than just a shared psychological construct?

**ASCHBACHER:** Yes.

**ZIERLER:** But I guess to go back to the lack of being able to prove a theorem, how do you know? How do mathematicians satisfy themselves that they know these things to be true? Or is that where you bump up against belief?

**ASCHBACHER:** Well, probably these days in particular, a group of mathematicians write down the proof of a theorem, and the community looks at that proof. Eventually, they either say it's incomplete or wrong, so you have to go back to the drawing board, or says, "Yeah, this proof looks right." And then, it becomes accepted mathematics.

**ZIERLER:** Maybe a more concrete way of asking the question, to come back to progress, because you work in so many different areas, when do you achieve particular satisfaction on a problem that you're working on, either that you can put at this time way for the time being, or you've done all that you can on it, and it's time to work for something else? How do you make those decisions?

**ASCHBACHER:** Well, breaking things up into little pieces. You try to prove theorems. This guy, Gorenstein, came up with a strategy for approaching the classification of the finite simple groups. The strategy is composed of proving a bunch of theorems. So if you can prove some important theorem within the context of some large strategy about how to approach your area of mathematics, that's an accomplishment. For me, I would get satisfaction if I did that. And at that point in time, I have to decide, "Do I want to pick out another theorem in this area that I want to work on? Or do I want to do something slightly different?"

**ZIERLER:** A more nuts and bolts question, not so philosophical, but what does a day in the life for you as a mathematician look like? Are you at the board all day? Is it pen and paper? Are you on the phone, are you talking to colleagues? What does a day look like for you?

**ASCHBACHER:** I'm thinking. Sitting here at my desk, thinking about something, maybe scribbling down a few things on a piece of paper. When I have enough scribbles, and they've coalesced into something which seems to be locally complete, I go on my little computer here, type it up, print it out, stick it away, and build up enough of these small pieces of work that they go into completing the proof of the major result that I want to prove.

**ZIERLER:** We'll talk lots about Caltech, but just at the outset, an administrative question. Being here, where there isn't a discrete math department, it's the division of physics, mathematics, and astronomy, is that meaningful to you at all? Does that make a difference in terms of how you work, who you interact with?

**ASCHBACHER:** No, not really. But politically, and from the point of view of the department, it makes a difference.

**ZIERLER:** Politically meaning how?

**ASCHBACHER:** For example, the division chairman is probably going to be the major person deciding how to allocate resources. Any given division chairman may or may not be sympathetic to mathematics and allocate what we think is our fair share of resources. So there are good times and bad times from that point of view.

**ZIERLER:** Because you don't need an expensive laboratory to do your work, what are the major budgetary requirements in supporting mathematics?

**ASCHBACHER:** Salaries.

**ZIERLER:** And obviously bringing on new faculty.

**ASCHBACHER:** You can't bring on new faculty unless they're willing to commit the salary for that person. That decision will be made primarily by the division chairman, but also by the provost.

**ZIERLER:** And broadly speaking, over your long career at Caltech, how has mathematics been supported in those dollars and cents terms over the years?

**ASCHBACHER:** Not as well as lots of places. Mathematics has historically not been a high priority at Caltech, but I don't know, maybe sometime around the mid-70s or so, we, the mathematics department, started to develop better strategies and had better taste than some of the previous faculty had. So the quality of the department has improved a lot since that time. I sort of lost track of how it is at the moment, since I've retired. But over a certain period, there are ratings for all departments. Physics, chemistry, mathematics, I don't know what we typically would've been pre-1975, but eventually, we at least got up into the top ten departments. Seven or eight, something like that.

**ZIERLER:** Let's go all the way back to the beginning. I'd like to ask about your parents first. Tell me a little bit about them.

### Upbringing and Education

**ASCHBACHER:** My parents came from a small town in Southern Illinois called Staunton. Maybe I have to modify that. My mother lived, at least when she was a small child, in New York. But her mother died at an early age, then she went to live with an aunt and uncle in Staunton in Southern Illinois.

**ZIERLER:** What about your father? Where does his family come from?

**ASCHBACHER:** From Staunton.

**ZIERLER:** How many generations back do they go there?

**ASCHBACHER:** Well, not that many generations. My father's father, I think, probably was from Staunton. But his father might have emigrated from Europe. I'm not sure. One time, I sat down for a day and tried to find stuff like that out. I found that some Aschbachers came over from I don't know where, but they were on the boat from Le Havre and came to the US. So I'm not sure. I've heard from Alsace or Switzerland. The name is not so uncommon.

**ZIERLER:** It's a German name?

**ASCHBACHER:** German or Swiss.

**ZIERLER:** Do you know what it means?

**ASCHBACHER:** Ash brook? Something like that.

**ZIERLER:** What was the highest level of education that your parents achieved?

**ASCHBACHER:** Oh, my father had a PhD in accounting. He wasn't very ambitious, but I think he was probably very smart. After leaving high school, maybe he went to some small college to play football, only he couldn't do that because he had low blood pressure. So for some period of time up until the beginning of the second World War, he was working in the post office in Springfield, Illinois, the capitol, and my mother was teaching school. Then, the second World War started, so he would've been drafted, I guess. But he took some exam and got into officers training school. And my mother was a WAC for a while. I was born in Little Rock, Arkansas, so everybody thinks I'm from Arkansas. But as I recreate it, my parents were at an Army base in Arkansas. And he might not have been around when I was born, I'm not sure. He was in the Pacific, eventually was on Okinawa, so that must've been after the Japanese surrendered.

**ZIERLER:** Did he ever talk about his experiences in the Pacific?

**ASCHBACHER:** No. And I never asked him. I never asked my parents lots of things. Didn't bother me not knowing. But then, eventually, when I got to be of an age when it started to be interesting, they were dead. All I had was my own vague memories of what went on.

**ZIERLER:** What was your father's career? Did he teach accounting?

**ASCHBACHER:** Yes. When he came back from the War, he went to school at the University of Illinois on the GI bill and got an undergraduate degree there, probably in business. Because I assume he got a master's degree at the University of Illinois in accounting, then had to write a thesis, which he wasn't all that interested in doing. But he got a job at Michigan State in East Lansing and eventually wrote the thesis. So I think we went to East Lansing in '53, and then '59, came out here. He got a job. He was a professor of accounting at Cal State Northridge. I don't know if that was because he didn't get tenure in Michigan State or whether he preferred to come to California, where he could play golf all the time, which was his major interest at that point.

**ZIERLER:** How old were you when the family got to California?

**ASCHBACHER:** I must've been 15 or 16. I was born in '44, so 15, I guess.

**ZIERLER:** And up until that point, you grew up in Illinois?

**ASCHBACHER:** No. Started in Illinois until '53, and then we went up to East Lansing, so we lived in a little farming community outside of East Lansing, a community where they were starting to build up a subdivision as was the American way after the war, and then came out here in '59.

**ZIERLER:** Do you have brothers or sisters?

**ASCHBACHER:** No, I was an only child.

**ZIERLER:** When did it dawn on you, or your parents, or your teachers that you had mathematical abilities?

**ASCHBACHER:** None of those people would've known what it meant to have mathematical abilities. You don't see what mathematics is until you get to university, and then may not see what it is unless you go to a university that has a good mathematics department.

