Prof. Neil Trudinger was born on June 20, 1942 at Victoria, Australia. He received his B.S. in 1962 from the University of New England, and his Ph.D. in 1966 from Stanford University. He was a faculty at New York University, Macquarie University, University of Quensland, and since 1973 at Australian University. In 2016 he was appointed as a distinguished professor at the University of Wollogong. For his important contributions to nonlinear partial differential equations and geometric analysis, he was elected a fellow of the Australian Academy of Science in 1978 and the Royal Society of London in 1997. The highly cited book, “Elliptic Partial Differential Equations of Second Order”, co-authored with his thesis advisor, Prof. David Gilbarg, has won the AMS Leroy P. Steele Prize.

TPL: Neil, I would like to start with the one question which everyone would be interested in. Namely, you’re writing the book with Gilbarg . The question is about your book with Gilbarg. When was that?

NST: When did we start? We started in 1971.

TPL: Wow! 1971. My goodness 40 years ago. I see

NST: And the book was the first edition, published in 1977, but it started at Stanford in 71.

TPL: I see. So how did it start?

NST: Okay, so I was there. I was a visiting associate professor for two quarters, spring quarter, summer quarter, and I taught the last quarter of the graduate PDE course and I taught a summer extension. And in that graduate PDE course I did Sobolev space theory and the $L^2$ theory of elliptic equations. And in the summer course I did quasilinear stuff. Dave already had some lecture notes on Schauder theory from many years before, and for the students I prepared hand written notes. I thought it was quite a good treatment of Sobolev spaces and so on and I said why don’t we put the Schauder and $L^2$ theory together and do a book and so we started. So that course of mine in the spring quarter became Chapters 7 and 8. Chapter 7 was the Sobolev spaces and Chapter 8 was the $L^2$ theory and Dave’s notes became Chapter 6. The rest came slowly, very slowly. The quasilinear stuff was slow, I wrote almost all of that and it was quite slow. We had some arguments of course.

TPL: I see, so as usually happens with collaboration of this sort, junior faculty with a senior faculty and it is really the junior faculty who did almost all the work. Maybe you don’t want to respond to my statement.

NST: We had, it was divided, and I wrote most of the book it was true. But every sentence in that book we sat together and read out aloud so we could get it polished, choose the right words, the language and everything. So Dave did all that as well.

TPL: I see. So this is important point. You two really went through every detail together.

NST: We went through every detail together.

TPL: A graduate level book has two types. One is standard material, doesn’t go to too much depth and use for graduate course. Or else, it’s like research monograph. But your book is unique.

NST: It’s quite in between.

TPL: It’s in between. It has serious research level stuff and yet is used widely in the graduate level course. That’s partly because you two spend so much effort on making sure the details are readable and so on.

NST: We wanted it so, most of it is self-contained to some extent. We did assume some knowledge of integration theory but we didn’t assume functional analysis and other topics as well. Other topics like potential theory, we developed there.

TPL: So this may answer part of the question I am about to ask, which is, what do you think the reason for the popularity, long time popularity of that book?

NST: I can’t explain that, I just don’t know why. I just don’t understand. Maybe it’s more popular than it deserves. Because when we wrote it, we were conscious of all the shortcomings. Don’t forget, we may have done things differently and of course Dave was more classical than me, and I wanted to put a lot of the real analysis at the start, so this was one of our big discussions. But Dave didn’t want it, he wanted to do things even more classically and I said things like, let’s do all things like estimates with compact support and then recover the general estimates with cut off functions. No, no, Dave wanted to use the old classical interior norms and so there was quite a bit of negotiation. But I think at the end of the day, Dave was a good expositor and I may have had a bit more modern approach, not by today’s standards but modern back then, and somehow we compromised quite well.

TPL: You two compromised, you’re much younger which means you have to insist on something sometimes.

NST: Basically, I took some advice on the ordering of the chapters.

TPL: Are you thinking about a revision or a new version of the book.

