Second interview with Richard Brualdi (February 5, 2007): Tapes 3-6

Richard Brualdi (#836) Transcript RL: This is tape number four, side one, of the ongoing interview with Professor of mathematics Richard Brualdi. February 5, 2007. Richard, in this three-year period leading up to you receiving tenure on the start of your career here, what were the essential components of your research program at this time? Even as you were starting your teaching career. RB: My research, basically, was along the lines of my PhD thesis at Syracuse University. It concerned, by and large, what I call combinatorial properties of matrices. In particular, there was this important combinatorial counting function which is called the permanent. And my research dealt with inequalities connected with this function, some relationships between math and sort of classical linear algebra in variants. And I mentioned one of the, this collaboration that I had with my two colleagues, Hans Schneider Seymour Parter was, dealt with this, but was a bit more analytic than I was used to. Dealt with what was 00:01:00 called the diagonal equivalence of a non-negative matrix to a doubly stochastic matrix. Let me explain what that is. So a non-negative square matrix simply means that the entries are not negative numbers. And you want to re-scale the matrix by multiplying all the elements in the rows by a certain positive number, and all the elements in the columns by another certain positive number. And ending up with a matrix where all the sums, the sum of the elements in each of the rows and the sum of the elements in each of the columns is equal to one. It's called a doubly stochastic matrix. And this has important implications for probability. It had important implications for some statistical analysis which I won't go into there, into that. I also wrote, I think I mentioned this before, a paper with a colleague, a very distinguished colleague, who 00:02:00 was visiting here for a couple of years. His name was Helmut Wielandt. He was a wonderful mathematician who worked both in matrix theory, which is one of my areas, and in group theory, which was not. And I had a conjecture concerning these stochastic matrices, which I mentioned to him. And together we solved this, we showed the conjecture was true. And I was extremely pleased to have a joint paper with such a bigshot in mathematics at the time. So that's, by and large what it was. Except there was one other topic which I became interested in at the time. And this was, which is related, but in a slightly different way to my other research, to what were called transversals, or systems of distinct representatives of families of sets. And matroids. And this sort of was a topic that even though it went back to the 1930s, all of a sudden became fashionable among combinatorialists. There was a lot of research going on in the 00:03:00 '60s, and this area was a popular area. I sort of got into this area and did some research in this topic for a few years, I don't remember exactly, half a dozen years or more. Eventually, as I talk about my research later, I came back to this. And it's been a theme of my research for many, many years. And it continues to be a theme. So one of the things that sort of surprised me when I first came here to Madison was that the Algebra Group, this is sort of the area, my area of mathematics, fell within the Algebra Group. And there may have been ten or twelve people in this group. Maybe a little bit more, I can't remember. But the Algebra Group, as a group now, had a National Science Foundation research grant. Those were the early days of NSF, I guess I would say, where there was less restrictions, less competition for grants. And much to my surprise, after I had been hired in the spring of '65, and I guess the 00:04:00 Algebra Group had put in a new research proposal, they included my name on the proposal. So much to my surprise, that first summer, I had National Science Foundation support. That, of course, would never happen nowadays. [laughs] It's so much more difficult. The competition is stiffer and the expectations in a proposal are a lot stiffer. I mean, I wrote absolutely nothing. Whatever they said, probably what was said, that I was coming to the University of Wisconsin and that I worked in the following area, and that I was a good addition to the department. And that was probably it. Okay? So they had this big NSF grant which continued for a number of years. I think it may have been renewed once more when I was still on it. But I think later the National Science Foundation decided that we really shouldn't be submitting such a proposal for such a large group. That the groups ought to split up into smaller groups. And so, in fact, my colleague, Hans Schneider and I, split up into a group. I think it was just the two 00:05:00 of us, although I have vague recollections of somebody else. But whom, I don't know. But it was by and large, in any case, Hans and me. We submitted a proposal and it was funded and we had NSF funding for quite a number of years. I'm trying to remember exactly the year that it stopped. But it certainly went from 1966 to 1996 or '97, something of that sort. So for thirty years, we had support from the National Science Foundation for our research. And that, of course, was extremely helpful. It provided summer support. It provided travel support. It provided support for graduate students. In those days, there was also support for supplies within the department. I don't know to what extent that continues anymore these days. But they're a little more stingy with the kind of money that NSF gives, because there's a lot more demands on the money that they have available. RL: And this, and what work was this supporting? RB: It was supporting whatever research we were doing. Combinatorics, matrix 00:06:00 theory. RL: Sort of open ended? RB: Well, no. When you write a proposal, you sort of highlight what you've done in the recent past, and what you hope to do in the near future. So that would include indicating