**ZIERLER:** What about your father with his expertise in accounting?

**ASCHBACHER:** That's not mathematics.

**ZIERLER:** What would you say that is, arithmetic?

**ASCHBACHER:** Not even that. I guess if you're going to say anything, you'd say arithmetic. There's accounting theory, I'm sure. But I doubt very much it's taught well. I won't speculate.

**ZIERLER:** What kind of high school did you go to in California? Was it a big school?

**ASCHBACHER:** Yes. Went to a big LA city school, James Monroe High School. 3,000 students, three years in the high school. About 1,000 in each class. But only a small percentage were in a track that would lead to going to college.

**ZIERLER:** Were you a strong student in high school? Did you do well?

**ASCHBACHER:** Yes. I didn't do as well as I should've. But yeah, I was about tenth in my class maybe, something like that.

**ZIERLER:** Did math come easy to you? Algebra, calculus?

**ASCHBACHER:** Everything came easy to me. School was boring.

**ZIERLER:** Did you read outside? Did you study on your own to keep yourself occupied?

**ASCHBACHER:** I read outside, but just genre fiction. I didn't read to improve myself.

**ZIERLER:** What about college courses during high school? Did you look for more education than you could get in your high school?

**ASCHBACHER:** No, I didn't.

**ZIERLER:** And you said your parents, your teachers wouldn't have had the tools to recognize…

**ASCHBACHER:** Well, we weren't doing what a mathematician would think of as mathematics. You're doing arithmetic and some low-level calculus maybe. The school at least had a half a semester of calculus. At Caltech, even, the majority of the people who came did not have a calculus course. So there were two sections for people who had some sort of calculus course in high school. Some things have changed. Nobody now is admitted as a student wouldn't have had a calculus course.

**ZIERLER:** When it was time to think about college, were you thinking about focusing on mathematics?

**ASCHBACHER:** No. I was probably thinking I'd focus on science, but I was actually thinking more in terms of physics than mathematics. But when I got here, I didn't do too well in physics, and I didn't feel so comfortable with it, whereas mathematics, I was doing well, and it appealed to me.

**ZIERLER:** After you graduated high school, was the draft something you needed to contend with?

**ASCHBACHER:** Eventually. But I was a student, so I had a student deferment. The draft ran until you were 26. So I think I got out of graduate school when I was 25. For a year, I was draft-eligible. But I had a big Los Angeles draft board, so they weren't likely to take some 26-year-old mathematician. They had plenty of other bodies. Fortunately, I wasn't drafted.

**ZIERLER:** Besides Caltech, where else did you apply?

**ASCHBACHER:** Well, I don't really remember completely.

**ZIERLER:** Was Caltech the dream for you, where you'd always wanted to go?

**ASCHBACHER:** No. I applied to Princeton. That was the place I wanted to go, but they didn't accept me. I applied to MIT. I was accepted everywhere else. At MIT, Caltech, and I must've had some backups like maybe some branch of the University of California. I don't remember. The two schools that were in the mix were MIT and Caltech. And I wanted to get away from home, so I wanted to go to MIT. But my mother had cancer in the summer before my senior year, and she said she wanted me to not go that far away. She wasn't sure that the cancer would not reoccur. So I agreed. They might not have financed me to be anyplace else, so maybe I had to say OK, I'm not sure. But anyway, I ended up at Caltech.

**ZIERLER:** Now, Princeton, MIT, Caltech. You're making these decisions on some vague notion that you were pursuing a degree in physics?

**ASCHBACHER:** No, not really. Of the various sciences, physics was the one that had somehow most appealed to me, that's all. I had not given serious thought to the matter. I wasn't even sure I would go into science. But coming to Caltech, that pretty much meant I was going to go into science.

**ZIERLER:** When did you arrive?

**ASCHBACHER:** '62.

**ZIERLER:** What are some of the classes in physics where you start to say to yourself, "Maybe this is not for me"?

**ASCHBACHER:** Well, at Caltech, you presumably understand that there's a core. And in those days, the core included two years of physics. So it wasn't even a choice of what you were taking. The physics that I wasn't doing so well in were these two years of physics. Plus, there was also a year or two of physics lab, in which I probably did even worse. Worst of all was chem lab.

**ZIERLER:** That didn't work for you?

**ASCHBACHER:** No, I wasn't disciplined or attentive enough to detail to do well in a laboratory setting. I couldn't have been an experimental physicist.

**ZIERLER:** Was there a specific professor or course where a lightbulb went off, where you realized where you belonged in math?

**ASCHBACHER:** Yeah, in my freshman year, there were these two special mathematics sections. But it turned out it was lousy to be in those sections. For example, the instructor in our section was an aeronautical engineer. I think he was a grad student, as a matter of fact, not even a post-doc. So that wasn't so great. But then, my sophomore year, I took two mathematics courses, the introductory algebra course and introductory analysis course and had pretty good instructors. As a matter of fact, my instructor in the analysis course was Donald Knuth. He was a student of Marshal Hall, who was the biggest name in the mathematics department at that time.

Hall had gotten him hired as an assistant professor, I guess. The way I've heard, he probably would've stayed, but he wanted to consult to make extra money. And Caltech, in those days, wouldn't let people consult. So he left and went to Stanford. But he was a very interesting instructor. And the other guy was pretty good, the algebraist. He was less exciting, but the course was interesting. I did well in those two courses and enjoyed them. I'd already said I was going to be a mathematics major because I wasn't going to be a physics or chem major. [laugh]

**ZIERLER:** What were some of the exciting ideas in mathematics during your time as an undergraduate? What were the professors working on?

**ASCHBACHER:** Well, the professors here, probably most of them weren't working on things that were too exciting. Marshall Hall was. He worked in combinatorics and group theory, both finite and infinite. And at this point in time, finite group theory was starting to take off. That was probably the most interesting thing that was going on here at that time.

**ZIERLER:** Was Caltech considered a leader in this field at the time?

**ASCHBACHER:** No, I would say not. Depends what you mean.

**ZIERLER:** Was it an intellectual center for the field?

**ASCHBACHER:** Yes, as long as you allow yourself to go down a number of universities for intellectual centers. But sure. Marshall was here, and eventually he collected some instructors that were group theorists, too, including me and David Wales.

**ZIERLER:** Were some of the physicists catching on to group theory?

**ASCHBACHER:** Well, you hear the eightfold way, Gell-Mann's thing described as being based on groups. But it's really based on lie algebras. There's a lie algebra, which is another algebraic object and has an associated group with it. But I think, to the extent that I know anything about particle physics, which is not at all, it's the algebra that they're using. But these are algebraic objects that are related to groups. And indeed, indirectly to the finite groups. The simple algebraic groups are related to the finite groups.

**ZIERLER:** Were you particularly strong in your classes? Did the professors single you out or recognize that you had the ability to go on to graduate school?

**ASCHBACHER:** Oh, certainly, to go on to graduate school. You gave me two interviews to read, and in at least one of them, the notion arose that the Caltech undergraduates were much stronger than the graduate students. Well, it was even worse for mathematics. There were probably a few graduate students that were equal to the undergraduates. But that was about it. So anybody that was a decent undergraduate here could go on to graduate school if they wanted to. I was one of the better ones. For example, there was a guy named Vern Poythress [?], who won the Putnam Exam at least one year, so he probably would've been thought of as the best student. And then, there were myself and Richard Stanley. Both of us went on to have very good careers in mathematics. If you win the Putnam Exam, you become a Putnam Fellow, so your way is paid at Harvard if you want to go there. And he went there, but he went to divinity school, not to the mathematics program.

**ZIERLER:** You said as an undergraduate, that's when you decided on pure math to focus on. Was it a specific class? Was it just a general sense that you enjoyed it, that was what you were good at?

**ASCHBACHER:** Well, you had to declare a major, so I declared a mathematics major as opposed to applied mathematics major. I assume there was no applied mathematics major at that time. Because of the fact that there were so many core courses, and in those days, it was even worse, you almost have to take the introductory courses in whatever discipline you're working in early on in your sophomore year, so you can build up the background to take graduate courses your junior and senior year and get out with some decent background. The mathematics courses I took were two introductory courses in pure mathematics.

**ZIERLER:** Does the focus on pure mathematics coupled with the decision to go to graduate school put you on a professional track where, if you're successful, you'll end up being a professor? Or are there other career options to consider?

**ASCHBACHER:** Well, there were certainly other career options to consider, but at the time back when I graduated in '69, that was a little before the job market turned sour. So the job market was pretty good then.

**ZIERLER:** '69 was the PhD. '66 was the bachelor's.

**ASCHBACHER:** Yeah. But getting into graduate school is nothing, coming from Caltech. Although, the University of Wisconsin, after Berkeley, was probably the best public university in the country at that time. In those days, it was easy to go on and become an academic if you got a PhD in mathematics. It might be a little harder to go to a good research university. Matter of fact, it would be harder. But a decent research university, you'd probably go to. Now, you might have run into the bad job market about the time that I got out, so you might've had some difficulties getting tenure. But at least for a lot of people, the academic life was one viable alternative. But you could go to work for an aerospace industry out here. There must've been lots of things. There's more now in computer science, statistics, finance. Some of those things weren't available when I got out of school. But I was thinking in terms of being an academic anyway. I'd seen that my father didn't have to work real hard. [laugh]

**ZIERLER:** Did you have a senior thesis at Caltech?

**ASCHBACHER:** No, Caltech didn't have a senior thesis. But they did have a competition. So I wrote a paper for that competition.

**ZIERLER:** What was the paper?

**ASCHBACHER:** It was a paper in combinatorics. I forget what the title would've been.

**ZIERLER:** What was the advice about graduate school that you received? Is it better to stay, is it better to go?

**ASCHBACHER:** No, it wasn't good to stay here. We're too small. So you've probably exhausted the resources if you've taken a lot of courses as an undergraduate. In special cases, if you want to work with somebody here and couldn't work with somebody comparable anywhere else, maybe. But you were counseled to go someplace else.

**ZIERLER:** What were the top programs in the country? Where were you considering?

**ASCHBACHER:** I'm not really sure what the top programs were in the country, and I didn't apply to the top programs. In those days, Harvard, Berkeley, Princeton, Chicago, probably MIT. And it's pretty much the same thing these days, except Stanford is in the mix.

**ZIERLER:** Did you apply to all of them?

### Combinatorics and Thesis Research

**ASCHBACHER:** No, I didn't apply to any of them. I had come here to this small science-oriented institution. I wanted to go to a different sort of institution. I wanted to go to a big public university, where there were different things going on. So the best of those universities, the one that I wanted to go to and could've gotten into without worrying, was Berkeley. But Marshal Hall didn't want me to go to Berkeley. He said he wouldn't write a letter of recommendation. I think he probably would've if I'd pressed him. But he said the department was too big, I'd get lost in the mix and everything. My estimation of the situation was that the second-best place amongst the big public universities was Wisconsin, so I went there. And it had the virtue of having Richard Bruck.

**ZIERLER:** Did you know him by reputation even before you arrived?

**ASCHBACHER:** Sure, yeah.

**ZIERLER:** What was Bruck known for?

**ASCHBACHER:** A certain type of combinatorics and non-associative algebra.

**ZIERLER:** And that's specifically what you wanted to focus on?

**ASCHBACHER:** The combinatorics. I didn't know anything about non-associative algebra. Strangely enough, later on, I did a little work in it. But at that time, I didn't.

**ZIERLER:** On the social side of things, Madison must've been pretty interesting in the late 1960s.

**ASCHBACHER:** Oh, yeah. The first day I got there I walked around the university, and came to a place where the police were pushing the crowds back. I waited to see what was happening, and in a while they set off tear gas. People were running around the university, trying to escape from the tear gas.

**ZIERLER:** Were you politically involved at all?

**ASCHBACHER:** No. This is probably too old for you to know about, but right next to the mathematics building was a small building devoted to Army research, or at least Defense Department research. And somebody set off a bomb in there, I think next to the computer. And the computer was someplace else. People were killed by the bomb. For example, I roomed with some astronomers, and one of the astronomers had his thesis on stellar interiors in this computer, and he lost his thesis. But of course it was even worse for the people who were killed in the explosion.

**ZIERLER:** Tell me about Richard Bruck as a person. Did you work closely with him?

**ASCHBACHER:** No. I had a problem I was working on when I was an undergraduate here, and I worked with it and solved it, took it to him my second year. He wanted me to do some more, so I did some more. And I had to stick around for the three years anyway. And I went to his seminar. He was a nice guy.

**ZIERLER:** How much of the curriculum is coursework, and how much is you working independently?

**ASCHBACHER:** Well, you had to take a certain number of courses. Maybe if I'd stayed around longer, I could've escaped some of the courses. But you had to take courses, and you had to pass qualifying exams. So for me, I always took two or three courses in each term. That left plenty of time to do work.

**ZIERLER:** Intellectually, being at a bigger school, bigger department, in what ways were you exposed to new areas of mathematics?

**ASCHBACHER:** Well, you had to take some courses and pass these qualifying exams. But most of what I needed to know in my qualifying exams, I had encountered as an undergraduate. And I took them the end of the first year, so I hadn't forgotten them by then. So I was in fairly good shape. And then, I was freer to take what I wanted to. For example, one thing about being in a reasonably good university like that is, they had visiting professors. One of the first two years, I took a course in infinite groups from a guy named Bernard Neumann, an excellent infinite group theorist. And the other year, I took a course from Helmut Wielandt, who was German, obviously, and at that time was thought of the excellent finite group theorists.

So I was exposed to some group theory taught by good people. Two people I went to work with at Illinois were Michio Suzuki and John Walter. And Suzuki had a student that came to Madison I think probably my third year, I'm not sure. But that's when I became aware of him. And he taught a course out of a mimeographed version of Gorenstein's book. Gorenstein wrote a book that was on modern finite group theory. So this guy, Bauman, taught the course out of these notes of Gorenstein, which really interested me, it was very pretty stuff. Bauman probably got me the job at Illinois, my year as a post-doc before I came here. I was living in Champaign-Urbana, which was not my idea of a fun time.

