On Saturday, February 7, 2026, at 10 am, Hung Tran and Dallas Albritton met up with Paul Rabinowitz on the 9th floor of Van Vleck Hall to have an informal chat about the history of PDEs at UW-Madison.
[DA claps once to synchronize the microphones]
HT: So today we're going to have an informal and relaxed discussion with Professor Paul Rabinowitz. And in the audience is Dallas Albritton and me. So I'll just introduce very briefly Paul, and then we dive right into the discussions.
So Paul Rabinowitz is the Van Vleck Professor, Schauder Professor, and Vilas Research Professor of Mathematics at UW-Madison. He got his PhD at NYU in 1966 under the direction of Moser. And from '66 to '69, he was Assistant Professor at Stanford University. And then he came to UW.
PR: First I was an Instructor.
HT: Okay, sure. Yeah, as a, first, Instructor. And then he stayed at UW for 40 years, right? From 1969 to 2009. And, of course, Paul has proved many important results. If I just need to name two, then his Global Bifurcation Theorem and Mountain Pass Theorem. The first was done in part with Crandall and the second was done with Ambrosetti. And Paul is currently in the U.S. National Academy of Sciences and has received a lot of prizes and awards, including the Birkhoff Prize and the Schauder Medal. So it's great to have you here. Do you have any quick thoughts, anything you want to share first with us before we dive in?
PR: No, no, just go ahead. It's my pleasure to be here.
HT: So very quick thing first, right? So do you have anything to share quickly about the general landscape of UW mathematics during your four decades?
PR: Well, of course, it's changed a lot. The university has changed a lot. The department has changed a lot. The composition of areas has changed a lot, which is to be expected.
DA: So one thing I'm wondering about is, when you first came, did you think of yourself as a PDE… I mean, did you think, "I do PDE", or "I'm a dynamicist", or just something in between maybe?
PR: Well, when I came there really wasn't a PDE group. So the first PDE person I know of was Kennan Smith, who had been a student of L.C. Young.
HT: Okay.
PR: And I'm not quite sure about his history here. He'd been here for a while. He went elsewhere. Then he came back. And then occurred the bombing next door. And then he left again. But also before me, at least a couple of years before, was Bob Turner. So Bob was a student of Friedrichs. And so we definitely called him a PDE person. Whether there were people who worked in PDE in the earlier days, that I don't know.
HT: So, for example, you mentioned Young, you know, he's, surely – if people don’t know, he’s famous for the Young measures, right? Did you interact with him when you first came?
PR: No, he was already retired when I came. I think he... maybe he just retired, but he was retired. There was an age limit in those days. And perhaps 70. Wisconsin was a little more enlightened than many other places. And they had a higher age limit. It might have been 70. And he had to retire at that age.
DA: So when you came here, Conley was already here, right?
PR: Conley was already here, yes.
HT: Okay. And you two had a lot... I mean, he was also a student of Moser and you two had a lot of interaction.
PR: Yes. Yes. So he had come. I had met him at NYU. He was actually a student of Moser's at MIT. Moser had been there for a few years before he went to NYU. And he was a postdoc at NYU when I first arrived. So I met him there. Bob Turner was a classmate of mine at NYU. As was Wayne Dickey, who was an applied mathematician here, but doing elasticity, PDE… Applied PDE. There were applied people who were doing PDE, in that sense, here.
DA: And how were PDE and analysis viewed by the department, do you think? By your colleagues? Was it valued, or just...?
PR: Let's see… Well, Rudin was in effect the leader of the analysis group, Walter Rudin. And I would say, he was one of the leaders in the department, one of the more well-known people in the department, so I think there was respect for it.
HT: Great. So because you started mentioning about PDE, so let's dive in. So, as you said, right, I mean, PDE, when was it officially becoming a caucus at UW? I mean, not precisely the time, but, you know…
PR: When I came, there was a sizable presence in ordinary differential equations. There were five or six people. I think three or four were former students of Norman Levinson. John Nohel, Jake Levin, Fred Brauer, Howard Conner. There was Dave Russell in control theory. So there was a sizable group there. That subject has totally disappeared. In Madison. It's morphed in some sense into symplectic geometry, the subject itself. So you don't find a lot of people working in classical ODE.
HT: I mean, in a sense, our caucus, right, I mean, that is still being called differential equation caucus…
PR: It was part of… so PDE was part of the differential equations caucus. Originally. Once the caucus system evolved.
HT: So you mentioned that that direction [ODEs] has kind of totally disappeared. What are the changes you see in the sort of people doing PDE in the four decades, you know…?
PR: In the '50s, '60s, say '70s, [there was] a lot of work in linear theory. So linear elliptic and parabolic PDEs were the object of workers throughout the world. And that subject, well, you can't say it's completed, but basic theory was well developed by the '70s. And then there was more of a movement towards nonlinear. And, even that – so there was a long period of, say, nonlinear… quasi-linear elliptic PDEs, and that sort of has died down. There's more of a movement now towards PDE intersecting other areas, so applied math, math biology… pretty much anything.
DA: Yeah, I agree. I mean, in terms of the PDEs I see now, I agree with that.
PR: That's where the action is.
DA: Yeah. And we should circle back to that. So I will say that last night I reached out to Tarek Elgindi to say, "Hey, we're interviewing Professor Rabinowitz tomorrow…” Okay, anyways, I think I said we're interviewing Paul. Yeah, but he always calls you Professor Rabinowitz.
PR: And he's very respectful.
DA: He is, yes. And he said two things we should touch on are the Army Math Research Center and also this thing about the styles of PDE—how the style of doing PDE has changed in terms of building theory versus solving problems. I think we should circle back to that. I think it'd be great to talk about whether there are different eras in PDEs in Madison, one of those eras being when the Army Research Center was active.
PR: Yeah. Well, we always referred to it as the MRC. The Math Research Center. The name "Army" was suppressed.
[HT and DA laughing]
HT: Exactly. I told you.
PR: There's a little book about it, by the way. I don't know if you know of it.
HT: I read it. Yeah. 100 page. You told me, and I read it.
PR: It was written by Jag Chandra, who was the Army representative of math at that point, and Steve Robinson, who was in Comp Sci here. So the history of it, that's interesting to read. Yeah, so that strongly affected PDE because the MRC had a mandate to do research in certain areas, one of them being applied math, another being computational math, statistics, and differential equations. So because of that, there were always strong ties between some members of the department and the MRC. And since MRC regularly had people coming, especially short term—people might come for a year or two—they would often be joint with math. There were some frictions within the department as to the preference that those areas might have. So that led to some tensions.
DA: So the MRC, was it mostly permanent researchers, or was it a few permanent, and people come?
PR: It was a core of permanent people. So in the computational areas, Ben Noble was director for a while, and he was also a member of the math department. And Carl de Boor, who was partly a member of the math department, also in Comp Sci. There was Mike Crandall, whose appointment was joint between math and MRC. People like myself were not infrequently partially supported by the MRC.
DA: So, okay, whenever I – if you just do a trivial Google search about MRC, all you get is bombing stuff. And I don't want to make this focused on the bombing. But the bombing happened one year after you came, right? Because you came in '69 and it was in '70.
PR: Yeah.
DA: And then, so what was the MRC like after that? It still exited, right?
PR: It still existed and it functioned well. It moved. It moved to the WARF building.
DA: Okay. It's a little ways away.
PR: Not that far. Remember there's bus service.
DA: Okay, sure.
PR: And well, there were people who would just walk it, say, it's a 15-minute walk.
DA: But it's not as close as Sterling.
PR: It’s not as close as Sterling. So I barely knew it in its Sterling days because of the timing. And I was totally accustomed to walking back and forth.
It was a major influence to PDE because the interest in PDE led to many more PDE people visiting the university, both longer term, say, for a year, and short term. So this made us a real center in non-linear PDE. So we had very good ties with, say, the French school, and regularly had young people in PDE.
DA: Do any particular visits come to mind? You mentioned the French school. Any particular members of the French school who…?
PR: Well, Brezis came frequently, Temam… Luc Tartar spent the year here, for example. That was unusual in that younger guys would generally just come for shorter periods. He spent a whole year.
DA: Ah, yes – Luc Tartar, the younger guy.
[Laughing]
PR: Younger Luc.
HT: Also, Lions also came, right?
PR: Lions had been here – both Lions, the father and son, yeah. Mike Crandall worked with the younger Lions. The father was good friends with some people here, like John Nohel and Seymour Parter.
HT: So, yeah, I want to touch base also on this thing – Crandall and P.L. Lions, they proved the well-posedness of viscosity solutions here, right?
PR: Yes.
HT: And how about the DiPerna-Lions theory? Was it done here, or was it done afterwards? Do you remember?
PR: That was probably after. DiPerna was a notable young guy that we attracted for a while. We often had problems keeping younger people – two-body problems. So occasionally there are cases where we solve them [gestures to DA], but often solutions were not satisfactory to one or more of the partners, and that's life.
HT: And I also remember Berestycki also said that he visited the MRC, right?
PR: Yeah. Berestycki. Many of Brezis' students. So, Coron…
HT: Coron also visited, great. So it was a great attraction point.
PR: Yeah. So I would say it's probably fair to say most anybody who was somebody in PDE visited it at some point.
DA: So I was kind of curious about the mandate of the MRC. I do know now that, okay, when I apply for grants, I’m really thinking about what I’m interested in, and some of it has applications as well. But I think mathematicians like to do the math that they want to do. And if they can fit it into something that gets them a grant, then that's extra good. So I'm wondering, at the MRC, how people thought about what they were doing. I thought I read somewhere that they were supposed to spend a certain amount of time on Army stuff and some amount of time on… whatever?
PR: Yeah. So some of the people, in particular the permanent staff there, were occasionally asked to consult at Army institutions. It was not so much the case in PDE, as far as I know. Mike Crandall maybe once or twice may have gone, but I don't think he went on such trips very often—rarely. It might have been much more so in something like statistics or Comp Sci. You could ask Carl de Boor about that sort of thing.
HT: So in terms of the… you came in '69, right, so, in the '70s, '80s, there were a lot of activities, but in terms of core members of the PDE group, let’s say, there were you, Crandall, DiPerna a bit… I checked, there was also, Rennardy was here.
PR: Turner.
HT: Turner was here, and…
PR: Yeah. Michael Rennardy… He had been a student of Klaus Kirchgässner, who had visited MRC before I came, actually. Rennardy was another case in point where a two-body problem interfered. He met his future wife at MRC.
HT: So how would you characterize the PDE group during the '70s and '80s? Because clearly it's different from Minnesota school, right?
PR: Well, it was not a big group, but it was supplemented by visitors to MRC. So I think it was a strong group. Minnesota always had a much larger group than we did.
There was a time before I came when there were even some regular meetings between the two departments. I'm not sure if it was the whole department or subsets thereof, but they would meet maybe somewhere in between…
DA: Oh, interesting.
PR: … once a year So I know it existed, but it had basically disappeared by the time I began.
DA: Related to those meetings, I think at some point we should talk about the Midwest PDE Seminar. But first I want to go back to Hung's questioning about who was around in the '70s and '80s, and maybe we should just talk about Mike Crandall a little bit.
HT: Yeah. Can you tell us a bit about Mike Crandall? Okay, he's my academic grandfather, so, of course, you know…
PR: Well, Mike is, was a very dynamic person. Very good expositor. He attracted some really first-rate students. And grand-students. [gestures to Hung]
[Laughing]
PR: And had broad interests. We worked together for several years. And he had strong opinions about things, which often didn't make him many friends. [Laughter] So, he was a crucial member of the department.
DA: But had you met before coming here? You'd met in California, or…?
PR: Yes. Mike and I were both junior faculty at Stanford. We did… we were certainly good friends there. I think we did some work together there. And there also was Amnon Pazy, who spent... made a couple of longer visits here. I think he was maybe a Van Vleck or something… He spent a longer period here. Various times. Yeah. So he [Crandall] was certainly the leader of the group in some ways.
HT: So, for example, maybe because of your style, right? You are working a lot in nonlinear analysis, PDE, and you work also a lot in the intersection between PDE and dynamical systems, right? I mean, what was the motivation for you to do that? And also – because Dallas and I also chat a bit – Minnesota school is really into regularity theory, right? So, Wisconsin…?
PR: Well, regularity theory was a major subject of research in those years, in, say, the 60s, 70s… And Minnesota probably was the U.S. center in that subject, so you had several people working on that. Madison didn’t… it’s a very technical subject, and Madison never had any... well, for a little while Emmanuele DiBenedetto was here.
DA: Oh, I didn't know that.
PR: Yeah. Another two-body problem.
HT: Oh, okay. [Laughs] How about any interaction of us with Chicago, UIUC, Purdue... I mean, like other places, you know... Michigan?
PR: Well, there were certainly contacts between places. I mean, in the end, the Midwest PDE Seminar evolved, which involved multiple places.
DA: But do you remember who had the idea for that? Who made it happen?
PR: There were several people involved. I don't remember the precise dynamics. But in Chicago, there was Felix Browder. In Northwestern, there was Avner Friedman. In Minnesota, multiple people. Here, myself, Crandall and Turner. And the original proposal was to alternate meetings between various places. And in the old days, Madison was a regular such visiting point. Minnesota rarely, because that was farther from the center of gravity. But it continues to this day.
DA: Yeah, so I read that the first one was 1977, at Northwestern, and the second one was here. And on the website, it says... There's a thanks to Paul Rabinowitz. And… because they have scans of the programs. You must have sent, maybe you sent that...
PR: Well, Jeff Lewis decided at some point to try to collect the programs, and they had a collection of some of them, and there were certain missing years, and I was able to provide some of the missing years.
DA: I see. … which is great. I just saw this last night. “Oh, wow, they have the programs…”
HT: I need to break news to you that it was discontinued.
PR: It was discontinued…
HT: … just a few years ago, I think because of NSF funding, but it’s something I want to revive.
DA: Yeah, we should.
DA: So I saw – I looked at the program for the first one – I saw you spoke.
PR: Yes.
DA: And with Jim Serrin, Hans Weinberger, I think Haim Brezis as well, and maybe Avner Friedman.
PR: Well, the strategy was always... most of the speakers would come from the Midwest, and in the early years it was senior people, and then there was more the younger people. But there was always an attempt to get some people from outside of the Midwest. So Lars Hörmander spoke, Louis Nirenberg, Jürgen Moser, etc., etc.
DA: I was wondering whether with sort of the waning of the MRC, whether the Midwest PDE Seminar and maybe also the IMA at Minnesota, were kind of evolved to fill in a little bit of the gap. Maybe it was not really related, but…
PR: Maybe in part. I think the NSF felt pressure to distribute these institutes geographically.
HT: Yeah, that was the goal, I mean, we did apply earlier for such an institute, but we didn't get it. Did we have a big hit after the MRC was closed down in terms of activities for PDEs and all these sort of lively discussions and everything, in your opinion?
PR: Well, it somewhat lessened the amount of visitors we might have in certain areas just because of the funds available for that purpose. There was an interim period when there was a milder version of the MRC; it was the Center for the Mathematical Sciences.
DA and HT: Mhm.
PR: Marshall Slemrod was the last head.
DA: I see.
PR: And they had a presence on campus near the Union. So it lasted for a while and then sort of disappeared. There had been changes in the department too, in that the earlier leadership in applied math retired, whatever, and then there was… newer people with different interests, broader interests. And that changed [the] applied math group after that, substantially.
HT: So I'm going to ask you a personal question before we move on. You have been here for your entire career, right? Have you ever at some point considered moving?
PR: Yes. There was a point when Crandall and I were approached by Duke, for example. We considered it. Yeah. And then Mike did leave – he went to Santa Barbara. There were a couple of occasions when I thought about it, but I was comfortable here, and I remained.
HT: At least Craig Evans told me that Crandall is always a little bit of a person who believes in spiritual… in the spirit, that's why he wanted to get back to California. I mean, that’s what Craig Evans told me. I'm not sure whether that's, you know…
PR: Well, he had come here from California. He had mainly grown up in California. His wife was also from California. So when he had the opportunity, he opted to go back.
HT: I see. Okay.
HT: And, you know, after the 70s, 80s, there is a new flow of also amazing faculty in the PDE group, like Angenent, Sigurd Angenent, Takis Souganidis, and then Misha Feldman, Sergey Bolotin. Did you see the dynamic change from the 70s, 80s to then, you know, the 90s, 2000s?
PR: Well, there are different people, different interests. After Mike left, there was no… the leadership was not as dynamic as it had been, and we just evolved somehow. We had good fortune in, say, getting Sigurd and Misha, et cetera, et cetera.
HT: Takis also has a strong voice, right?
PR: Well, Takis, yeah, Takis made a difference, but another two-body situation.
HT: Yeah. So it's clear that the PDE caucus has never been big, right? I mean, I think that at this moment we have six. Has it ever been bigger than six during your...
PR: No, I think it's maybe as big as it's ever been.
HT: Okay.
DA: Yeah. Uh-oh, shouldn't say it too loud. Have to hire more PDE!
[Laughing]
HT: Oh, sure, I would love to do that, yeah.
PR: But many fields which historically were big have diminished or disappeared here. So, I don't know what the current situation is, but a traditional area was logic. That was usually... We would have the biggest logic group around, us. And I suspect that's probably much smaller now.
HT: Yeah, we only have three now.
DA: But one thing that... Of course, we can count people in the PDE group, but then there's, at least here now… What's nice is that there are people who really belong to two groups like Hao Shen, you know, is in the probability group, but he does stochastic PDEs. And of course, Sergey Denisov does analysis, he works on PDE sometimes…
HT: Laurel Ohm.
DA: Yeah, Laurel Ohm, my wife, is in applied math, but proves PDE theorems. And I like that we have... okay, then I also think about some applied math, and Cole Graham, our newest PDE hire, who cited you in his job talk, he also does probability. I think it's great that we have this cross...
PR: Yeah, that’s wonderful. There has been pressure over the years to have more of such interactions, both within the subject and also with external areas too. It's a good thing. It's a good reason to have to hire more mathematicians.
HT: Because also, like, Slemrod and Tzavaras, right, I mean, they were in applied math group, right, but they also did a lot of PDEs.
PR: Yeah, well, we used to have, at one point we had a joint applied math-PDE seminar. So, Sergey Bolotin was by training a dynamical systems guy, but he had strong interests in applied math.
HT: I was told that we have the new... well, the PDE and Geometric Analysis Seminar was created by Sigurd and Xiuxiong Chen. But earlier our seminar was joint with the applied math seminar.
PR: Yeah. Well, Sigurd was a good case in point. He's at the intersection of PDE and geometrical analysis and many other things, too. But people change, the fields change, et cetera.
DA: Sigurd told me that... I can't remember if it was you or Steve Wainger, but, up until maybe 10 years ago, someone was calling him the "new guy."
HT: Steve Wainger.
[Laughing]
DA: It might have been Steve.
HT: Yeah. I think I'm running into Steve, and he's still calling me a new guy now, so…
PR: It sounds like Steve.
[Laughing]
HT: So, you know, like, you have talked about fields changing and have sort of evolved into certain things. I mean, in a sort of broader perspective, do you think that as us mathematicians working in the field, we are driving it or, you know, like outer forces driving it, or…
PR: A mixture, a mixture. Yeah. It depends on all sorts of things. Successes, serendipity, external forces.
DA: When we were leading up to this, I was reading an interview you did with Tai-Ping Liu, okay, and someone else whose name I forget (sorry). [It was Chao-Nien Chen.]
PR: That was Tai-Ping Liu.
HT: There’s one more.
DA: Yeah. Anyways, in Taiwan. And in that interview, you talked a little bit about a kind of symbiotic relationship between applications and developing theory—that, okay, you have some application you want to solve, and then you develop a theory which encompasses some class of problems, and this leads you to discover new problems. I was wondering if you could comment on that, or if you have any examples that you can think of in your own work.
PR: Well, that's at least in part the way I function myself. So you may come across a particular problem. So in my case, say, there were bifurcation problems. And you manage to solve it, and then that solution may get you to think about the theory a little bit, and that may lead you to work on that, and that may lead to, say, the global bifurcation theorem. Or also with the Mountain Pass Theorem. Ambrosetti and I were independently working on various kinds of non-linear eigenvalue problems. I went on leave and spent three months in Pisa, and by chance I shared an office with him, and the rest is history, in some sense. We even... originally there was a little bit of a language problem, and our common language was German. So it started out in German.
DA: Paul, what's your second best language, after English? I'm assuming English is your best.
PR: English is my only good language.
[Laughing]
I guess in terms of my command of it, I know some German, which I almost never use, and I know some Danish. Those would be my second and third.
DA: Okay. So despite the French visitors here, not as much French?
PR: French is funny. When I was a graduate student at NYU, you had to know, you had to show proficiency in two languages to get a PhD. And my first... I studied German as an undergrad there. And I needed a second, and so I started to take a course in French. And I found the teacher so pedantic that I couldn't take it. And I studied myself. And I failed.
[Laughing]
But then I tried again, and I studied more seriously. I only had studied two weeks the first time.
DA: I see.
PR: That wasn’t enough. But the second time I studied more industriously, and I passed. But I never learned French, therefore. I mean, I knew some vocabulary, but I didn't learn how to speak it.
DA: Just quickly about that. So was it a little bit of a shock to you to fail? I mean, I think all of us... I've definitely had instances where I didn't prepare enough, and I thought… and it really stuck with me. I remember once I gave a really bad practice talk, to my advisor, and…
PR: Well, I use that as motivation to really work harder the next time around.
HT: Yeah, I did fail many times. And, you know.
[Laughing]
DA: It's good to do some failing.
PR: Especially early on. Then you take it seriously.
HT: I mean, because we were talking about sort of how you found problems and what your thoughts about math, and it just recall in my mind that there was an article in the AMS talking about sort of "birds and frogs" in the culture of doing research in mathematics, where birds are sort of flying and seeing the landscape and creating theories, and frogs are sort of problem solvers… Do you agree with such sort of assessment?
PR: Well, there certainly are people of each type and people of both types. So I guess "flying frogs".
[Laughter]
HT: I like that. I like that. This is great, I like that.
DA: But, Hung, do you see yourself in… Homogenization actually has a sort of theory, and you’re established in that…
HT: I mean, sometimes we try to see the landscape, and trying to see a little bit, right? But at the end of the day, like what Paul said, we need to sit down and solve problems. To go further, I don’t know, because we just try to see and solve something. But you know for Paul, he has a better assessment than, I mean, maybe 20, 30 years from now. [???]
PR: Yeah. You should do what's fun. If it's not fun, it'll be painful.
HT: Yeah. I agree.
DA: So you wanted to ask about... I remember you mentioned you want to ask about students.
HT: Yes. That's important.
DA: Do you want to do that now or come back to it?
[HT gestures at DA]
DA: I said you wanted to ask about it!
HT: Yes. So I think that grad students... that's an important part of UW-Madison. Of course, we are not talking about the general grad students at UW-Madison math, but can you tell us about the grad students in PDEs during your four decades and certain observations that you have seen?
PR: Well, so to put things in perspective, you have countries like France or Italy, where you have nationwide universities, national… Grandes Écoles, and where you have wonderful students. In Madison, it's much more mixed. You get some students who are very good, and many more students are pretty good. And so we occasionally have major successes, but we get lots of solid people. And so one has to be happy when you get particularly good students. And we've had several. Mike Crandall has had a couple of very good students. Charlie Conley had some excellent students. Some of your teachers in Minnesota probably. [PR gestures to DA]
DA: Those are the ones I'm familiar with. Dick McGehee and Rick Moeckel, who I never had for anything.
HT: How about your students?
PR: So my students... I think my most successful students have been from other countries. So you may know Yiming Long. Became a member of the Chinese Academy of Sciences, which is much harder than becoming a member of the U.S. National Academy.
HT: Don’t think so, but yeah.
PR: Patricio Felmer. They've probably been my most successful students, but there are others who have done very well. Some students... Well, in the U.S. there are good periods for students and bad periods. I'm not sure how it is these days, but a couple of my students went out in the world in the bad periods, and they opted to go into other areas.
DA: I see. So when you're talking about bad periods, do you mean sort of the broader academic job market? Or do you mean bad in terms of, I don’t know, something else?
PR: Well, the job market. Opportunities. So I had one very good student who had a junior position in Minnesota. And he didn't like it. He felt they didn't take teaching seriously enough. He liked to teach. He left there and went to a small college on the West Coast. And then he discovered that he enjoyed it, but it didn't pay so well. And he had two kids and he became an actuary. And so it happens. Not everybody has to be a researcher.
HT: Right. Because later in life, there are a lot of constraints, like what you say, right? With families and lots of other reasonings coming into the game. So you're saying that all these oscillations will really affect all the students, right? I mean, we are also seeing it now, this year, for example, the postdoc market seems very hard. Tenure-track market…
PR: How many applications do we get this year?
HT: I mean, we have MathJobs. So every year on the order of magnitude, we have more than 1,000 applications for both tenure-track and postdoc positions..
DA: And for PhD, we had, what, 900 last year? This year is similar…
HT: This year is 825. And it's double compared to four or five years ago. And we have less number of positions. So it's fierce on all front.
DA: But part of it is also that the application systems are more centralized, so it's easier to hit apply. But yeah, definitely lots and lots of applicants.
Related to students, one thing I've noticed as a relatively new professor is that I can't tell... actually, I can't tell how people are going to do. And also some people in my graduate student cohort... the people at the beginning who we thought, “They're the strongest,” maybe in the end decided to do something else or didn't enjoy the research or the problem didn't work out... Some people get really excited once they start researching. Is that something that you've noticed with your own… in your time here?
PR: Well, just looking broadly, there are people who are really smart and they're interested in everything. And it takes a while before they really focus on a single subject, say, and then they're very successful. And there are lots of people who are smart and do okay, but as you indicated, they maybe never fulfill the promise that you see in them. So it's a mixture. Sometimes it's a matter of luck, too. You just hit on something that's right for you.
DA: Do you feel you were lucky in some of your own… in your early progress?
PR: Serendipity never hurts.
[Laughter]
Well, I was together with Mike Crandall at the same time. And that was very helpful.
DA: And sharing an office with Ambrosetti.
PR: Yeah.
HT: Yeah, so, there are a lot of things have changed, right? Because in the past, you need to be in the same room or kind of meet people. And gradually now, you know, I think Dallas and I have Zoom meetings every week with people all over the world. There's still a sense that if we were in the same room together, it would be better, because on Zoom, it's very hard to explain your intuition and everything. But I see that there are a lot of changes, but I also cannot predict what's going to happen.
PR: Well, also the use of the computer and now AI. It's early days for AI.
DA: Right. Right. Is that something on your mind? AI and the kind of ChatGPT, sort of…?
PR: Yeah. Interesting to see how that develops in math. It's a… Little bit skeptical, but you got this resource next to you that has an enormous memory. So it can… Well, in a sense, it restricts you to what's known. So it can connect two points, but maybe it's not a good two points.
DA: Sure. Yeah. Related to this about the development of technology and how we do math, I have a kind of theory, but not really well thought out, that, okay, the way that we record the math, either by handwriting or typewriter or directly into LaTeX, affects how we choose to write and maybe even a little bit the math that we do in that, when you're handwriting an article, you may have to choose... you can't just write so many words. You have to choose your words carefully. And when you're typing, you can just copy, go very quickly, copy and paste.
PR: Well, that's funny. I had just the opposite feeling about writing or typing. I mean, I started out in the day when you just did things on paper, and I wouldn't mind writing a draft and then five minutes later starting a new draft or whatever. Whereas when I'm doing something on the computer, I don't like to have to do it multiple times.
DA: I see. Is it because it's sort of a lot of work to write every integral?
PR: So I'm not proficient in typing. That’s probably part of it. I still like to write things.
HT: Me too. But I do see the changes in the new generations because there are students, or also I have seen that also from our postdocs and others, that they really like to write on iPad, or, right... When teaching, they're also not writing on the board, writing on the iPad and projecting. So there are big changes. I think that because the way we think, the way we do is wired into… like what you said. But I also prefer writing on the paper and checking things first.
PR: Also, giving talks – I’ve almost never give[n] a non-chalk talk.
HT: Right. I did all the chalk talks earlier. Then after the pandemic, kind of, like, visiting places, and sometimes that they are not prepared for me to do the chalk talks.
DA: I've noticed that too. I went to give some mini course at a, you know, a school for math. And they didn't have a board, so I thought was…
[Laughing]
HT: So things are changing quite fast, you know. So I want to ask you a little bit – this is, kind of, Wisconsin – but a little bit about all this, sort of... because the teaching load and everything earlier was 2-2, right? And then all this committee work... you know, before the age of computer, the age of internet. And after this... Because, you know… Do you see, you know, teaching 2:2 was hard, right? And, I don't know what, I mean, doing committee work, was it a lot of commitments in all of these combinations?
PR: These things are all relative. Go back earlier. When Fritz John came to the United States, I think he went to Kentucky. His teaching load was 5 courses a semester.
DA: Wow.
PR: So, and then he had some promising student or students, and he gave some extra lectures to that student. So, we're spoiled. On the other hand, there weren't the same sort of distractions. And administration will always fill up all of the available spare time.
HT: Yes. That's what I feel, because with all this new technology, emails and everything, the amount of paperwork and other things are piling up.
PR: I would always notice the difference between summer and then the semester starts and you’re a week into the semester and your use of time has totally changed.
DA: So, since we're on the subject of teaching, I remember I was asking Sigurd, have we always had – you know, I'm teaching a 290 person class right now – and I asked Sigurd, “Have we always had such large classes?” And he said, ever since he can remember. But what about ever since you can remember – have we always had large classes?
PR: There… I think so. I mean, if you look at the size of the largest classrooms, it tells you what went on. The economics has changed things a lot. And Wisconsin was always slower to change things than many other places. So now there's some classes that are even bigger than 300…
DA: That’s right.
PR: … and they have to move outside of Van Vleck, so… Now the current teaching load is less than 2-2, isn’t it?
HT, DA: Yeah. 1-1+epsilon.
PR: So it was not that the case when I taught.
HT: I mean, as a whole – as you said you have seen that the department is evolving, it's changing a lot, right? And, do you think, you know, as a member of the National Academy of Science, right, what's the importance of mathematics overall in this new age of things?
PR: Well, there's been a consistent uptick in the amount of math required by other departments and that in recent years has driven the growth of the math department. I don't remember who it was – it was Biddy Martin or after her – who among the chancellors who decided that the size of the department should more closely reflect the number of students they taught. So for many years, well, back in the 60s, there was… enormous numbers of students, so math, I think, had order of magnitude 300, 350. [FACULTY?] The government was supporting things strongly in those days. The economy was good. And math reached a local max in size. And maybe global max.
[Laughter]
And then it started to shrink. I remember there was even a tough dean named Kleene who told math that they're going to have to shrink. And… tricky situation for the administration. When you're in a period of financial constraints, and you have two departments who have people retiring... one is a department which is pretty sizable, and another is a small department of five people. So are you going to let 20% of that department disappear, without replacement? We'll take it from the big guys; they're not going to notice the difference. But after you do that ten, how many times? It's felt. And when, I think it was Shi Jin who was chair, maybe, a point was reached in which the department couldn't really function anymore under those… And then the curve turned. There were chancellors who were more willing to change the rules of the game. And now it's more of an upswing or at worst a leveling.
HT: I think it’s leveling.
PR: So the future looks good for math in the sense there's going to be a demand by students to take the courses. And math modeling or whatever is used in more and more areas.
DA: Yeah, so I don't have too much more to add. I do, actually, since you mentioned chair, and you're obviously very attuned to these kind of issues and global picture of these fluctuations and things and the needs of the department… But you were never chair, right?
HT: [Laughing] He’s smiling!
PR: Yes. Some people managed to avoid it. I know my limitations. I even had a generous friend who had been chair. who offered to be the associate chair if I ran. But I decided I didn't want that.
HT: But I assume you had some other tasks, like, were you ever director of graduate studies or things of that nature?
PR: No, I certainly did my share of committee stuff. I was always willing to do that.
HT: Maybe, you know, because we are doing sort of, like, gentle and random chats, right? So I think that in my mind maybe there's one final point I want to ask you. This is a deep point I want to ask you about. You have been sort of publishing and doing a lot of important works and also you have been in the editorial boards of many important journals, right?
PR: Yeah, I've done a fair amount of it. Almost none now, but still a little bit.
HT: So how did that job evolve, and did you see some changes in those paths?
PR: Well, the editorial end, I think the biggest changes have been technological. So it used to be if you were on an editorial board, okay, there was some mechanism by which papers were sent to you and you handled it. And then when it was finished, you sent the information to the managing editor or whatever. Now it's the use of these, quote, "editorial managers" by the journals, which I do my best to avoid as much as possible.
HT: So was it centralized or decentralized in the earlier time?
PR: It was more decentralized, I would say, since, that, you ran your own show.
HT: Which one is better in your opinion?
PR: I don't know. I guess it depends. Maybe it depends on how you grew up. If you're used to something initially, you get comfortable with it. Certainly the most freedom was if you run your own show. But maybe people like it if they just get the papers sent to them and they do something…
DA: I assume that you're seeing just a huge increase in the amount of papers that one must take care of.
PR: Yeah, there are more mathematicians, especially… There are more countries that, say, 20 years ago or whatever would send their students to the U.S. to be trained. Now they're doing their own training. They have many more departments. They have many more people doing research. So the numbers have grown immensely.
DA: But I suppose also given the influx of papers... I mean, I think in math now we have... There's more specialization because so much math, even PDEs, exists. And I assume in the old days when you were an editor, you would need to be attuned to a much broader spectrum of... You might receive papers that aren't so closely in your area.
PR: Yeah, well then you might just say this should go to X or Y. But when fields grow more mature… I remember being on a committee, an AMS committee that was deciding on topics, maybe the CBMS lectures, and there were collection proposals. And I remember one of the… Of course, there were proposals coming in from areas that only a couple of members of the committee may know something about. I remember one member making some very critical remarks about the kind of proposal, and he had been looking at it, and he said that this field is a very mature field and the kind of questions people are looking at now are a little bizarre. That’s… He couldn't see supporting that kind of thing. So fields get mature, and then maybe the questions become more artificial, less interesting. So, that… By the same token, that's one reason why PDE keeps expanding. There's more and more good sources of problems that are ripe now.
HT: That's great. So, I mean, just to follow up, I think that there's one more question on this is that, as you said, right, PDE is broadening and, like Dallas said, things are getting more and more technical and more specialized. Papers are getting longer.
DA: Yes, much longer.
HT: So, I mean…
PR: That means that you’ve got to expect more time from referees, etc., etc.
HT: So, do you see that as a plus or what? Because I mean, it's great, right? But there's always sometimes that I have a little bit worried that things are just getting too big.
PR: Yeah, well, again, problems can be attacked now which are more technical and take more pages. So, if you want something shorter, maybe you’ve got to look in a new area, where they haven't done the simpler things yet.
HT: It's just a natural course of things.
PR: Yes, it’s a natural course.
HT: Okay.
DA: But, so related to areas maturing, and coming back to Tarek [Elgindi], so, would you say that what you might call nonlinear analysis is fairly mature?
PR: The actual analysis itself? Well, in some sense, yes, it's fairly mature. But there are different sides of nonlinear analysis. It used to be called nonlinear functional analysis. And then the term broadened to nonlinear analysis. So there are computational sides, there are combinatorial sides, there are algebraic sides, there are analytical sides. One area where there's maybe much more room is topological. Probably. I mean, there's more computational topology these days. So, it's pretty mature, but there's certainly room for more.
DA: Related to the topological aspect. One thing I've noticed, just among my peers, people in my sub-area, is that these kind of topological methods—something which can give you a global theorem... Okay, maybe it doesn't tell you how many solutions there are, but at least it tells you there's a branch going forever. They’re not… I don't feel that they’re as well known now. I know a little bit of it because there's a lot of study of the structure of steady states in Navier-Stokes and things like that. I remember talking to Tarek and he's saying, I have these old notes from Professor Rabinowitz, and we should really teach a topics… we should ready study and internalize them, otherwise that kind of area, which is actually well-developed, gets lost a little bit. Do you… have you noticed that those kind of topological ideas are not as in vogue now or not as well known?
PR: I guess I don't know what exactly is taught these days. When it comes to problems in the analysis end of nonlinear analysis, you have… there’s the contraction mapping theorem, or the implicit function theorem is its smooth cousin. And that’s how you… Those are the tools for local problems. There are special situations with maybe special methods. And if you start looking for global methods, there's degree theory, and there's variational methods, some other special tools. There's not a lot. So, you do see them used. I don't know. It would be good to ask somebody like Henri Berestycki, who's more active, and in working in biological directions in particular.
HT: But I agree. There's still a lot of room for global results.
DA: Well, I've never actually used the full degree theory in any of my papers. I've sometimes used the... I forget the name. It was a kind of a softer version. Maybe it's called Schaefer fixed point or something.
PR: Yeah, there are variants of the Schauder fixed point theorem. Well, it's good to know a little bit about that stuff. And from the point of view of analysis, it's always going to be crucial to get those a priori bounds.
HT: [Laughs] Absolutely, a priori estimates. Yeah, so I mean, so maybe a final point I want to ask you now is that if there is a young student – I know that there are students who ask you this – but if they come and ask you if they want to be major in mathematics for undergrad, or if they just started grad school, and, they want to… Because lots of changes now, right? So if they want to look at the total landscape and go forward, what would be your advice to them?
PR: What particular areas in math to work on?
HT: Yeah, or let's say, let's start with undergrad first, right? I mean, if they come and they say, hey, I love math, I want to be a math major…
PR: Get as broad a background as you can. In science, there is a lot of action in the biological directions. So, I don't know. It used to be the case that one out of every four students who came to the university wanted to work in the biological sciences. I don't know if it's still the case.
DA: It's still big.
HT: It seems so, yeah.
PR: But AI is a big area now. That's where huge sums are being thrown at it, much of which will be wasted.
DA: Did you go in the new building? The new computer science building yet?
PR: No.
HT: You can come and have a look. It's interesting. How about, let's say, okay, let’s maybe be more specific about… Let’s say a student coming to UW and want to do PDE. What would be your advice to them?
PR: That’s fine. Learn tools where you're more likely to see growth, like stochastics, probability. Those are probably good areas to think about. Look at interesting biological applications. Yeah. There's plenty of room for stuff.
DA: Well, maybe do you want to save the rest for another time?
HT: Yeah. I think that today's great. We have been touching on various important points, and I think that we should save the rest for the next time.
PR: Well, you should maybe try to catch by telephone Bob Turner. Because he's been here longer than me, and he's older than me. Or in the Comp Sci direction, Carl de Boor. Or Carl's student, former student... what's his name? Has work with him…
HT: I don't know. But at least when I was a grad student at Berkeley, I was in a course called Numerical Functional Analysis taught by Olga Holtz and she used Carl de Boor’s lecture notes. That was cool.
PR: She was a student here.
HT: Yeah, she was a student here.
PR: She took PDE, too, I think.
HT: Oh, she took PDE, that's great. Yeah. So, I really enjoyed her class when I first came. So, thank you for being here.
PR: Thank you.
HT: So, do we clap?
DA: No.
[Laughing]
PR: You should talk to Sigurd!