NST: Always, because we had a second edition that was a few years later where we included the LP theory and the Krylov –Safonov theory as well as fully nonlinear stuff. Not long after that there was a Russian translation, I wrote new material for that translation in 1984 but it wasn’t published until into the late 80s, and then Springer issued another version which was called a revised printing of the second edition, which was really the Russian translation, the new material back to English. That was in the 90s. Then they made the paperback version. But I’ve have a dialogue with Springer now for more than 20 years about a third edition which will be two volumes. Volume one would be a textbook on the basic theory of linear equations with some expansion into parabolic equations as well in systems. And the other would be the non-linear theory. But this volume 1 and volume 2 has never materialized because I never get time. And there’s not enough time. It took a lot of time for that first book. That first edition took a huge amount of time, and I wasn’t writing papers at the time, just had to put all my effort into the book. And to turn around now for a couple of years to revise the book will be hard. Maybe there’s lots of time but I just have to give up research altogether if I am going to do that.

TPL: Oh, that’s amazing thing you just said.

NST: Of course, if there was no Xu-Jia Wang, I would have definitely done the book. But having this guy in the next office to me, I can’t escape being involved in research.

TPL: I can understand that. It used to happen to me with Shih-Hsien Yu. But it’s amazing that you are reversing the order, because usually it’s the people who are most senior feel that I need to summarize things and write book for the next generation. But actually you did that while you were younger.

NST: But I had a similar view, I had just taken on an admin job. I was a department head at a young age of 31. I thought the subject was dead, that is the quasilinear stuff was dead, so I said okay, I can write the book, and it’s done. And then I don’t care what happens in my life, I can change my area, I could stay with this stupid administration for the rest of my life, but I will have finished something. So you were sort of saying what the old guy does when he is sixty, when he wants to record their subject and see it written but I started that when I was 30 with the same sort of reason but then things changed. There was a big explosion in applications to geometry and other things and everything changed, the stuff became very important. Even now it’s important and many young people don’t realize that to work on non-linear stuff, you better learn techniques on the quasi-linear case first.

TPL: So, for the non-linear one, for the geometric analysis, the proof provides the foundation.

NST: I think yes, those foundations in the quasi-linear stuff you don’t see so much anymore, I think the techniques though are still important. I mean if I went back I would put more emphasis on the techniques than the theorem statements. You need to make theorem statements but there’s hundreds of ways you determine a theorem in PDE. The statement should somehow be just reinforcing the ideas, the techniques. But I’d have to do it differently, the statements could be used in a better way with more flexibility, if I went back.

TPL: There’s of course this book by Ladyzhenskaya and Ural’tseva which is before your book.

NST: Ladyzhensaya and Ural’tseva was before and I learnt a lot of the subject from the Russian version of their book, actually when I was a graduate student. So, yeah, I knew their stuff. But we thought we could do a much better job.

TPL: But this book is always on the top of the list, of the best seller.

NST: Top of the top 10.

TPL: I think it’s number 1.

NST: Number 1 every year. I have a theory about that.

TPL: Yeah, what is that?

NST: Because you know most people in PDE work in the semi-linear stuff, the Laplacian equals f of u. And they write lots of papers and do lots of citing, and this community seems to cite that book.

TPL: Just one question on that book, so David Gilbarg, he says that in public and I heard him personally say after a certain age he started to feel bored with doing research but he is a good citizen. He always attends seminars, makes comments. So my question really, you have been so intense in doing research these days, so can you understand the psychology. What makes some people work research on and on and some people just feel bored?

NST: People do research because they want to discover things, that’s the main reason. But that may be different now as people want to write papers. They are more interested in writing papers than discovering things. But in the old days, you did research because you want to find things out and to answer questions. So I guess that’s always there for everybody. Now, it’s nicer if you have questions and someone else answers them. But if no one else is answering them then you are forced to answer them yourself.

TPL: Okay, so that’s a key thing

NST: I think it’s good to do research but not to feel pressure. I would have thought you should write a paper in a year or one or two papers in a year, good papers. You find that people don’t do that anymore. Somehow they want to do a lot, they want their papers in top journals no matter whether they deserve to be or not, it’s all different. The marketing is very important now.

TPL: Some sense of fatigue could set in with such a mental state.

NST: People probably want to protect their reputations by being seen to keep on doing things. So the older guys like us, we end up working with young people, with a lot of helpers do the work. I don’t think it was like that before. The old people dropped out more, like David, and the young people were very good before because more top people went to mathematics proportionally among the younger students before. So, the old people did not compete as well as they do now. Let the old compete. Much better. A lot of PDE was developing when I was young, so I could be an expert after my PhD. I was an expert on what I worked on. Now if someone does a PhD on non-linear equations, they may not be an expert; they might not be an expert until 10 years later. It wasn’t like that before when there wasn’t so much to know.

TPL: Australia is somewhat unique; first geographically it’s somewhere outside right?

NST: Yeah, it’s far away, don’t tell me that. I am sick of the travel.

TPL: And so my next question is, how does mathematical research evolve in Australia?

NST: Yeah, well I am a bit cynical about Australia these days so I would say it’s not evolving very well. But I am sure Taiwan is similar. I mean our problems are the same as places like Taiwan, in that there aren’t enough participants. Population is not large enough to have the massive centers like you have in the US and Europe. We have an advantage in that we can draw lots of people from other countries for our jobs. We fill jobs from all around the world. We don’t have to rely on our own population, so we have some advantages there. But it is still much harder for us to get the young people to come back than before, much harder. They are not coming back, they don’t have to so much because people travel much more now and they can keep their cultural identity anywhere in the world, watch their sport and keep up with current affairs on the computer. They can readily travel every year to visit their relatives. This wasn’t the case 40 years ago. It was hard to preserve your identity somewhere else. It’s changed now, so people don’t come back as much to countries like Australia.

TPL: Here in Taiwan we have a rapid train, and people think that with rapid train, small town people stay there and come to big city to work. As it turns out with rapid train, people say I can live in big city; it’s very easy for me to go home on weekends by rapid train. So the small towns are actually losing population.

NST: Sure, sure. It’s like that for us, it’s very easy for a young person from Australia in the US to come back every year or more often. Very easy. And of course they have got internet, skype, everything else. They can completely be involved with their old associates. So yes, it’s unfortunate. Whereas before, a lot of us gave up good careers to come back which seemed like the right thing to do.

TPL: And the kind of lifestyle you enjoy.

NST: The lifestyle, you couldn’t have the lifestyle. The visual lifestyle, seeing the trees and things like that. It’s hard to give that up when you’re used to it.

TPL: When was the time Australian mathematical research began to have a substantial amount of it? When did it start?

NST: Okay so when I first went back in the late 60s, the situation wasn’t that good, but then universities expanded, more young people got jobs, much better. Then there was a falling off period, almost like a generation that wasn’t getting jobs, I think now the jobs are coming back but the top young people are not coming. I think one of the biggest problems for us was that we got into a situation where mathematicians got regular grants. This is very dangerous. That means that instead of just working for their own enjoyment or for their international reputation, they are now working to get the next grant, and I think this completely corrupts a lot of the research and the way it is presented. For the grants they need papers, they need to make people think they did something good, they need friends who will write grant assessments. We don’t do that well in Australia compared to here in non-linear PDE, as we have a too much of a concentration in Canberra. So it’s a big problem with the grants because we don’t have a good network to write the referee reports for the grant applications. In small countries some of those areas of mathematics which are not so popular elsewhere can get a lot of attention. We have all these problems too.

TPL: This problem sounds rather universal.

NST: I think this is common to small countries, probably Taiwan as well. You can’t say mathematics is important because it is important. It has to be important because there are people in Taiwan doing it. Because they are the ones that are going to say that things are good in this building.

TPL: Let me go to some mathematics. First, your own research, what are the things that you have done that you feel most happy about? I know there are estimates named after Trudingerand so forth, but how do you think of it yourself? What are the happiest moments in your research?

NST: I think once you do something, maybe you are happy. You could be happy when it’s not done. You might think you’re finished and you’re not done, you know? When it’s done, it might be a relief. You didn’t waste all your time. So I don’t really know at the end of something, because I hate the writing up so that’s unpleasant, so I don’t really have this sort of feeling, this is done, so great result. No I think by the time something is written and you kind of lose interest because you’re more interested in some other thing. Once you finish something, you want to find something else immediately. You don’t stop, it has to be something else, and another question is there. You don’t understand everything, but want to understand everything. Well I don’t know, but like many discoveries, that exponential imbedding was truly accidental and later Moser found the best constant.

TPL: It was not the most difficult thing you have ever done.

NST: it was like a diversion. I was not interested in that. I was interested in avoiding John-Nirenberg lemma in the Moser Harnack inequality. That was my main motivation, avoiding the use of this complicated covering and BMO and so on. That was in that paper, showing how I could get this exponential by using a power series and getting control of $L^p$ norms. That was the paper and then I realized the same technique gave a sharpening of Sobolev. So that was kind of the add on to what I thought was important from an expository point of view, so that was a kind of accidental find.

TPL: But when looking back, you have a few decades of research up to now, what are the things after a few years, you feel, I’m happy I have done that,

NST: Right, no this is interesting. Because for grants sometimes you are forced to say what the best papers are. Usually I take a mixture, the papers that should look like the best papers but not necessarily the best, they might be the ones in the top journals and I might mix up the topics and so on. But the things I like more, of course are in the present thing. Most of my recent stuff is with Wang and so some of that stuff I really like. Because there are actual solutions to conjectures and open problems, not just mathematics which cannot be necessarily be measured or a problem being unresolved or something. So there are lots of things in that answer. There was the Chern conjecture, the Monge problem, other things in affine geometry, the fact that our conditions for regularity in optimal transportation had a lot of impact. I mean this is kind of pleasing. At the time I wouldn’t have regarded that as a big work. It was only when it became important it was pleasing afterwards.

TPL: I have heard that various people, Ambrosio and others, they have to mention your name.

NST: Oh yeah, we don’t really care about that. We care more when they don’t mention it.

TPL: I see. But how do you feel about partial differential equations or the research you did in PDE in particular? How do you feel, what do you think of the future of it?

NST: I don’t know. I always keep thinking maybe, it’s time to end. Always, I’m wondering what else can be done. It’s kind of strange.

TPL: But keep going on.

NST: Keeps going on. Well, what’s going on is the paper writing. The injection of new ideas is probably not that substantial, really new ideas, breakthroughs. Probably not happening that much, but certainly there is a lot of paper writing going on, a lot of generalizations and extensions.

TPL: I have this thinking all along and let me try it on you. Namely, certain subjects of mathematics there is a fundamental, basic, the core subjects in mathematics, you can even say, of human creation. So for example, complex analysis is one, linear algebra is one and I would say that among the PDEs, elliptic PDE is one. And so, for example Hörmander had written about several volumes book but wouldn’t it be nice if Hörmander with his broad understanding of linear PDE, write one volume not two big volumes, so people in the future can use it to teach a course in one semester and get the basic ideas that, because we cannot, most of us, cannot afford to read so many volumes. So how do you feel, would it make sense to write a book, not that long, suitable for one semester and I think it can be done and has been done for subjects like complex analysis. But how about for elliptic PDE.

NST: Elliptic PDE, sure, you can do that. You can make a semester course with a book for elliptic PDE. I don’t think that’s so bad. This is something that’s not thorough. For example there are books that are meant to be thorough. Some limitations. For a one semester course of course you have to pick and choose topics that try to convey the essential ideas for the whole area.

TPL: Yeah, not thorough. Do you think you are suitable, you will write such a book?

NST: No, getting too late. I wouldn’t mind. I need to give a few more years of my life and I’m not sure how many years there are to give up anymore. Also, I like relaxing by the sea, and relaxing by the sea is fine to think about mathematics but sitting down and having to write it is a pain.

TPL: Oh, I see. This is a new statement to me. I thought writing is easier than thinking about new maths.

NST: The thinking is nice, but the writing is horrible. You think something and okay, you think the picture through, get most of the essential things, techniques and so on. You want to write, you’ve got to do things carefully. That can be much, much longer process

TPL: You did between 1971 and 1977

NST: Yeah, that was a lot of writing, a lot of checking.

TPL: We usually ask this question, how did you get yourself interested in and decided to do mathematics research as a living, or even to be major in mathematics.

NST: Oh, good question. The majoring was easy because it didn’t take as much time as the other subjects. I had more time to enjoy my life. So majoring was quite easy. Made life easier.

TPL: A lot of people wish they can say what you have just said.

NST: Well, I found some undergraduate physics boring and frustrating. When I was a sophomore, I had a course on thermodynamics. I couldn’t stand it because there were different quantities there, and they all relate to each other but they are all mixed in. They don’t say what’s held constant and what’s changing. Very frustrating. Yeah it was, it’s all cleaned up now but I found it frustrating. It looked interesting but nothing was precise.

TPL: So that’s not your topic.

NST: So I found physics a bit boring in this respect. Also I was terrible in the laboratory; I was dangerous in the laboratory. If I kept doing chemistry and physics there would have been a disaster. I could have blown the place up or anything. I really was bad in the laboratory. Very clumsy. I also tried humanities too, in my first year, but I thought some of my teachers may be insane. So I started in philosophy and French and I worried some people were sort of mentally defective. I didn’t want to join this community. And then I switched to physics and chemistry and I found these people not as interesting. The opposite thing. They were probably not insane, they were reasonable. They could get out the university and they wouldn’t be locked up but they were very boring. So in between there was mathematics. And also not as much work.

TPL: Now you have come across many mathematicians. Would you be willing to offer your impression on some of them?

NST: I think we do have a lot of interesting characters. Some of the people I was associated with early in my career I got to like a lot and some I found interesting but you may not want them to be the best friend. I talked to Courant a bit when I was at the Courant institute. That was quite interesting. Because Moser was my mentor. I would sometimes go to Courant’s house, and he used to talk to me in a very frank sort of way. He’d tell me first of all I’m stupid to go back to Australia. Just like that. You’re stupid, crazy. Things were very direct. So he was an interesting person. I saw Friedrichs a bit and also Fritz John, in those days. Those people were in a very interesting world. And Jürgen Moser I liked so much. We really respected him, an extremely good person.

TPL: Who was you advisor?

NST: At Stanford it was Gilbarg. And then when I left Stanford I went to Courant for one year as a Courant instructor, the second Courant instructor ever. But my main contact was Moser because Moser had been a visiting professor at Stanford. So a lot of my thesis work and so on arose from talking to Moser as well as Gilbarg. So I had quite a good relationship with Moser and this continued in New York. Our families were often together, we went skiing with Moser and were often round at his house. So this was a very nice relationship. He was a fantastic person.

TPL: Moser was very famous for his KAM theorem. But he has some important work in PDE.

NST: that’s why he was my contact because of his Moser’s Harnack inequality.

TPL: One time I heard him give a talk, and he said that now we are going to talk next about PDE, which is a harder subject. Look at what he did in KAM, it’s very deep stuff. Now, I have heard the name Michael in Australia.

NST: Jim Michael . He was Simon ’s supervisor.

TPL: Is he still around?

NST: No, he died some years ago. I think about 10 years ago. He was the supervisor of Leon Simon.

TPL: So he’s one of the earlier, main figures in analysis in Australia.

NST: Yes, although he was a fairly quiet person, very modest, he’s from Adelaide. He had a several good students at Adelaide that stayed on to do PhDs there with him. Leon was one of those.

TPL: So you are working quite a bit with Xu-Jia Wang, on what kind of problem?

NST: Well, a whole variety of problems. He came as kind of post-doc to Canberra in about 1995, and then we started some collaboration sometime after that in Hessian measures and that collaboration just kept going. Just kept going and going. And it was all pretty non-linear elliptic PDE applications. But Xu-Jia’s really a terrific person. He’s really a top mathematician.

TPL: Yeah, he’s a very nice guy. Because you have been so active in doing research, so you’re not thinking about revising the book with Gilbarg.

NST: Oh, I am thinking about it but I just don’t know if I have to somehow make that break from everything else. I don’t really care about publishing papers anyway. What I don’t like is that I kind of develop something and I don’t want to publish it and I see that someone else has published it. Personally I don’t care about publishing myself. And I don’t like making papers out of simple observations like a lot of people do. Then again if you don’t record your observations, other people will record them afterwards for themselves, so in some sense I would like to make a good break and just do the expository stuff when I have a chance.

TPL: So you may yet to do it, to write book and so-on.

NST: To do some revision of some lecture notes of some course I taught at the University of Tokyo almost 20 years ago. A lot of people use that for knowledge in PDE’s and I am doing some revision.

TPL: Are you still having Ph. D. students now?

NST: No, I haven’t had any by myself for a while. We just had a couple of very good students and but Xu-Jia Wang was their main supervisor. They were Chinese students so everything was done in Chinese. But before that I had a student who went into physics afterwards. The physicists wanted to get someone with expertise in optimal transportation. I had some students before but not in the last couple of years.

TPL: When did you get your PhD?

NST: In 66.

TPL: 1966, that would make it what, 45 years ago right? You have all the spirit, that’s really great.

NST: I have to admit my garden is going well. I am more excited to getting success in the garden than in mathematics theorems. If I have something growing very nicely in the garden, this is a good feeling.

TPL: You can say whatever you say, but you still keep going on. I saw Nirenberg in India last year, he has been at it even for longer period.

NST: Oh yeah. There are people like that and I guess Peter Lax, Eli Stein and so on. They just go and they go, I don’t know, I mean, perhaps they don’t have distractions. I mean what else would you do in New York right? You go out, take in the culture, the theatre and so on or you do mathematics. What else is there?

TPL: I think if you ask people on the street, they can tell you that there are many things you can do.

NST: Whereas I find that in my life, I can just enjoy looking at nature, watching animals like kangaroos and so on. Watching the sea. These things are so nice. Different from people talking to themselves in New York. Of course I love New York by the way, I really enjoy it there very much as long as I am not there for too long. A visit is great.

TPL: But Australia, I can be happy there for a long time. Last time I went there.

NST: You can be bored too. I think if I didn’t have the mathematics, I may find I like the other life, the simple life. But I may like that because I have the mathematics in my mind as well. But if I took the maths out, everything could be completely boring. So that’s another worry. You can have things and take them for granted.

TPL: I knew that you would say that.

NST: But I don’t know, I haven’t tested this out. My wife believes that if I didn’t have my job I wouldn’t know what to do with myself, I would have nothing to do. But for me, I feel like, I don’t want to do the job, I want to sit and look at the sea. Look at the whales, things like this.

TPL: I think your wife is right.

NST: And think a bit about mathematics, but not feel as though it’s a job, the pressure, you know.

TPL: That’s a very upbeat note. Very good.

NST: You gave up the US to be here. You have a life; you have a purpose, right? You are not just one sided right? You have a much bigger mission right?

TPL: I don’t want to go into that. But it’s okay here because there are some young people here I can get along and be happy.

NST: I’m quite respected of you. That they do something for their own environment that they came from and so on.

TPL: Even though Taiwan is not a big place, you still find young people with different background and it’s quite nice to have that mixture. Of course Australia, huge place and people must be very different right? Different background.

NST: So you would see many Taiwanese mathematicians quite regularly. So in Australia, when I’m not travelling, I am not seeing anyone except my immediate colleagues and most of them I don’t necessarily associate with socially, I prefer other people.

TPL: So we have these trilateral meeting between Australia, Italy and Taiwan.

NST: In Wollongong the next one.

TPL: That would be nice, I look forward to that.

NST: They were so happy when they came to Taipei at the end of 2009. The Australian delegation was really happy. Someone who was there, an Australian applied mathematician, Tim Marchant, now is a senior person in the University of Wollongong and he is supporting the next meeting.

TPL: That is very nice.

NST: I don’t know when it is. Did you hear something?

TPL: When was the last one? You could not make it last time. When was the last time?

NST: I think that 2009.

TPL: 2009, December.

NST: So if Wollongong is doing something, they will try to get a webpage out or something. You haven’t had a contact?

TPL: Not yet, but still one and half year from now. So were you one of the people who start it.

NST: Yeah I guess.

TPL: I guess Fon-Che Liu . Last time was very nice and I have a strong impression that Australia does have advantage with this big geographical area in that you can see people really work on their own subject for quite a long while. And there are some people who are a lot interested in application last time when they came.

NST: Well, we mix up the applied people with the others. That’s partly because we are not like the Italians, we don’t have thousands and thousands of people working in PDE so we try to make the PDE a bit broader and include the applied people and a range of people, numerical people. Just try to have this cross interaction. And those people seem quite happy being with the purer people.

TPL: That’s nice to hear. We really appreciate this kind of meeting because we are geographically small and we tend to have a problem of everyone doing more or less the same kind of thing. So you can see the contrast of the cultures of Australia, