傳播數學知識．促進數學教育

Interview with Prof. Charles Newman

**Interview Editorial Consultant:** Tai-Ping Liu

**Interviewers:** Tai-Ping Liu (TPL), Shih-Hsien Yu (SHY), Jeremy Quastel (JQ), Fraydoun Rezakhanlou (FR)

**Interviewee:** Charles M. Newman (CMN)

**Date:** July 11th, 2011

**Venue:** Institute of Mathematics, Academia Sinica

Prof. Charles M. Newman was born at Chicago on March 1, 1946. He received his B.S. from M.I.T. in 1966, M.S. in 1968 and Ph.D. in 1971 from Princeton University. He has been a faculty member at Indiana University, University of Arizona, and since 1989 at New York University. He has contributed to numerous fields where probability mixed with physics. He was elected a member of the National Academy of Sciences, for his unique views on statistical mechanics and probability.

TPL: Actually I did some research about you, not that I didn’t know you but…

CMN: It’s actually a news article appeared some time no less than a year or so in the Proceedings of National Academy of Sciences because I had an article there a few years ago which is called an inaugural article. You are invited to do that when you become a member. Actually I waited for maybe 4 years after becoming a member but so was allowed… to do an inaugural article and then together with an inaugural article they do a little kind of extended biography which is actually done by some science writers and because you got an interview and that came out another 2 years after the article itself, so it appeared relatively recently that might be … so there is a discussion about ex-history and things like that.

TPL: I read one of these articles this morning. People interviewed you, but some interesting mathematics you said was aired only for few seconds.

CMN: That’s right. I was interviewed a few times. I am at NYU somehow I am the listed people when they get the questions from news journalists about related probability or statistics even though I know almost no statistics, so I am one of the people who sent person to. So I was once interviewed for a local tv station after somebody won the New York State lottery for the 2nd time, the same person. He wanted to know how unusual was that and how surprising it was. I agreed to have this interview. Earlier that day I went out to buy a lottery ticket because I didn’t even know how it looked like and did some elementary calculations so I could make some discussions. I spent a few hours because it was a little bit interesting to decide whether it’s surprising or not. It turned out it was not surprising because the lottery has been going on for 20 or 30 years, I think twice a week and each time there are usually several winners shared the prize, so thousands and thousands of winners have lots of money to buy more lottery tickets. Then one of them should win a lottery isn’t so surprising. In fact that happened before. This is maybe the second time. The thing that I was most proud of was my prediction, there would never be a three time winner. But all that interesting discussion was eliminated as you reported I appeared for only about 7 seconds on the TV scene. I don’t quite remember, that’s just something completely silly. It was just an excuse in the news program in order to show a stroke of lightning, “Isn’t this less likely than being hit by lightning?”

TPL: A particular person won twice is rare but someone would.

CMN: That’s exactly …

TPL: There was this thing about how many people in the room in order to have a probability greater than one half of at least two of them having the same birthday. The answer was like what?

CMN: It’s something around 28 approximately. I don’t remember exactly, 28 or 27. That sounds surprising but it’s the same reasoning because it’s some pair of people, not usually you think psychologically somebody having the same birthday as you. Actually I had a discussion exactly about that phenomenon with somebody who I met just a week ago and whom I heard before. A few years ago, I got an e-mail from my brother in law congratulating me on having a letter to the editor of New York Times appearing. So I looked up to New York Times and there was a letter to the editor from Charles Newman talking about some political things. But it was not me. It was another Charles Newman. Then I decided to try to figure out who he was. I discovered there was an attorney in New York not only Charles Newman but even the same initial, Charles M. Newman. He was the one who wrote this thing. By accident in the last few weeks I have some dealing with a law firm, and I came into the law firm and the receptionist asked my name. I said Charles Newman. They looked at me very strangely. They said, “We have a Charles Newman here also.” So it turned out the Charles Newman is in this law firm. The next time I went there ending up meeting him. I had a very nice discussion with him. It turned out that he was an engineering student when he was younger and he knew Courant Institute and he asked about this birthday problem.

TPL: But it seems to me that from this little story I heard and also in one of the articles I read that you have broad interest, something comes about and then you are willing to spend time to think about it. This is not true with a lot of mathematicians.

CMN: I didn’t really start as a mathematician. I began as a physicist. So I was an undergraduate physics major. But I began taking a lot of math courses. I was in MIT as an undergraduate, everybody there was pretty good at mathematics but many people back in those days quite a few decades ago, did not have calculus in high school then but many people did. I went to a small high school which didn’t have calculus. So on the one hand, I was one of the people who had to take calculus but I also scored very high on the math entrance examinations and so some of those people who did not have calculus but had scored high were invited to take a special calculus course with James Munkres , a professor and a topologist, who had just started this course a year before. He did calculus not like the usual course which was mainly designed more for the engineering students that was not rigorous, but he designed potentially for math majors and did things rigorously from the beginning. Because of that course, I got much more interested in mathematics than I probably would be as ordinary physics student and started taking more and more mathematics course. But I ended up doing both physics and mathematics and gradually have moved from physics towards math. But I think because of the background that I have, maybe wider point of view than some of these strictly mathematics from the beginning.

TPL: But I think we also have seen people who were engineering majors and just discovered their true love is mathematics. Some of these people actually go to very abstract thing, Ukai in Kinetic theory, for example.

SHY: Raoul Bott … he is a EE.

JQ: Harish-Chandra he was a physicist who decided there’s not enough rigor in the theory he learned from Dirac so he decided to make things rigorous. This is after he got his degree. He was a physicist when he graduated.

CMN: I’ve always been interested in the conceptual mathematical issues that arise from physics problems.

TPL: Can you offer an example?

CMN: The talk I will give later this week is something which is a subject that I first began working on just after my Ph.D., and only now beginning to get some results, which is understanding Euclidean quantum field theories in the cases of two dimensional space, which is really 1 dimensional space, 1 dimensional time originally, but then became Euclidean when you go from real physical time to pure imaginary time like in the Schrodinger equation to the heat equation. In particular, those are the kind of things arise as scaling limit of critical statistical mechanics models like Ising model. So this is a subject that I was really quite interested back in the 1970s and did some work there. But because of the recent developments in two dimensional critical models related to formal invariants and Schramm-Loewner evolutions, now one has many more tools than one used to have. So now we have a kind of stochastic geometric representation for these kinds of scaling limit continuum field objects, which is sort of thing that I was always interested and tried to understand back from the days when I was mainly interested in the quantum field theory side. I think that’s an example. It’s also an example of sometimes things take many decades before actually get anywhere or get to somewhere where you like to get to.

TPL: That’s very nice.

FR: You received your degree in physics?

CMN: So my undergraduate I began as a physics major. I started taking more mathematical courses. I was taking more courses than normal anyway that actually happened because I discovered that I did calculation when I was a freshman in MIT. If you take the tuition divided by the number of lectures hours per week you attended, how much each lecture cost? I thought it was terribly expensive. This is still the case but those days you pay one fixed amount of tuition and there was no a priori maximum on the number of courses you could take so I figured the only way to lower the tuition per lecture was to take more courses so I started taking more courses, to make it more reasonable price per lecture. Since I had decided I like mathematics because of this special calculus course, most of the extra courses were in mathematics. And sometime after a year or so I discovered because of all these extra courses, I could get two degrees not one degree with two majors but officially two degrees by just taking a few extra math courses that I might not have chosen on my own so I ended up doing that and ended up getting one degree in physics and one degree in mathematics as an undergraduate. But I still regarded myself as primarily a physics major who’s also interested in math. So in fact I was in the PhD program in physics at Princeton and my Ph.D. is officially in physics although it’s really very mathematical. No self-respecting physicists would accept me as a physicist.

JQ: Who was your advisor?

CMN: Arthur Wightman.

TPL: What kind of person is he?

CMN: He was a very nice, very pleasant person. He was not somebody who micromanaged his students. He gave some general advice things like that and would be happy to talk to you, at least in my case, that maybe depend on what problems that you were working on. That works fine for me. I was able to decide what to do mostly on my own. I just had to report to him every so often. That worked very well.

TPL: You talked about getting your money worth by taking more courses. I remembered J. J. Stoker one time said that education is the only business where the customers want to have less. So you are a counter example to that. You are not a usual customer. Can I turn the subject a little bit to something else? You were a director of Courant Institute for how long?

CMN: Maybe four years.

TPL: What was your experience?

CMN: It was a mixture of some very pleasant things, among the most pleasant things were being able to represent the Courant Institute at two Abel Prize events. The first one was Peter Lax when I was still officially director. And the second one was Ragu Varadhan. I actually just stopped being director so Leslie Greengard was the director then and would normally be the one attending, representing Courant. But he already had something for the NSF and could not get out of and he had to go to that so I was happy to go and replace him. Those were extremely wonderful, interesting and elegant kinds of events. Things like this are very nice to be involved in. Being directors gives you that opportunity, otherwise you don’t have so many opportunities to shake hands with kings. Although I think at the Peter Lax event the king actually did not attend the main things, they said he was not feeling well although there seems to be some questions about that because apparently he was involved in some kind of sail boat racing right around the same time. But the queen was there so I shook hands with the queen the first time that was at the state dinner at some castle. As for the one for Varadhan, the king was actually there. I shook hands with the king, some things like that are very nice. Other things are not nearly so pleasant, not all dealings with faculty are always positive. Especially when faculty wants something that you don’t have resources to give them and that was not very pleasant. All in all, it’s a good thing to experience for some time but I am happy that Courant the tradition mostly is that the directors should rotate every 3 or 4 or 5 years. So I don’t have to continue for too long. I was happy to step down.

FR: Before that you were the math chairman.

CMN: Yes, but the math chair is a much smaller position in the sense of what you need to do. In fact, it’s even smaller than it’s now and it becomes actually somewhat more like a normal department chair which involved certain kinds of things, recruiting and so forth normally. But in Courant it’s less than a usual department chair because the director did many of those things so the chair was a position that did not take up so much of one’s time but the director takes up so much more.

TPL: Courant Institute now is very strong in probability.

CMN: Yes, that’s correct.

TPL: Maybe now it’s the period where Courant is the strongest in probability ever.

CMN: I think that’s probably true. Probability itself has become much more popular I would say the last 5 to 10 years.

FR: Those people who win Fields Medals even if they are not probabilists, they are dealing with problems more or less related to.

CMN: So in the last two sets of Field Medal awards quite a few in probability or closely-related things. So I remembered learning from Monroe Donsker when I first came to Courant Institute. Probability had a kind of complicated history within mathematics and some years before I got my PhD he was describing that many people did not regard it as really mathematics. Maybe because of its history, it kind of came out of gambling, original motivation was lottery or gambling. That’s why the famous elementary probability was called the gambler’s ruin problem, how long it takes before whether you use all your money or you break the bank, what’s the probability of that. That problem usually is used in the elementary course.

TPL: That’s important for the gambler. But I think probability is a difficult subject for a lot of people.

CMN: One reason is that a lot of people, for example, students when they take probability course they have trouble even they are good in other mathematics courses because a lot of thinking is somehow different.

FR: It’s mostly the intuition really, otherwise probability does what an analyst does PDE or ODE. But you have different intuitions

CMN: Maybe someone here remembers this quote. There was somebody who describes probability theory as measure theory with a soul.

TPL: That’s true. A lot of us don't know what the soul is. Actually here in Taiwan I think we have a little problem, we don't have enough people who really know that soul. For some reason, it is so important in the modern science, we are certainly not good at that. You can see that in this meeting there are very few speakers from Taiwan. You can count only one or two. Everybody realizes this is a shortcoming but we don’t know what to do about it. But anyway, can I go on a little bit about that soul? What is it?

CMN: Well. In many situations in probability theory, you can try a problem sort of focusing on a kind of analysis techniques behind it. For example, you study Markov processes, take continuous Markov processes, those are described by semi-group operators on space which have generators you can use all the tools in functional analysis, or you can try to study it from a more probabilistic rather than analysis point of view, try to understand probabilistic interpretation of the various questions that you are asking rather than reformulating them as analysis questions. Sometimes doing analysis is a faster or better way to get results. Sometime it’s the probabilistic way of thinking about it that gives you the right idea. So I would describe, of course I am biased somehow, I would describe that the soul of it has been more probabilistic when you think about it rather than turning it into analysis problem where you can forget what the probabilistic meaning of the various quantities are.

FR: The baby example is the semi-group property in probability theory becomes the Markov property which has that intuition.

TPL: I have one time taught a course in probability and the student asked me: I have no trouble with calculations. You give us two numbers but you did not tell us which number is $n$ and which number is $k$.

CMN: In teaching elementary undergraduate courses, almost the worst thing you can do to many students is a problem where they have too many choices. They prefer problems where they are forced to do something. If you say prove that something are linearly independent, they start doing something and do it fine. But if you say give me three vectors that are linearly independent, they have no idea how to get started it because it’s too much freedom.

FR: When I teach calculus if I give true or false questions, then everybody stuck. You give them statement and they have to figure out this is correct or not. And if this is not correct, then they have to give a counter example. No matter how simple it is, that’s a killer. As opposed to give them a formula to derive or verify something, that’s a very different ball game.

CMN: For my introduction to serious true/false questions was undergraduate analysis course while I was at MIT. They had a tradition there in that course and also the undergraduate topology courses having T/F questions but very difficult ones where a lot of danger if you didn’t, for your own benefit, either find a proof or find a counter example. And there were typically 10 questions. You got +10 if you got the right answer, 0 if you did not put any answer, -10 if you have the wrong answers. When I took it, there was no time limit. You were allowed to stay as long as you want with books and notes. Some people used to keep it overnight and handed it in the next day. But each question you have to be extreme carefully and make sure everything… Really I thought this was one of the closest things to research for undergraduate because you don’t know the answer to start with. There was a famous case I think a year before, not in the analysis, but in the topology course where the class average on one of the exams is -7. I think that meant if you did not answer any questions and have a 0, they gave you maybe a B.

FR: So I eventually quit as I get older and more conservative. I am not giving them true/ false questions anymore even though they don’t answer it right I don’t take any point there, but still there are so much complaint about it.

CMN: Last semester I taught undergraduate analysis course, not at the same level as I took at MIT. But I did get some true/false ones and also these days I would not take off the points. You really find out in those whether you understand the material or not.

JQ: I directed the prelim exam using this idea.

FR: That’s the best way to measure.

CMN: True/false. You sure found out what they understand. That’s for sure.

FR: For undergraduate courses, I already knew the answer. The majority of course they don’t know what’s going on. Unfortunately there is nothing I can do about it so I quit.

TPL: You are afraid to find out the truth. This reminds me about someone said that the best weather prediction is that today’s weather is the same as yesterday. For 0 point one gets a B grade.

JQ: Were you already doing probability in your thesis time?

CMN: So actually my introduction to probability is kind of funny. I tried taking an undergraduate probability course and it was a disaster. It was an extremely old-fashioned course and the person taught it was using his own book written 20 years before. And all I could remember was that I ended up dropping the course after a week or two. All I could remember was that he began by talking about different types of $\beta$ distributions that was the first topic, some explicit formulas and so forth. So I really didn’t have probability class, I tried but didn’t like it. Well, when I was in graduate school, I was in physics I took some math courses but they were mostly analysis things of that sort, but then I started doing thesis on mathematical physics which is supposed to be about quantum field theory. But I did a funny work on a funny model which quickly turned out to be really more probability than anything else, so I began having to read some material for that. The first book I read that was kind of probability, but not the normal first probability book. It was the volume 4 of Gelfand and Vilenkin’s Generalized Functions which talks about generalized random fields, so at the time I did not know what random variable was but I knew what the generalized random field was. Actually generalized random fields actually that’s what I am going to talk in this meeting. And then I didn’t actually find out what the definition of random variable was until a few years later, when I taught undergraduate probability course I had to learn it. But then during my last year in graduate school I took a course that Ed Nelson gave. The first semester was quite advanced probability, it was about Brownian motion and fine properties of Brownian motion, mostly just learned standard things about stochastic processes and things of that sort. And then the 2nd semester was at the time he was developing Euclidean field theory which was really a probability theory motivated by quantum field theory and as I said later on I actually learned more conventional probability theory when I actually taught it, so I had a very kind of backwards introduction to probability theory. So actually it would be natural for me to be one of the people who focuses more on the analysis side than the probabilistic side. But somehow I ended up when I finally learned it, I really liked the probabilistic explanation of things and I think about things much more in that way than focusing on analysis.

CMN: Among Varadhan’s many features are in fact he is amazingly strong both in probabilistic way of thinking things and analysis and other stuff as well.

TPL: I tried to teach probability and in fact the undergraduate teaching I have taught probability quite a number of times. I must admit that I was never really into it. But I did an apparently okay job that for several years they always wanted me to teach probability. It’s a difficult subject I find somehow.

CMN: The intuition is different.

TPL: One time I learnt from Mark Freidlin that if you have a random walk in 2 space dimension then almost surely you come back to this given boundary, in 3 space dimension there’s a positive probability that it will never come back. It was later on proved analytically through PDE methods but I think it was discovered first through probabilistic thinking.

CMN: That’s probably true, I think that’s correct. I mean the standard argument you would look at the expected return times, you can think of that either analytically or probabilistically that would be something natural to look at and more natural to think about probabilistically. That’s one of the classical facts and remarks. There are of course those kind of dimension dependence in statistical mechanics models and how different things vary depending on the dimensions, also for P.D.E.s.

TPL: Recently general public became more aware of the importance and centrality of probability in science or in mathematics in particular.

CMN: From a practical public relations point of view, probably Werner getting a Fields Medal made a big difference. I think that’s the first Fields Medal in probability somehow gave a kind of stamp of approval to the subject. Gerard Ben Arous told me that he remembers when he was chair in ENS (Ecole Normale Superieure) in Paris, they used to have trouble; algebraic geometry was what everybody wanted to do, always the best students went into that; they have trouble convincing good students to go into anything else, once in a while they will get somebody to do maybe PDEs or analysis or some kinds, and great once in a while in probability. But more recent years in France in particular, probably Werner getting a Fields Medal made a big difference. They have the top one or two students coming into ENS often going into probability so they have extremely large numbers of extremely good young probabilists coming up in France. The Fields Medal and Varadhan winning the Abel Prize I think that has big influence on what students and postdocs think are the important areas but as you said people in the field always felt that they were important. Besides there’s another thing which I don’t know why that is more recognized now than the past, but probability does seem to pop up in all other areas of mathematics, in surprisingly lots of situations.

TPL: But probability is very international, you can count on all the major schools say in Moscow, France, Japan, U.S., very important probabilists in almost all places. It’s not localized.

CMN: It’s often localized sometimes within countries but many different countries are very strong in probability.

TPL: It’s a bad question. How do you predict the future of probability study?

CMN: Oh yes, that’s not an easy question to say anything coherent about. I guess I kind of agree with the way where things have been developing that because of the fact that probabilistic things are using besides within mathematics also in many other fields, physics of course is an old story, but now probabilistic and statistical things become extremely important in biology, and of course in recent decades in finance and for long time courses in engineering. So I think that means it will continue to be helping growing area both in mathematics and in applications. So I don’t think that it has reached the peak yet. There’re always people in it and influence, etc.

TPL: How about you? What are you up to?

CMN: I am still interested in the kind of mathematical structure of probabilistic models that typically arise in physics, so one thing we are talking about here is about models that are examples arising in critical points of statistical mechanics models. Here will be basically about 2 dimensional models where most of the progress has been made in recent years, mathematical progress, much earlier progress was in higher dimensions. Things actually get simpler like they do in topology once you get above some dimension, so for one particular class of models like Ising models the place where things started getting simpler is four dimension or above four dimension, but in two there are very special two dimensional methods which lead to very specific things like the Schramm-Loewner evolution and conformal invariance. But in between 2 dimensions and when they start getting easier are the most difficult, for example 3 dimensions. So there the least is known and there it’s completely wide open to start developing some techniques. One of the reasons the thing that I am going to talk about 2 dimensions even though we only do things in two dimensions, the point of view seems like it’s potentially applicable in 3 dimensions as oppose to conformal invariants methods that are really specific to two dimensions. The program doesn’t play a major role in the approach that we have, it does technically but not in the overall point of view, so that’s one reason I like it is that it might eventually be useful say for 3 dimensions, but that’s something which is completely wide open, and nobody has many ideas. Then another class of models that I am not talking about here but a number of people here have already mentioned it today, the general area, this sort of systems, are spin glass models that has something with extended history in the physics literatures, and mathematicians especially, mean field type models for which people have done major works in recent decades or so. But again in short range models very little progress has been made, that’s also very wide open field where I am very interested in but I suspect the time scale which major results are being obtained; I am not sure I will see many of them in that area.

TPL: That’s the area that Parisi was one of the persons who initiates it.

CMN: One of the questions that I am very interested in … Parisi’s approach was mainly initiated in…mean-field models are ones in a way simplified versions of models in some spatial dimension with short range interactions, in which roughly speaking you take a limit which you have longer and longer distance interactions but weaker and weaker and in the limit you get something where everything interacts with everything, kind of equally strongly. Even then Spin-glass model is extremely difficult to analyze that Parisi made the Ansatz for, which much isn’t in the last few years proved rigorously. But the question of whether various features of that Parisi Ansatz for the mean-field models will be applicable to short range models, that’s a pretty wide-open question which we are very interested in. But in general relatively little rigorous results, actually not just rigorous any sort, have been gotten for this real short range models in any dimensions. So that’s something which we are looking forward for eventually some developments in that. We made some progress in very specialized questions in two dimensions but it’s very tiny compared to what you would like to know.

TPL: That’s a very happy situation.

CMN: Yeah, that’s right. That’s very happy for future generations.

TPL: We were talking about probability is a subject with a soul. How does Parisi fit into that statement?

CMN: I probably shouldn’t say too much again … I would get myself into trouble. I will say something that will get me into trouble in a different way, but it has to do with a soul … very little to do with mathematical physics or mathematics … so this has to do with New York University and its activities in Abu Dhabi. So something has been going on for a number of years, and it’s very interesting and may develop very well. There are lots of questions that some people are concerned whether NYU should be doing that. People from outside the NYU for the last few years sometimes assumed that the only reason for NYU to do this is because they are getting a lot of money from Abu Dhabi. Therefore, they are doing this and these people think this is not a very good idea, and NYU is only doing this for money. The truth is that even though Abu Dhabi is really paying for pretty much everything happening there, they really have not given very much money besides that. So people made comments like “NYU has sold its soul.” That’s the usual phrase. You have a soul and you sell it to the devil in exchange for some kind of short-term advantage. The basis of the question is that NYU got a lot of money they did something in the exchange that has been done. Where I am not sure whether the exchange has been done or not. Time will tell, but they actually did not get a lot of money. So I tell them that if that were the case that they actually had gotten a lot of money therefore it would be a worthwhile discussion to decide whether they should have sold their soul. But that is not the case, they actually gave away their soul. I have to hope that this is in the article that doesn’t get read by... NYU is also starting a campus in Shanghai so that will get into trouble…

TPL: Of course, this has nothing to do with my original question about Parisi and soul.