specific problems that you wanted to work on. Often it would be the case that you would say that you had some partial results in this area and you wanted to continue the research in this particular topic. And that's the way it is today. You review what you've done. You look forward to what you hope to do. And these things are reviewed by people at NSF. And they send them out for peer review within the community of the United States, etcetera. RL: Were these three-year grants? Or five-year? RB: They were general three-year grants. There was a time when NSF said they'd consider five-year grants. And I think maybe Hans and I did have a 00:07:00 five-year grant at one time, although I don't remember exactly. So if you wanted to get continuing money from NSF, you had to continue to do research and publish good papers in quality journals. And that was looked kindly on by your colleagues also working in this area because they were the ones that were going to be reviewing your, some of them were going to be reviewing your proposal there. I did mention that I got interested in this topic of systems of distinct representatives, transversal theory and matroid theory. And there were two people at the University of Sheffield, England, who were prominent in this area. They were Leon Mirsky and Hazel Perfect. And it was because of that that I applied for a NATO post-doctoral fellowship to go to the University of Sheffield for a year. And I did get that fellowship. And I did go to Sheffield for the year '69-'70. Summer of '69 to summer of '70. But maybe I should go back a little bit, because the years of the late '60s, of course, were the years where there was a lot of upheaval in the United States concerning the Vietnam War. And that upheaval certainly was at the University of Wisconsin here at Madison. When I was living in university houses at the time, in the '60s, first years here at Madison, there were also a lot of young, radical English professors [laughs] living in university 00:08:00 houses. One of whom is still my friend today, and is still here in Madison. The others, I think, never got tenure at the University of Wisconsin. So I was involved, to some extent, in what was going on on campus, and what was going on among the faculty here at the university. I remember going evenings to certain marches. I remember, as a group of faculty, marching within a demonstration. I remember one instance of going down to City Hall where the city council was meeting. I think Bill Dyke was mayor at the time. And they were considering some proposals concerning what was going on on the streets of Madison, etcetera. So we marched down to the city/county building. And I think, I remember Bill Dyke coming out, meeting us, and saying that he would allow one of us to come into the meeting--okay, I guess it was crowded--to speak to the group. So I remember that time was sort of my first taste of tear gas as well. Okay? Which was, of course, rather common at the time with the demonstrations here in Madison. And I remember the big demonstration just outside of Van Vleck Hall there between Bascom Hall 00:09:00 and what was then Commerce Building, now called Ingraham Building. That was the first instance where the police went into a sit-down demonstration there and forcibly removed the people from that demonstration. I wasn't in the building, but I certainly saw what was going on in the building. I remember demonstrations within Van Vleck Hall, okay, where students would go into classrooms, into large lectures, and disrupt them so that people could not teach their classes. The halls of Van Vleck, at least at the basement levels where the classes were held, were often filled with shouting demonstrators. RL: Were your classes interrupted? RB: You know, I don't think I was teaching a large lecture at that time. And there was, for the most part, I think the demonstrations took place within the large lectures. This was also the time of the black student movement. Again, I'm afraid I don't remember the exact dates, but I 00:10:00 believe that was in the '60s, late '60s, as well. And frankly, the demonstrations for the black student movement and the demonstrations for the Vietnam War, they were related, and they're sort of mixed up, all jumbled in my mind. I don't remember, can't distinguish between one and the other. But it was a very difficult time within the university here. And as I say, I participated somewhat. Now one of the targets of the demonstrators were demonstrating against the Vietnam War, was the Mathematics Research Center. Also known as the Army Math Research Center because it was, the finances from that came from the United States Army. And that was housed over in Sterling Hall, which is adjoined to Van Vleck Hall at the time. And there were a few of us in the department who were concerned about the support that the army was giving to the research center. I don't think we really believed that the people were doing 00:11:00 secret research that was going to be of benefit to the department. Their view was that the research they were doing was open. Of course, they, people who were funded by the Army Math Research Center, did occasionally, from what I understand, go consult with people in the United States Army. Okay? They wanted to know some information for design of whatever, okay? Or for whatever they were doing. So there were certainly instances where people at the research center did consult with people in the United States Army. But the Army Math Research Center was a symbol on campus, okay? It was a symbol to attack. If you were attacking the Vietnam War, it was a symbol that students and faculty, etcetera, could attack. So there was some opposition to the research center. I was part of that opposition. And, in fact, there were a number of us who did something that, I don't know whether I regret or not, but certainly it was very upsetting to some people in the department. The research center was, as they usually did, they 00:12:00 sponsored conferences occasionally. And there was