**ZIERLER:** What did you work on for your thesis research?

**ASCHBACHER:** A certain type of symmetric block design. And the big result was constructing a new symmetric block design of this particular sort. Still the largest one. But it's not like discovering a sporadic group.

**ZIERLER:** How long was your thesis?

**ASCHBACHER:** 50 pages.

**ZIERLER:** Is that about average size?

**ASCHBACHER:** For mathematics, yeah.

**ZIERLER:** And how much of it is the equations, and how much of it is prose, you explaining what you found?

**ASCHBACHER:** Well, you would say that most of it is prose, but it's not explaining what I found. There's probably some introduction where I'm trying to make what I did look good. But a proof in mathematics is basically English.

**ZIERLER:** What did you find? What were the conclusions of your research?

**ASCHBACHER:** That there existed a symmetric block design of a certain type. To have found it, I had to have some body of theory which would lead me to this thing. It didn't pop out by accident. So also, part of the thesis was developing a theory of automorphism groups of symmetric block designs. Already, I was dealing with finite groups there.

**ZIERLER:** To the extent that there are fashions or fads in mathematics, was this a hot time for this area of research? Were there lots of people working on it alongside you?

**ASCHBACHER:** Yeah, but it still wasn't a central area of mathematics. There was a fair number of people, particularly in Germany. Bruck had worked on these finite geometries. Some of his students worked on that.

**ZIERLER:** Was there an oral defense?

**ASCHBACHER:** Probably, but I don't remember.

**ZIERLER:** That's probably good, it was uneventful.

**ASCHBACHER:** Oh, I don't think I had to worry about defending my thesis.

**ZIERLER:** Besides Bruck, were there other professors at Madison you were close with?

**ASCHBACHER:** No.

**ZIERLER:** As a graduate student, you could really work on your own?

**ASCHBACHER:** Well, with Bruck, you could. Like I say, the biggest part of my thesis, I did before I even knew him. But then, I became a member of his research group, and he had a weekly seminar. And I got to know him a little bit better.

**ZIERLER:** Is a post-doc standard after the PhD in math?

**ASCHBACHER:** It is now. In those days, I don't think it was, no.

**ZIERLER:** Were you on the job market, and Illinois was simply the best offer at the time?

**ASCHBACHER:** I was on the job market because I was going to graduate. But I didn't apply very many places. I applied to Caltech. I liked living in Pasadena, and Marshall Hall was here. I applied to Berkeley, probably. Probably the only reason I applied to Illinois was Baughman, because I wouldn't have known there was a job available. To be truthful, I don't remember exactly where I applied. But this might've been the only job that I was accepted for. It was a funny type of post-doc. There was only $6,000. A year later, when I came here as an instructor, I think my salary was $11,000. But I didn't have to do anything. So in that sense, it was good. It was probably better to have the $6,000 and not have to teach. Besides, at Illinois, it would've been more than one course.

**ZIERLER:** Were you happy to come back to Caltech?

**ASCHBACHER:** Yes. I lived part of my early life in Champaign-Urbana, but later on, I wasn't so pleased to live there.

**ZIERLER:** It was good to be back in Pasadena.

**ASCHBACHER:** Yeah.

### The Classification of Finite Simple Groups

**ZIERLER:** What were you working on when you arrived back at Caltech at that point?

**ASCHBACHER:** I was working on permutation groups. Marshall Hall thought I was going to be the combinatorics instructor. David Wales was here at the time. He was supposed to be the group theory instructor. But I wasn't that interested in the combinatorics. Some of the group theory problems I worked on had combinatorial associations.

**ZIERLER:** And how did you become involved in the classification of finite simple groups?

**ASCHBACHER:** Well, Hall had a seminar, and the seminar was going over papers on sporadic groups. And there was a visitor named John MacKay, and he had a mimeographed copy of a preprint by a person named Bernd Fischer, a German mathematician in Bielefeld that was groundbreaking, very novel. In it, Fischer discovered three new sporadic groups. So I was assigned that paper. And the way I read it, there were some tricks that he had that could be extended to a much more general situation. I took that as a problem to see if I could determine the groups that satisfied this more general property and then was able to do that. And that turned out to be important for the classification. And I started going to meetings those people went to and already knew a little bit about it from working with Walter and Suzuki.

**ZIERLER:** But you were somewhat outside of this field, initially?

**ASCHBACHER:** Oh, yes. Sure. The insiders were people who worked with the–there was a new technique introduced by a guy named John Thompson called local group theory. And the people proving most of the good theorems were proving them using local group theory. And so, those were the graduate students of the good local group theorists like Thompson, and Feit, and Gorenstein. Suzuki was such a person, but the year I was there, Suzuki was writing a textbook, so he wasn't all that accessible. And Walter was well-known to be a terrible communicator. Very nice guy who had good ideas, but he couldn't write them down.

**ZIERLER:** In what ways did you have to play catch-up in this field, coming from the outside?

**ASCHBACHER:** Well, I had to learn the basics. But I learned them pretty fast, so I never felt at a disadvantage.

**ZIERLER:** What were some of the overall goals of classifying finite simple groups?

**ASCHBACHER:** Well, you have some objects, some finite simple groups, and you want to prove that your list contains all of them. You want to prove that each one of these things exists and is unique up to isomorphism. And moreover, if you have any finite simple group, it's on this list of groups. Now, this is a little difficult because remember, during this period, a couple new groups are being discovered every year. But that's OK. You can somehow work that into the mix. I should say, it's important to know what these things are, not just from the point of view of having something to work toward proving, but because the proof works inductively.

So you consider a finite simple group that's of minimal order subject to not being on your list of groups, and that has the nice consequence that any finite simple group that's properly involved in this potential group is on the list. And if you know things about the groups on the list, then you can use that knowledge to help in your proof. As a matter of fact, that's the only way you have a chance of doing anything, unless you come up with some brilliant idea that's probably not out there. That's what the program was. And the most important part of it, the part that's most problematic, is, if you're given a simple group, why does it look like one of these groups that you know about? What does it even mean to look like? What general nice properties did these simple groups share that are useful in your proof?

**ZIERLER:** Tell me about the Duluth Conference – where you were presenting your findings.

**ASCHBACHER:** Actually, George Glauberman was presenting his findings. The National Science Foundation, in those days funded meetings. The typical meeting that they would fund, there would be a principal speaker, and then some small group of established mathematicians. And so, the meeting would consist of a series of talks by the principal speaker and maybe one talk by each of the supporting mathematicians. And then, there would be a bunch of graduate students or post-docs that would sit out there, and take notes, and learn stuff. So that's what this Duluth meeting was. And the principal speaker was George Glauberman, an important figure in the field. And I was one of the other people.

**ZIERLER:** What clicked for you in this field? How do you understand your success in being able to accomplish all of these things? Does it come from combinatorics?

**ASCHBACHER:** Yeah, in part. For example, I mentioned this thing about Fischer's paper. Fischer was using combinatorial ideas. He didn't know any local group theory. He used some combinatorial ideas, and he used some facts about small file Coxeter groups, whatever those are. In this more general setting, you lost the ability to use the information from the Coxeter groups, but it turns out, you could keep the combinatorics. So I put together the combinatorics with a little bit of local group theory, whatever I knew at the time, which turned out to be enough to solve this problem that I was working on.

And more generally often, I think, coming from this combinatorial background, it gave me a different viewpoint, which I think probably was advantageous, too. But one important thing was, this local group theory was new. So the old timers like Marshall Hall and Wielandt wouldn't have known anything about it anyway. I wasn't really at much of a disadvantage, vis-a-vis I was young and willing to learn. Maybe the students of some of the good local group theorists had a slight advantage, but after a few years, that disappeared.

**ZIERLER:** What are the overall goals in the classification? What are you trying to achieve?

**ASCHBACHER:** You're trying to prove the theorem that this list of groups is the list of all finite simple groups. What does it mean to be a simple group? Well, it means you can't break the group down into two smaller pieces, whatever that means. So if you have a general finite group, if you know all the simple groups, then in theory, you know this group–well, that's not true. You break it up into pieces, which are simple groups, and then there's an extension problem, which asks you to paste together these groups. And what ways can you do that? Well, it turns out there are just too many ways to do that. It's not feasible to try to classify all finite groups because this extension problem is too complex.

But what seems to be the case, what works out surprisingly often is, if you have a problem involving some finite group, you can reduce it down to a problem about maybe not a simple group, but something that's very close to a simple group. So if you know the simple groups, and you know the right properties that those groups have, then you can say something about this group, which arises in some context.

**ZIERLER:** On the question of complexity, how do you convey these ideas so that your colleagues understand what you're talking about, given the fact that these are such difficult concepts to understand?

**ASCHBACHER:** Well, the way Gorenstein did it was to say, "Well, ghd proof of this theorem involves thousands of pages." Maybe he said 10,000. "And hundreds of mathematicians. Took years to do." That was a measure of the complexity of the problem.

**ZIERLER:** These require books. These are not just papers.

**ASCHBACHER:** Hundreds of papers and some books, too. Michael Atiyah, a great mathematician, held a meeting in London under the auspices of the Royal Society on proof, and basically, I think, the real reason for it was what you were talking about, computer-aided proofs. But what I talked about was exactly this fact that, "Here's a theorem that is terribly complex, the proof of which is very, very long and involves lots of people. Is this proof really correct?" And the answer is no, it can't possibly be correct. There are going to be all sorts of local errors. But what you expect is that these local errors will be small, and you can correct them more or less immediately locally. Now, these days, there are theories of computer-aided proofs, and somebody will probably try to write out this proof in the right language and use the machine to verify it. And no doubt, it will discover patches where it's not so easy to fill in the details. But I think there's enough robustness in the proof and what's gone afterwards that it's probably correct. But I wouldn't bet a large amount of money on the nonexistence of any more sporadic groups, for example.

**ZIERLER:** Tell me about your time at the Institute for Advanced Study in the late 70s. Why'd you take the sabbatical? Just a change of scenery?

**ASCHBACHER:** Borel, who was one of the members of the Institute at that time, organized a year on finite simple group theory. So that meant that he was organizing sort of a super one of these NSF things. He invited some core group of simple group theorists who stayed there the year or some large fraction of the academic year, and then brought in lots of other people for shorter periods of time.

**ZIERLER:** Did you enjoy the intellectual atmosphere there?

**ASCHBACHER:** Yeah, sure. Like I say, I tend to more work on my own stuff. So it was a mixed blessing. It was nice to hear all the talks and talk to the people, but on the other hand, it also means that people try to take a certain amount of your time.

**ZIERLER:** Was the Cole Prize the first major award you won?

**ASCHBACHER:** Yes, that's right.

**ZIERLER:** What did it feel like to get the Cole Prize?

**ASCHBACHER:** It felt very good. [laugh] Validation of your work, in some sense. The mathematical community thinks that some portion of the work was good.

**ZIERLER:** What kind of administrative service did you do at Caltech?

**ASCHBACHER:** I was Executive Officer for Mathematics for three years in the 90s.

**ZIERLER:** Is that sort of like a quasi-department chair, if math was a department?

**ASCHBACHER:** It's exactly like a department chair, except lots of duties are done at the division level, not at the department level. For example, salaries are determined at the division level, which is good. If you're executive officer, you don't get in trouble with someone who feels their salary isn't what they'd like it to be.

**ZIERLER:** Did you take on a lot of graduate students?

**ASCHBACHER:** I never took on a lot of graduate students. I had graduate students most years, initially, as long as there was sufficient activity in things I was working on at a high enough level, so that if I had a graduate student, they could get a good first job. But it reached a point maybe in the 90s, something like that, that there weren't any faculty members at good American universities who would work to get one of my good graduate students. For example, my last graduate student came from China, Beijing University. He was on their olympiad team. He was very quick and wrote a good thesis.

And normally, the only person who was left at that point in time at a good university was Walter Feit at Yale plus a couple people at Chicago including Glauberman. Anyway, my student went to the University of Illinois at Chicago and accepted a job there. And then, unfortunately, something like a week later, a job turned up, I can't remember whether it was Yale or Chicago. But it might not have turned up. So he spent a year at Chicago Circle, and then his wife got pregnant. He got a high tech job in San Francisco and eventually started his own hedge fund.

### Key Collaborators

**ZIERLER:** When did you first meet Stephen Smith?

**ASCHBACHER:** He was an instructor here in the mid-70s. Actually, I probably met him before that. He was a student at Oxford with John Hall working under Graham Higman. I would pass through Oxford at that point in time fairly frequently. And I'm sure I must've seen him there. Actually, I was Hall's de facto advisor.

**ZIERLER:** And is Stephen Smith that rare person where you did find opportunity to collaborate, not just work on your own?

**ASCHBACHER:** Oh, yeah. We collaborated on this quasithin thing. It took five years approximately.

**ZIERLER:** And what is it either about Stephen or the problem that compelled you to collaborate and not work alone?

**ASCHBACHER:** Well, it's such a big problem.

**ZIERLER:** What's the problem? Why is it so big?

**ASCHBACHER:** It's asking to classify a certain subclass of the finite simple groups part of–well, it wasn't exactly what Gorenstein wanted, but related to what Gorenstein wanted. So these quasithin groups have a certain property. This is a class of groups that Gorenstein thought was being done by this guy at Santa Cruz, but after a while he wasn't doing it. This was an important chunk of the classification. People were getting restive. Serre was saying bad things. I had worked on similar things, worked on a subclass of this class called thin groups, and there were certain classes of argument that one could use to approach this problem that I was familiar with. So I recruited Steve to work with me. He wasn't familiar with these things, but he learned. Actually doing the mathematics probably only took a couple of years. Most of the time was spent writing it up. I have a certain style, and he has a certain style, and they weren't too close. We'd fight it out, and the style that emerged was, I think, a good compromise.

**ZIERLER:** What was the division of labor? Did you each do everything and then double check? Or you did some things, and Stephen did other things?