CMN: Yes, that’s true. I have tried to avoid that question.

TPL: Maybe you gentlemen have something to say about it.

JQ: These guys have amazing ways of guessing formulas and what’s going to happen something like that. It’s nothing wrong. It’s just not rigorous

FR: Yeah, very important and useful.

JQ: Yes, it’s extremely important what they do. It’s not rigorous enough but that’s fine. That serves big jobs for other people.

CMN: No, no, I have no complaints about nothing rigorous. I think certainly it seems to be the case what he predicted for the mean-field models is quite correct. People are proving that. The things that we have problems with are whether claims that the same qualitative behavior should happen in short range models whether or not that’s correct. There’s much less evidence of… or let’s say there’s conflicting evidence. There’s at least one case that Parisi was involved in one of these non-rigorous, very interesting but not well substantiated claims in a different area, also in the disordered systems having to do with something called the dimensional reduction, which has to do with whether you should compare the disordered system in a certain dimension to a non-disordered system. Some people by one set of arguments claimed that would be non-disordered system one dimensional less, other people claimed that would be two dimensional less and whichever was the one Parisi claimed turn out to be wrong. So it’s not always the case the physicists’ predictions are correct. Everybody remembers the ones that are correct but forget the ones that are incorrect. It was mathematical point of view but not conceptually, not only non-rigorous but often a little bit weird, that was not based on physical argument but based on weird mathematical argument.

FR: I think that’s the point. You made these non-mathematical arguments and there are two possibilities either there’s a way to get around that, find another way to do it or even if you fail, you still learn something you know that why you should not make that assumption, something looks correct why it’s not correct. This is a good starting point.

JQ: You know it’s funny we are talking a little while ago about probability having these strange reputations inside mathematics because it started out from gambling and a lot of non-rigorous arguments. And this is just a modern version of that but it’s probably a sign of big health of probability, would they stay close to some problems that are interesting enough that there is a wide degree of methods, some of them are very rigorous, some of them are extremely non-rigorous which can help you understand what’s going on.

FR: The other reason that why probability was not so popular among other mathematicians was because this kind of part of the probability that were so abstract for non-probabilists was impossible to follow. I heard that from several people after my talk that usually probability talk we quit after 5 minutes as soon as we hear the word Martingale or something like that because we have no clues of what these things are.

CMN: I remember just after my PhD I bought some, because it was some sale cheap at the bookstore, some book about Markov processes where the first sentence was a Markov process is a sextuple of … absolutely nonsense when I read.

TPL: There’s a physicist C. N. Yang, the Yang of Yang-Mills equation. Yang said that he used to read the introduction of mathematical papers, now he cannot even understand the abstract. I think we seem to have no other choice because the journals demand that rigor. Also it’s the easiest way to present material...may not be the most understandable way but the easiest way to present the material.

FR: Which is unfortunate …

TPL: You should come back in a better weather to be my guest and relax a little bit.

CMN: Go hike and see the mountains.

TPL: I don’t believe in too many lectures. One time Peter Lax told me that he was going to Beijing and they asked him to give something like 16 lectures in a week. And I said no, no, I will bargain it down, how about 8? Eventually he said 6.

CMN: Something not quite that many but something like that happened to me one time when I visited Moscow maybe 25 years ago. When I first got there, I met Sinai maybe my first talk which was one more occasion and he said that we would like to have you to give talks in this seminar in the university. What could you talk on? So I listed four or five possibilities and thought he would pick one of them. Instead he said okay, good, you will talk this one on Monday and that one on Tuesday…

TPL: Well, you are thinking about the next lecture so maybe we will stop here. Thank you.

- Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.
- Shih-Hsien Yu was a faculty member at the National University of Singapore and is now a faculty member at Institute of Mathematics, Academia Sinica.
- Jeremy Quastel is a faculty member at the University of Toronto.
- Fraydoun Rezakhanlou is a faculty member at the University of California, Berkeley.