a conference coming up that they were going to sponsor. And a few of us, and I was not the ringleader in this, but a few of us actually wrote a letter to some of the people who were invited to come to this conference, telling them not to come. Okay? Because of opposition to the war, etcetera. And opposition to the center being on campus during those times. So that upset some people in the department. This must have happened in the year '68-'69, after I got tenure, I think. [laughter] Not that I think it would have influenced whether or not I had gotten tenure. But I believe it was after I had gotten tenure. And frankly, I guess I don't know, if I hadn't had tenure at that time, would I have gotten involved in something of that sort. So having tenure certainly promotes free speech and activities that possibly you'd think twice about getting involved with if you didn't have tenure here at the university. So that was, to some extent, an exciting time at Madison. RL: Did you all in the department know which of your colleagues were supported by the Army Research Math Center? RB: Oh, yeah. That was well known. RL: Common knowledge. RB: That was common knowledge. There was no question of who, there 00:13:00 were people who were sort of considered permanent members there, who were just at the mathematics research center. Some of them had tenure within the department. But they were either not teaching, or teaching one course a year. And that was the expectation, that they would be supported by the army, and their affiliation with the department was small. Okay? One semester, or one course a year. So this was a difficult time. And it was also, as I say, an exciting time. So that was '68-'69. I participated somewhat. But I never spoke to student groups or anything of that sort. By and large, I was a quiet participant, except for this one instance where we did write this letter, we did upset a number of people in the department. RL: Was the letter effective? RB: I don't think so. I think everybody who had planned to come to the meeting did come to the meeting. And this, of course, continued into the year '69. But that was the 00:14:00 year that I went to the University of Sheffield, England, on this NATO post-doctoral fellowship. My family, my wife and my two young children at the time. And this was an exciting thing for us. It was the first time out of the United States for me. And we did it big, so to speak. We crossed the Atlantic on the SS France, leaving out of New York City, and crossing the Atlantic. RL: How exciting. RB: Nowadays, it's very rare for people to cross on a steamship. But this was a four and a half day trip across the ocean to England, arriving in Southampton, picking up a car there. We had actually made arrangements to buy a car there, but it wasn't ready. And so we had a different car at the time. But of course the steering wheel was on the right side, it was driving on the left side of the road. Driving in Southampton, and driving up to Sheffield, England. So that was, it was a very good year. Two new colleagues there, Hazel Perfect and Leon 00:15:00 Mirsky. And then there was another fellow there, John Pym, who was interested in, partially interested in some of these things. So I had a very good year there. All I had to do was to do my mathematics. I had no duties, no obligations. Nobody was checking up on me. I came in in the morning. I worked on what I wanted to work on. I collaborated with Hazel Perfect and John Pym. I don't, I never did actually write Leon Mirsky. It turns out Leon Mirsky was actually at the time writing a book on this topic, transversal theory systems of distinct representatives. So we would often talk. And some of my work figured prominently in his book. And he also thanked me for any other contributions that I may have made to the book. So it was a new culture. I enjoyed it very much. My kids went to school. RL: Oh, in the public schools there? RB: Well my son went to the public school. My daughter wasn't quite old enough, and she went to a Montessori school. And my son experienced what was called at the time, and maybe still 00:16:00 is called, open classroom, where the students don't sit at desks facing the teacher. But there are various areas within the classroom that students go and they work on these areas with other students. And the teacher comes around and helps them and talks to them, etcetera. And he thrived in that sort of situation. I guess it was first grade that he was in at that time. And we traveled a bit around England. Went to a conference at the Oxford University. I visited there at a later time. Went to a conference in Brest, France. We took a trip to the continent, going as far as, see, the first time, I guess, we went only to, as far as Salzburg, I think it is. We went down through France into Italy, over into Austria, to Germany, and then up into Holland, as I believe, and then got back to England. The second trip I took -- so that was sort of a trip just for pleasure. The second trip we took went all the way to Budapest, Hungary, by train. Where I had given a talk. I had another colleague, Mike Bleicher, at the time, who happened to be spending a year in England. In London, I think it was. We met up on 00:17:00 this trip and we both went to Budapest together. So that was an interesting experience, to be in a communist country. RL: Oh, that's right! It was still those days. RB: It was still under Soviet domination at the time. We traveled, as I say, all the way by train. We traveled back by train through East Germany. As we crossed from Munich to Berlin by train, you had to go through East Germany at the time. And I remember every couple of hours, the passport control people would come on the train, and would want to check your passports, etcetera. But there was nothing special about that except being wakened during, I think it was an overnight trip, every couple of hours by passport control. And we went to Berlin, saw Berlin. And we did travel, while we were in Berlin, to East Berlin. We went through what was called Checkpoint Charlie at the time. And spent an afternoon walking around East Berlin. And then back to West Berlin, and then continuing our journey back to Sheffield, England. I also during that time spent a summer, that summer, the summer of '70, in Copenhagen, the University of Copenhagen, Denmark. Not for any particular reason, other than my colleague, Mike Bleicher, was also going to spend some time there. And we just thought well, let's go somewhere different. I had freedom to do my research wherever I wanted to do it. So 00:18:00 we spent a month in Copenhagen before coming back to the United States. So I-- RL: Did you go at all to the ancestral homeland of Italy? RB: Well we did go to Italy, but we didn't go to the ancestral homeland. We visited Rome and Pisa and Florence and Venice during these two trips. But we didn't go to the ancestral homeland, to Pesaro, where my family all came from. But it was a very productive year. I did a lot of work. I wrote a lot of papers. I had time to think. I didn't have teaching obligations. I didn't have committee obligations. And so, under those circumstances, you can accomplish quite a bit if you want to. And I did, okay? RL: Were you making the kind of connections here with other scholars that stood you in good stead as your reputation grew nationally and internationally? RB: Do you mean scholars in England at the time? RL: Yes. RB: Yes. Certainly. I had these connections with Leon Mirsky, Hazel Perfect. Another person who was prominent in this area at the time was a fellow named Dominic Welsh in Oxford University. In fact, he wrote a book on matroid theory, I mentioned that word before, that topic before. And so I had connection with him. I made some connections with people in Budapest. And that will come back later, as you'll see. I spent four days in Budapest, and I got to know some of the people there. So after Copenhagen, we flew back to the United States. We didn't take a 00:19:00 boat back. Through Iceland, which was typical at the time. Okay, Icelandic Air was cheap. People often flew and stopped, and we stopped at Reykjavik for a few days there. And landed in JFK airport. And were greeted by my wife's parents with a copy of that day's Daily News with the bombed out Sterling Hall on the front page. This was a shock. [laughs] RL: The day you got back. RB: The day we came back. I guess it happened the day before. And it made the front page of the New York Daily News. And there they showed us this paper with, there it was, the bombing had taken place, the person had been killed. They showed what Sterling Hall looked like at the time as a result of the bombing. And that was a tremendous shock, okay, to see that. So we spent a couple of days in New York, I can't remember how many days, and then drove back to Madison. We had a, our car was shipped back that we had bought in England. And drove back to some chaos, to some extent, I guess you would say. I don't remember the 00:20:00 exact day we came back, but within two weeks of the bombing, that's all I can say for sure. My office was a bit of a mess, because my office was facing Sterling Hall. So some windows were broken. There was glass in my office. A chair had ripped as a result of some glass, I guess, flying into it. And so those were difficult times within the department. RL: Richard, were you completely surprised that the bombing had taken place? Or was, in your mind, a certain inevitability about it? RB: No. I was completely surprised about that. There were violent demonstrations on campus. But if there was property damage it was, I would say, minimal. You know, people wrote slogans on buildings, etcetera, that kind of damage certainly existed. But I certainly was very surprised to see that some people were so against the mathematics research. And as I say, it was really a symbol. They were really protesting the Vietnam War by attacking this symbol which existed on campus. So it was a huge surprise, a huge upset for me. RL: And what was the reaction within the department to that? RB: Well, I mean, I think it was similar to my reaction. If you're asking whether somehow those of us on the faculty who had been a bit vocal concerning the Vietnam War, who may have sent this 00:21:00 letter about the Mathematics Research Center, whether they were blaming us to any degree, I would say no. Nobody, there were no repercussions of any sort against me. I should say, because I mentioned property damage, while I was away in Sheffield that year, there was a conference sponsored by the Mathematics Research Center. It's a conference that I would have participated in had I been here, because it was in areas that I was interested in, in my area of mathematics. And there was this one incident, maybe you're aware of it, where students, during one of the lectures, this took place in what is now called the Pyle Center on Langdon Street on campus, students came into one of eth lectures, disrupted it. They had red paint. And they splattered red paint over the room, over the participants, to disrupt this conference. RL: Really? No, I don't remember that. RB: So I know people that I knew had gotten paint splattered on them in this particular incident. As I say, if, so that's interesting, since I opposed the, was opposing the Mathematics Research Center. I said I probably would have 00:22:00 participated in it. But maybe I wouldn't have. I don't know, because of the situation at the time. I may have decided to boycott it. I'm not sure, frankly. RL: This concludes side one of tape four. [pause] RL: This is side two of tape four, Richard Brualdi. Richard, you returned to Madison at a traumatic time, 1970. And where does your career on campus and in mathematics go in the decade of the 1970s? RB: So returning from my year abroad, I resumed my teaching career and university career. I'm sorry, research career. In the early '70s, I did decide to develop a new undergraduate course. It's Math 475, which was introduction to combinatorics. The first time I taught it I used what was basically one of two available books at the time. I didn't like the book, so I decided that I would try to write my own book. So I taught that course several times, maybe once a year, or maybe even twice a year, I can't remember, for a few years, developing my notes. And eventually finished the 00:23:00 writing of this book, Introductory Combinatorics. I went it out for review to I think three publishers. Only one was interested, and they were willing to publish it. So I signed a contract with this publisher. This was Elsevier Science Publishers. And it was published in 1977. So this was an important activity during my time, besides doing my research. I've mentioned my colleague Hans Schneider. In 1968, there was a journal, a new journal, called Linear Algebra and its Applications, which was developed. And this was actually one of the first sort of specialized journals in mathematics. For the most part, journals in mathematics had been general journals, any topic, you could submit a paper to journals, to a journal. But this was one of the first journals where it was on a particular area of mathematics. And my colleague, Hans Schneider, was an original member of the editorial board of this journal. And in 1972, he took over as editor in chief of this journal because the founding editor in chief wasn't doing a satisfactory job. He was a bit too organized to be an editor in chief of a journal. And within a few years after becoming editor in chief of this journal, Hans asked me to 00:24:00 become a member of the editorial board, which I accepted. And I have to say that this has really had a profound effect on my career. So you know, being, at least certainly at that time, a member of an editorial board of a journal in the area or one of the areas that you work on is a big deal. Your name is prominent, okay? People read the journal. They see you're a member of an editorial board. It must be you're a bigshot, you're good, okay, or something of that sort, okay. So it was a big deal. And I, you know, worked in the journal by handling papers, either refereeing them myself or getting other referees to the journal. And in 1979, I'll come back to some other things in the '70s, but in 1979, or it may be before '79, but Hans asked me to join him as co-editor in chief of this journal. So '79, I became with Hans co-editor in chief of this journal Linear Algebra and its Applications. And to this day, we still are co-editors in chief of this journal. And we also have a third one, a Volker Mehrmann who works out of Berlin. He's at the Technical University of Berlin. So I've been co-editor in chief of this journal with Hans for, well, I guess I'm in my twenty-eighth year right now. And this, of course-- RL: Impressive. RB: --has had an even more profound effect on my career. I mean, if being a member of the editorial board means something, then being an editor in chief of the journal means a lot more. So I think this has certainly helped me considerably in terms of advancement and prominence within the field of linear algebra. RL: I assume that that keeps you in touch with the developing new talent in this field. RB: Oh, absolutely. You know, in the '70s, the 00:25:00 journal was relatively small journal. I can't remember how many pages were published every year. But it was certainly initially 500 to a thousand pages a year, probably. Something of that sort. Now we publish about six thousand pages a year. And last year, the number of submissions of papers for possible publication in our journal exceeded eight hundred. And our publication or acceptance rate is somewhere between 40 and 50 percent, I would say. Something of that sort. So this field has developed considerably. In fact, I think having that journal played a role in the development of the field, and that it becomes more visible. People see the journal, see the [area?] and maybe read some stuff. Maybe say, "Oh, I can do some of that stuff," or, "That's interesting." And so there are a lot of people that are now working in this area and the area has just expanded considerably. So this was a big thing in my development here. I went to Sheffield, as you know, from '69 to '70. I also, four years later, took another trip abroad. My wife and I at the time 00:26:00 thought it might be nice to go back to Budapest, to go to Eastern Europe. And we found there was the possibility of a program that we could apply to, that I could apply to, to try to get funding. And this was the National Academy of Sciences Exchange Scholar Program. So I applied to this program to go to Budapest. There was an Institute of Mathematics in Budapest. And there were a number of people prominent in the field in which I was interested in there. I remember at the time because going to Europe, I'm sorry, going to Eastern Europe, was not a trivial thing. You just don't pack up your bags and go to Eastern Europe. And one of my colleagues that I mentioned previously, R.H. Bing, was a member of the National Academy of Sciences. And he had to interview my wife and I to see whether he thought we were up to living in a communist country. In a country where we didn't know the language, where living would not be so easy. So he interviewed us. I guess he gave us an okay that we could do it. And we, I was awarded this six-month fellowship. And we went to Budapest, [England?] in January of '74 and stayed there through June of '74. That was a very interesting 00:27:00 experience for us. When we arrived, they didn't really have an apartment for us. They were supposed to make arrangements for us there. And so we actually lived, the four of us, my wife and two young children. So my son was about eleven at the time, and my daughter was about nine, eight or nine. And so we lived in two rooms in a private home, sharing their bathroom, sharing their kitchen. [laughs] And that was a bit hard. This was for two months. RL: Was this another academic? RB: No. I didn't know, in fact, I don't recall what these people did at all. They didn't speak English too much, although the daughter spoke a bit of English. And so we had to talk to them, we basically talked with the daughter, with the daughter there. So that was very difficult time. Not knowing the language. It's a big city, of course. You had to get used to the surroundings. You had to, just sort of shopping in a grocery store to pick up food. And then having a shelf on a refrigerator to put your food and to cook. So we 00:28:00 didn't eat too well during that time. [laughter] And I suppose we ate out a bit in some of the restaurants. I didn't mention, but let me just digress a bit. When I was in England in '69-'70, I mentioned Hazel Perfect. Hazel Perfect had been a vegetarian for most of her life. And I had always associated vegetarianism with very thin people. She wasn't fat, but she wasn't thin. Just normal. And my wife and I had had vegetarian tendencies. We decided to become vegetarians when we were in Sheffield, England. And we sort of changed overnight, okay? Here you are a meat eater. Tomorrow, you're not eating meat. And that was difficult to get used to. That was not so easy at that time, to be a vegetarian. So I brought that up because, so here we are in Budapest. England, we were vegetarians, okay. And they're big meat eaters in the Hungarian people. But we managed. And my daughter went to the school of the British Embassy. And my son went to a school of the American Embassy. They had just started the school. It was going from fifth grade to eighth grade or something like that. And the 00:29:00 British school was kindergarten to eighth grade, something of that vicinity there. My son's school might have had eight or nine students, total. I can't remember. So they did okay there. We used to go by Metro. We got on the Metro and eventually my daughter and I and son, at some stop he would go off on a different Metro and go to the American school, and I would go with her on the other Metro and leave her off at the British Embassy and walk to the Mathematic Institute from there. But, so that was an interesting time, living in a different culture. It wasn't easy. We took some lessons in Hungarian in order to try to get by in the marketplace and restaurants. Once a week for eight weeks we had a tutor that taught us something about Hungarian, which is a very different language than English. And eventually we got an apartment of our own. It was glorious! [laughs] It was very small, but it was glorious to have basically three rooms and a little kitchen. Not having to worry about anybody else. And your own bathroom. The institute was very good for me. The accommodations there weren't all that great. I shared an office with two or three other people. And one of them was a big 00:30:00 chain smoker who smoked all the time, who was a very nice fellow, I liked him a lot, but he smoked. So I was working in a smoke-filled room. And it turns out there was, at the same time I was there, another person who was coming from McMaster University spending time in Budapest who in fact had been a Hungarian, and had left during the 1956 uprising and had gone to Canada. And so he was on some special deal, I don't know, some special program being funded. And he was there at the same time. And he also had assigned the same office. [laughs] So there were four of us in that small office there. And we had, I had more common interests with him than I had with the Hungarian people in Budapest, the other Hungarian people. so we worked together. So that's primarily what I did in Budapest as to work with him on certain mathematical topics of common interest. RL: Was English the working language for mathematicians? RB: Yeah. It was the working language. The people that was in my office, they spoke English. They had 00:31:00 traveled outside. They wrote papers in English. I remember giving, maybe this was on my first trip to Budapest, where I gave a talk there. I gave it in English, but it was translated spontaneously by somebody there to Hungarian. [laughs] So I gave it in English, I wrote it on the board, but there was somebody translating it into Hungarian. But the people at the institute spoke English. They had to speak English somewhat, because otherwise they wouldn't be able to participate in international mathematical activities, frankly. So that was a very good year, or very good six months in Budapest. I came back to Madison, resumed my career. During the '70s, I also developed some other courses. These were on the graduate or almost graduate level, in one case, course in what is called error correcting codes. Coding theory. Which is used for reliability of data transmission. Even though I am not an engineer, I don't 00:32:00 know exactly how these things are implemented by engineers. But it was a combinatorial topic. It used linear algebra. Both of the things I'd been interested in. so I developed two graduate courses there. This, of course, was Math 641, which is in that gray area between undergraduate and graduate. And I talked to the people in electrical computer engineering, because of the connection there. And they wanted to joint list the course. RL: Oh. RB: And then there was a second course, 842, which was Topics in Error Correcting Codes, which was also joint listed. And I also developed a graduate course, called Introductory Combinatorics, which I've taught at many times during my career 00:33:00 here. And I think my, I think good job that I must have done in teaching and the development of these courses and some of the other things that I may have been doing for undergraduate, getting involved with the undergraduate math club. There was a program called the mathematical talent search program, which was a program for high school students sending out five or six times a year problems, mathematical problems that didn't require a lot of background to solve, but required some ingenuity. This was a program that had been developed earlier by my colleague, Elsie Young. First time I've ever mentioned Elsie Young. Which was modeled, actually, after a Hungarian program. And this program still exists to this day. But I worked on this in the'60s and '70s. Frankly I don't remember the exact dates. And in fact was instrumental in getting a similar program started in Alabama. RL: This program works 00:34:00 searching for smart kids in the States. RB: In the States. But the problems now, the problem sheets are actually on the Web. So anybody can access them. And they do get sometimes solutions from people outside of the state of Wisconsin as well. And in fact, we actually get now solutions from middle school students. We have some very bright middle school students in the state who do extremely well on some of these problems. Do well enough to get invited to the Honors Day that there is for this program every April/May, where they invite the best participants to come. And give them some prizes, give them some recognition, etcetera. RL: Are these kids then on the fast track for admission here? RB: Well, there's no special brownie points or whatever you want to call them that they get. I mean, they could put this down on their application that they participated. But it doesn't give them any other special privilege. RL: I see. RB: And unfortunately, of course, some of the very best students go to Harvard, go to Princeton, go to MIT, rather than come to the University of Wisconsin. That's often the case. But we get quite a number of them coming to Madison. So I think that there are a number of things that I can, I guess, because I don't know for sure, that contributed to my getting the chancellor's distinguished teaching award in 00:35:00 '76, I believe, that I received that award. I paid a lot of attention to my teaching. I paid a lot of attention to my students. I developed courses. I wrote this undergraduate textbook. And presumably the word of mouth was that the students thought I was a good teacher. In fact, even today I still get, there's some fellow in Madison whom I've met at the Overture Center who was in my introductory combinatorics course. I don't know when. In the '70s, I guess it was. But when I saw him, and I didn't recognize him, but he recognized me. He said he still remembers that course. It was the best course he ever had. And then another time, subsequent to that, I met him and he was with his son. And his son, I'm still teaching the course, and his son wants to take the course. His son is a high school student who comes to the university and takes mathematics courses. And he asked me when I was going to teach that 00:36:00 course again, because he wanted to take that course from me like his father did. [laughs] RL: You like teaching. RB: I like teaching, right, very much. I like all aspects, well, most aspects of teaching. The aspect of teaching I don't like is grading papers. [laughs] Writing examinations. That's no fun, frankly. You do it. You have to do it. It's an important part of teaching. But I don't know anybody who says they like grading examinations. I like teaching small classes, and I actually like teaching large lectures, too. there's something, I guess I would call exhilarating, about being on a stage with 150, 200 students there listening to you or interacting. Well, there isn't too much interaction with lectures. But there's something, as I say, whether it's a particular showmanship that I have or whatever, but I really have enjoyed lecturing in large, large lecture classes. RL: What do you do 00:37:00 with smart students who don't have aptitude for math but are interested? How do you teach them? RB: Well, it's a difficult question, of course. The students who don't have aptitude certainly don't go very far in mathematics. But I taught, for instance, four years ago, I think it was, a course, Math 114. It's a five-credit course, three lectures, two discussions a week. It's college algebra and trigonometry. It's basically our college algebra course Math 112 and our trigonometry course, Math 113, rolled up into one. And students, to get into that course, because it's a five-credit course, have to have, first of all, they need both algebra and trigonometry. And they have to do a little bit better on the placement exam than those students who are taking 112 or just taking 113. I think--so, what do you do with students who, like these students, who maybe don't have an aptitude-- who maybe have had similar material in high school, because most students take college 00:38:00 algebra courses in high school, and usually some trigonometry--and didn't succeed on it. Well, I think there are several things. one, first of all, is you've got to teach at the level of the students. Okay? Now of course in these courses, the level varies a bit, okay? You can't gear it at the lowest level, and you can't gear it at the highest level, but you've got to find some compromise level, okay, that you hope that students of all aptitudes will be able to understand, to be able to learn from. So I think that's extremely important, is to know where your students are. Okay? And then, I think an important part of teaching is enthusiasm for the subject. You've got to show that enthusiasm. Especially true in large lectures. If you're bored, okay, with talking about this material, or think it's beneath you, or think you shouldn't be teaching this kind of stuff at the college level, the students are going to find that out, and they're going to turn off. So I, you know, I have no trouble in being enthusiastic about teaching college algebra and 00:39:00 trigonometry. Early in my career, I taught a course which was called Math 101, which was intermediate algebra, which is a course, it's really high school algebra course. And I enjoyed teaching that a great deal. These were students who really were very poor in mathematics, for whatever reason. They didn't learn the material, they weren't taught it well, or whatever. And I had no trouble getting down to the level of that course, to being enthusiastic about teaching stuff like the quadratic equation. Okay, the quadratic formula. How to solve simple equations. How to do simple graphing, simple inequalities. And we, at the time when I was teaching that course, the lecture that was associated with this course, sort of a laboratory where the laboratory was staffed by either the teaching assistants or the faculty teaching, lecturing in the course. And students could go into this at the times it was open and get help. In that sense, it was a laboratory, okay? There was no lecturing. But it was sort of one-on-one 00:40:00 with students, trying to find out what their problems were, etcetera. And so as a lecturer, I was expected to do that two hours a week. And I did that. And I taught that course for four consecutive semesters, this Math 101, intermediate algebra. And I, in my fourth time teaching that course, I noticed in the laboratory associated with that, I was starting to get a little impatient with the students. Okay? You could sort of say, "Didn't I just teach you this?" After four consecutive semesters of teaching that stuff, I noticed I was getting a little impatient. And I decided it was time to take a break from that course, and I did take a break from that course. So I think, you know, getting down to the level of the students is extremely important. Showing enthusiasm is extremely important. I think one has to be organized. I think one has to choose examples very carefully to illustrate the consequent ideas that you're trying to get across to the students. And I think I've been a success at teaching at elementary levels. And I hope at graduate levels as well [all the way 00:41:00 there?]. [laughs] RL: Have you noticed a change in the level of preparation of undergraduates coming through the department in the decades you've been teaching here? RB: Yes. I think, well, it sort of goes both ways. When I first started teaching here, we were teaching this course, Math 101, Intermediate Algebra, to several hundred people a year. I taught a large lecture with a couple of hundred students. There were probably other large lectures. And this was every semester. So maybe, I don't remember exactly, over five hundred, maybe as many as eight hundred students who had to take this course because they really didn't have the level of mathematics to even start on the college level courses. It was basically a remedial course. The university, of course, over the years now, has increased the mathematics requirement. I think officially it is three years of high school mathematics. It's either the integrated math or two years of college algebra and geometry. So that's the minimum mathematics requirement. I think it's hard for anybody to get into this university without four years of high school mathematics, okay, no matter what their major happens 00:42:00 to be. Because by and large, admission is not based on what your planned major is going to be. it's just based on I guess what college you're going to go into. And if you're going into the College of Letters & Science, you're a potential science or mathematics major, or what. So the number of those students who are now taking that course has really decreased over the years. and now it's just maybe one or two sections of twenty students a year that take that course. Because the students are better--they're a bit higher than that. There's still a lot of students taking this college algebra course, which is basically high school mathematics at a somewhat more advanced level, at a college level. But by and large, what really should happen is that this course should be, people shouldn't have to take that course. They should, if they're going to take any further mathematics, they should be able to start with calculus, okay? But that doesn't happen. We have a large number of 00:43:00 students taking still college algebra and trigonometry. And what I hear from my colleagues, because I haven't taught calculus much, is that the algebraic facility of students has really gone down. They don't know how to do algebraic manipulations. RL: Really?! RB: And there's some question whether that's because of the reliance on calculators and software programs to do these things for you. I mean, graphing, you can get graphing calculators that do these things. You can get answers by using calculators. And their ability to do algebraic manipulation seems to have decreased. But that does not mean this is true of all students. We're getting some wonderful students that come here. We get high school students coming from Madison West. We've had several who basically have almost completed an undergraduate math major here while still as a high school student. RL: Really. RB: Yes. And I've had experience with at least three of them. One is this fellow, you may have heard, seen something in the newspaper about him, Daniel Kane, who's now going to be graduating from MIT, undergraduate, BS degree. He has already published about ten research papers 00:44:00 in mathematics. And he took my introductory combinatorics when he was a high school student at West. There is this remarkable Loh family, L-O-H, in Madison. RL: Yes. Yes. RB: Father is a statistics professor. Mother was a mathematics teacher. There are at least three children in this family. And they're, I had the son for sophomore level linear algebra course when he was a high school student. He went off to Cal Tech as an undergraduate. And then spent a year in England. I think he's now getting a PhD from Princeton. His sister was a student in my introductory combinatorics course. She's also at Cal Tech. And I don't think I had the third child, member of the that family. I think also may have gone to Cal Tech. But they, I mean, when I taught this Math 475 combinatorics course when the Loh child was there and when Daniel Kane was 00:45:00 there, they were the best students in the class. And they were high school students at the time. So there's some remarkable students these days. And we get some remarkable undergraduates who come here who haven't been here from high school. But, and I mentioned the student who's currently in West High who wants to take my introductory course next September when he will be a senior in high school. And