**ASCHBACHER:** We worked on parts of the problem and communicated about those parts of the problem. It wasn't as if he went off somewhere and proved something here, and I proved something there. It was more that we had a picture of how he wanted to proceed, a strategy, and we proceeded through that strategy.

**ZIERLER:** Given the size of the project, how did you know when you were done? What was the concluding point?

**ASCHBACHER:** Well, it depends what you mean by done. After maybe two years or so, we knew that what was left to be done was something that we could do. The significant obstacles had been overcome.

**ZIERLER:** What new work was possible as a result of this collaboration?

**ASCHBACHER:** Well, this completed the classification. So all the things that you need the classification for became real.

**ZIERLER:** How do you define real in this context?

**ASCHBACHER:** Well, people just assumed the classification was true, even though they knew the proof wasn't complete for many years. I don't remember how many years, but 10, 15.

**ZIERLER:** Are the association with the National Academy of Sciences, the American Academy of Arts and Science simply professional honors? Or are they useful in terms of meeting people and being involved in that intellectual atmosphere?

**ASCHBACHER:** No, I don't think that, at least in my limited experience, the latter is the case. But there's something in the former. Besides being an honorary society, it also gives you a chance--which I don't take--for service in mathematics and more generally in science.

**ZIERLER:** Are there decadal type surveys in mathematics the way that the Academy does that for the sciences, where there's a review of the field every ten years? Or something similar to that.

**ASCHBACHER:** I don't know. I've never encountered such a thing. But it would be surprising if there weren't.

**ZIERLER:** I'm curious if there's any sort of systematic effort to sort of take the overall temperature of the field every decade, or five years, or whatever it might be.

**ASCHBACHER:** I'm not aware of such a thing. With mathematicians, it might be difficult. Who would undertake this? That would be performing a service, for example. The way things work, I assume it's the same for each section, but for the mathematics section, there's somebody that manages the section for a period of, I don't know, two, three years, something like that. I suppose such a person could appoint a committee.

**ZIERLER:** It does beg the question: how do you keep tabs on what's happening in the field? How do you know that what you're working on is not redundant or has already been done? Conferences, papers? How do you keep abreast of what your colleagues are doing?

**ASCHBACHER:** Well, for somebody in my position, it's just automatic. Email, conferences, rumors. But a bigger question is keeping track of things that are going on in other disciplines.

**ZIERLER:** That's important, too.

**ASCHBACHER:** Well, it can be important mathematically. But for example, one place it's important is, if you're trying to hire people, which of course, we always are. You have to know who has done something that's important, and maybe more important than that, who's likely to continue doing important work? What's their current situation? Are they likely to be movable for some reason? Do they have a spouse or girlfriend that has to be accommodated? And some people are tuned in to that stuff. I'm not. But virtually my whole career here, I've been on the committee that hires people and looks for people to hire. And to do that, you have to read all the letters of recommendation. And you can learn a lot by doing that, even in a field that is far from your own field, simply by knowing the letter-writers and having some sense about whether you can believe them or not, where you can believe them, and where you can't.

**ZIERLER:** Beyond the letters, what do you look for in assessing whether somebody's going to have a successful academic career in mathematics? What are some of the human qualities or characteristics that you see?

**ASCHBACHER:** Well, you don't necessarily see them. Hard work, quickness.

**ZIERLER:** What about intuition? How important is intuition?

**ASCHBACHER:** Oh, very important, yeah. But hard to measure. Whereas hard work is a little bit easier to get your hands on. Accomplishments. If you have somebody whose opinion you trust, you can get some sense of what they've accomplished. The way we started to raise the equality of our department was to use a strategy whereby we would identify young people and offer them tenured jobs, occasionally tenure track jobs. But if there were good enough tenure jobs, normally the other universities, particularly, the very good universities, would be willing to do. So we would get some very good young people. Of course, the down side was that as soon as they got to a point where their reputations were established, often Harvard, Princeton, or someplace like that would hire them away. But at least we had some good young people for some period of time in their career. Often, maybe the most important time.

**ZIERLER:** And the attraction would be more prestige or salary?

**ASCHBACHER:** Tenure.

**ZIERLER:** But what if they already had tenure here?

**ASCHBACHER:** We're talking about trying to hire somebody that has a faculty position, or is a post-doc, or a graduate student even. Get them to come to Caltech.

**ZIERLER:** But generally, has losing faculty been an issue for the math division?

**ASCHBACHER:** Turnover has been, yes. Attrition.

**ZIERLER:** How has PMA dealt with that over the years?

**ASCHBACHER:** Well, it depends on who the division chairman is. If they're sympathetic to mathematics and realize what's going on, they let us, for example, make more offers than some other division chairmen might. The rationale being--which is carried out by experience--that you're trying to hire very good people. You may make three offers and get one acceptance. So if the only thing you can do is to make one offer for each position that's open, you're going to have difficulties getting good people.

**ZIERLER:** On the undergraduate side, what have been your favorite courses to teach?

**ASCHBACHER:** Well, I haven't taught many recently.

**ZIERLER:** Over the course of your career.

**ASCHBACHER:** Recently, I taught the core course in linear algebra. There are two branches of this course. The one I taught was the more mathematical one. I taught three courses a year. One of them would be this freshman course for one term, and then the other two terms, either the introductory undergraduate algebra course or the introductory graduate algebra course. When I was younger and it made sense, I would teach group theory course every once in a while. Sometimes the faculty needed something special. I once taught an algebraic geometry course, which I knew nothing about, but I had a decent textbook, so I could do it. There was a period, I think when I was Executive Officer, as a matter of fact, and just before then, the core courses had been taught by older faculty members who weren't too active.

But they were retiring about this time, so we needed to get bodies to teach these courses, which were not courses that people wanted to teach. Maybe the first year, I cracked the whip and managed to get bodies to fill these courses. But after that, it was too unpleasant, so at least one of the courses, I taught myself. So I taught math 1 or math 2. I did that maybe a couple years.

**ZIERLER:** The demography of Caltech has changed very much, certainly since you've been an undergraduate. There are students coming from all over the world. Do you see any particular value, or are there cultural influences that students bring from other countries that have been of value to math here?

**ASCHBACHER:** Sure. A cultural affinity for mathematics, or more generally, the intellect. Americans don't seem to have that. So our good students are usually either coming from abroad, or perhaps their parents or grandparents were immigrants. They still have the immigrant values. Also, some countries, China, in particular, identifies and funnels their students into a small number of places, so it's easier to have some confidence that if you accept a student from one of those places, either undergraduate or graduate, that they're going to be good. For example, we have to decide which ones are going to be good. In the graduate program, for example, when we came up with a short list of Chinese students we were considering, we wouldn't make the decision. Our graduate students who are from China would make the decision. Because they knew what the letters meant. I'm not sure it's true anymore, but in those days, the letter would be written by the student and maybe read by the professor, who was supposed to have written it.

**ZIERLER:** Have you come across undergraduates in your teaching career that just blew you out of the water, whose intellect was really something special to behold?

**ASCHBACHER:** I wouldn't put it in such strong terms, but almost every class in the undergraduate or graduate algebra course, there are people who are very strong. You could tell that, if they chose to do so, they would have no difficulty going on and having very successful careers as mathematicians.

**ZIERLER:** What do those attributes look like? Is it quickness?

**ASCHBACHER:** Well, the homework's always perfect, the exams are almost always perfect. For most of my time here, every summer, I've had a SURF student, often a very good student, from one of these classes. So I give them some program to work on and suggest some ways to approach a problem. Some of them do extremely well. Some don't. It's hard to tell in the ones that don't whether it's lack of smarts or because they don't work hard enough. It usually is not working hard enough.

**ZIERLER:** Recently, before you retired, you were recognized with a slate of honors. It was the Rolf Schock Prize, the Leroy Steele Prize, and the Wolf Prize all in 2011, 2012. Do you have a sense of why all of this recognition came in such a clustered way around this time?

**ASCHBACHER:** I think it might have something to do with this quasithin work. That was published early 2000s. And it takes a while for that to percolate through the system, I think.

**ZIERLER:** And what was it about your quasithin work that was so impactful in the field?

**ASCHBACHER:** Well, like I say, it filled this large gap in the proof. But I'd already proven lots of big results. Maybe not quite that big, but comparable results. But this was back in the 70s. The 70s are too far in the past to affect such awards, I think.

**ZIERLER:** When you get these awards, is there opportunity to give a public address?

**ASCHBACHER:** In the case of the Wolf Prize, it's not only an opportunity, but a demand. A number of public addresses. One of them was a short talk aimed at high school students, I forget what short meant, five minutes or something, on what you'd done, which was impossible, of course, for a mathematician. I did something, but I don't think it worked.

**ZIERLER:** And is the goal to speak at a technical level, or for a more broad audience?

**ASCHBACHER:** Well, for the high school students, what could you do? Just expose them to the problem and give them some sense of what it's about. But for a mathematics problem, it was impossible. And in the other cases, I don't remember anything addressed to a general audience. A general audience of mathematicians.

**ZIERLER:** An overall question about enrollment. In the way that you fell into mathematics, not really knowing that that's what you wanted to do when you started, what is the best means to capture the imagination of undergraduates, so many of whom, nowadays, for example, might go into computer science because that's where, perhaps, the lucrative opportunities are? How has the enrollment in math changed over the years, and what efforts have you and your colleagues made to affect those enrollment numbers?

**ASCHBACHER:** I'm not sure me and my colleagues have done anything, but the community at large has. Basically, to expose people to what I would call real mathematics. The Russians did a very good job of that back when there was still a Soviet Union. They would have what are here called math circles. Maybe that's a translation of what the Russians called them. Little groups that very good professional mathematicians would meet with once a week or once a month, so they would see real mathematics, the beauty and the rigor that was demanded. And for some people, that was so appealing. For me, for example, before I discovered mathematics, I was bored most of my life. And then, I found these problems to work on, and it's extremely enjoyable. In my experience, the better mathematicians all feel the same. So if you can somehow introduce somebody to these real mathematics, and at least some percentage of the ones that have some talent will be captured.

**ZIERLER:** You used the word beauty. Where do you see the beauty?

**ASCHBACHER:** Well, if you can prove something that's important or unexpected and do it simply, or maybe even not simply, but somehow, the argument flows and each step is simple, that's attractive.

**ZIERLER:** When you went emeritus, having shed administrative responsibilities, did that give you an opportunity to be more productive in certain ways?

**ASCHBACHER:** No, I don't think so. I just allocated my time differently. Probably allocated more or less the same amount of time to research. But the time that was spent with teaching or committee work….went somewhere else?

**ZIERLER:** Now that you're a senior scholar in your field, how do you compare the energy that comes with youth versus the wisdom that comes with age? How do you apply those as they relate to doing the math?

**ASCHBACHER:** Wisdom means having seen all sorts of different problems, so that if you encounter something that harkens back to something you know, that can be an advantage. I'm not sure I would use the word wisdom to describe that. I think the drive and the newness is something that comes from youth, and that's an advantage over the older mathematician.

**ZIERLER:** Are there mistakes that you would've made 30 or 40 years ago that you wouldn't make today as a way of coming to the idea of wisdom or acquired knowledge?

**ASCHBACHER:** Yes, that's probably true, but not for the reasons you say. The reason is that, like many mathematicians, I had a very sparse style. Left out details. It was hard to read my mathematics. It's hard for me to go back and read it sometimes. But in my experience with Steve, he was just the opposite. He was very careful and very attentive to detail. So he would write something up, and in a sense, it's more readable, but in another sense, it's not so readable. Because the proof is broken up into little packets. And if you can look at a packet and say, "I know how to do that," then you don't have to read the packet. If you break things down like that, for the right person, the person who knows how to fill in these details, it can be easier to read than the proof that has all sorts of details, many of which, for some people, would be felt to be extraneous or necessary to the argument formally. And that division comes someplace all the time. It's a matter of taste where it comes. But I had my style, Steve had his style. When we both wrote this stuff up, we had a compromise, which I think was the best of both worlds. And it taught me to appreciate detailed proofs. If you include more detail, you're less likely to make small errors.

**ZIERLER:** It can also be more comprehensible to your readers.

**ASCHBACHER:** Yes, that, too. But I worried about that less.

**ZIERLER:** But you do need your colleagues to understand what you're doing to some degree. Or you're unburdened by that?

**ASCHBACHER:** No, it's not a matter of understanding. I think I have a good perception of what the important steps are on the proof, the major things. My presentation would often lack connective tissue details. Somebody that really wants to read, they sit down, and they fill in those details, cursing me for not helping them out. So if there's some incentive to read the paper, it'll get read.

**ZIERLER:** We began our talk discussing your current work, and now that we've worked right up to the present, for the last part of our talk, I'd like to end where we began, with some general questions in retrospect. The first one, at the broadest level, what do you understand about the world of mathematics now that you didn't when you were a graduate student?

**ASCHBACHER:** Well, I didn't know much about the world of mathematics when I was a graduate student. What I knew was what I'd been exposed to in a few classes and what I'd been exposed to in the limited amount of research that I'd done working on a particular set of problems. So the difference is vast. But mathematics marches on. For example, since I retired, like I say, I used to sit on the hiring committee and had to read lot of letters. And I could learn lots of things from the letters, see what problems were thought of to be important, who was doing the important work solving those problems. The last eight years or so, I haven't had that. So my knowledge of the current mathematics is far from what it used to be.

**ZIERLER:** Just in terms of keeping up with the literature?

**ASCHBACHER:** I wouldn't say the literature because, for example, the people we hire, none of them are finite group theorists. It's not enough of an active area anymore. So I couldn't really read the papers in detail. But by reading these letters, I could understand what the important problems were and who was doing the important work on them. I understood them at a superficial level, but that was good enough for having a picture of what mathematics is like.

**ZIERLER:** As you say, if finite group theory is not as active right now, is that simply a function that you did your job, you did the work that needed to be done? Or are those political and trend kinds of considerations?

**ASCHBACHER:** Well, this classification problem was a very visible problem. Almost all mathematicians were interested in it. The sporadic groups coming along like they did made it even more interesting. So lots of people were going into this field because it had such great visibility, and it was well-funded by the United States, by NSF and other countries' research programs. And the good universities, meaning the ones with good math departments, were hiring people to do finite group theory. Now, when this problem disappears, when it's solved, it's a knife in the body of this discipline.

**ZIERLER:** But that's a funding issue, you're saying.

**ASCHBACHER:** Oh, it's not just funding, it's sociological. What students are going to go into the area, the field? If they have a choice, they'll probably choose something more current. Now, just knowing the simple groups isn't enough to do things with them. You need to know properties, most importantly, the subgroup structure of these groups and the so-called irreducible representations of these groups. And you have to come up with ways to use this information to do the various things that you want to do. So quite frequently, somebody from some other area of mathematics will reduce their problem down to a problem in finite group theory and interest some finite group theorist in the problem. What you do is try to reduce this problem down to essentially a simple group and use this knowledge about subgroup structure and representation theory to solve the problem. There's important work going on, but it's not exciting.

### Eureka Moments and Short Papers

**ZIERLER:** Have you ever had what you might call a eureka moment, where something was just not clicking, and then it did?

**ASCHBACHER:** Small moments, yeah.

**ZIERLER:** What does that look like? How do you know when you've achieved a eureka moment?

**ASCHBACHER:** Well, some idea occurs to you, and you see that this idea is exactly what you need to remove some obstruction that's been causing you problems.

**ZIERLER:** Is there a top moment like that, if you search your memory?

**ASCHBACHER:** No, I don't think so.

**ZIERLER:** Not so much drama?

**ASCHBACHER:** Most of the work I've done is quite complicated and appears in very long papers. Almost by definition, if you have sort of the eureka moment that you're talking about, it doesn't turn into the long paper, it turns into a short paper. And besides, if the problem is sufficiently inherently complex, one little idea is not necessarily going to be sufficient. Now, what could be sufficient is this idea is this idea is the basis for all sorts of stuff for some theory that you have to build up. So in retrospect, maybe, that moment was important. But maybe at the time, you don't realize it.

**ZIERLER:** Is there an effort to unify seemingly disparate theories in mathematics?

**ASCHBACHER:** Sure.

**ZIERLER:** Is that a notion like in physics that there's possibly a theory of all of mathematics?

**ASCHBACHER:** No. [laugh]

**ZIERLER:** Because that's an absurd notion, or because it's unachievable?

**ASCHBACHER:** Well, some theory built on analytic mathematics and analysis, some theory that's built on algebra, well, there are geometries you can approach from an algebraic or analytic point of view, and there would be a lot of commonality there. But by and large, the two universes that are being considered are so divergent that you can't bring them together.

**ZIERLER:** Because we're limited by our intellect, or because they simply don't belong together?

**ASCHBACHER:** They don't belong together. The fundamental problems, ideas, notions are in disjoint.

**ZIERLER:** Which makes this not very connected to physical reality.

**ASCHBACHER:** Oh, I don't think so. All the theoretical physicists are trying to do is build some mathematical model they can use to make calculations in the real world, the physical world. But underlying that is the mathematical model. And as the questions they ask become more and more subtle, it seems a necessity that the mathematical model is becoming more and more sophisticated. The mathematics it's using didn't exist 100 years ago, 50 years ago maybe.

**ZIERLER:** I asked you about eureka moments. Conversely, what about problems that, no matter what you do, you always hit a wall? Do you come up against that?

**ASCHBACHER:** I don't work on such a problem. If I don't have a way into it, then I'm probably not going to try it. Now, if I have a way into it, and it's important enough, then I've always been able to solve it.

**ZIERLER:** How do you determine at that early stage if you have a way in?

**ASCHBACHER:** Intuition. Beauty. I have some notions that seem to do things in a simple way.

**ZIERLER:** You said earlier, until you discovered this level of mathematics, much of life was boring. That would suggest, and this gets me to my last question, you'll do this for as long as you can.

**ASCHBACHER:** Right.

**ZIERLER:** It keeps things interesting for you.

**ASCHBACHER:** Sure. During COVID, I'm sitting at home. What would I do?

**ZIERLER:** Golfing, fishing, doing the crossword doesn't cut it for you?

**ASCHBACHER:** No.

**ZIERLER:** Simply because it doesn't occupy your mind in the way that you want it to?

**ASCHBACHER:** Well, it could occupy things other than my mind. For example, I used to play golf in high school, I was on the golf team. My father loved to play golf, so of necessity, I had to play golf, too. But some period of time, 10, 15 years out of high school, I just lost interest.

**ZIERLER:** Do you establish a research plan? Do you know how long you plan to work on a given project?

**ASCHBACHER:** No. Until it's completed, or I decide it's not worth working on.

**ZIERLER:** So every day, essentially, is a bit of a surprise. You don't know what it's going to look like.

**ASCHBACHER:** Well, no. Some period of time is spent developing some overall strategy. And the strategy consists a bunch of modules. You need some tactics that approach each module. After you've built up that picture, you pretty much know, unless there's something you didn't foresee, which is pretty rare, for some period of time, what you're going to be doing is more or less there to be done.

**ZIERLER:** Which gets me to my last question. And that is, for as long as you're able to do this, what do you want to accomplish? What are some of the problems you have not yet worked on that you want to work on?

**ASCHBACHER:** Well, I have this big project that I'm working on right now. Who knows if I'm going to get it done before I die? I'm going to get that done if I last long enough. After that, I'll worry about it. I must admit, it's getting a little boring. Back in the old days, when you had so many people working in finite groups, you could take something to a place where you're not done, but what needs to be done is obvious, and there's enough machinery there to do it. A thesis for a graduate student, so you could stop where the interesting stuff is done and still see your program brought to a successful conclusion. But I don't have those bodies now.

**ZIERLER:** You're saying there's a generational problem at play here.

**ASCHBACHER:** Yeah.

**ZIERLER:** There isn't the next generation that might be prepared to pick up this work?

**ASCHBACHER:** Right. That's part of it. And another part is that these small problems are probably beyond the ability of most graduate students.

**ZIERLER:** Why?

**ASCHBACHER:** They require you to learn a lot of stuff and either be sharp enough to have your own approaches or read something where you see an approach done in some slightly different context.

**ZIERLER:** Does that increase a sense of urgency on your part to leave something complete, more complete than you otherwise would? Or you don't worry about those things?

**ASCHBACHER:** Yeah, I think there's some urgency. For one thing, I'm sticking with this program, even though the parts that are left don't interest me as much as the earlier work that I was doing. Normally, I might say, "OK, I'm going to take some time off, find something else that looks like it's more fun, and come back to this every once in a while." But I don't think I have that luxury now.

**ZIERLER:** There's a lot to do.

**ASCHBACHER:** Well, not a lot relative to what I've already done.

**ZIERLER:** A lot on this particular project.

**ASCHBACHER:** Not a lot compared to what I've done in this particular project. But like I say, it may be ten years I've been working on this off and on. So it's at least a few more years.

**ZIERLER:** On that note, Michael, it's been a great pleasure spending this time with you. I'd like to thank you so much for doing this.

[End]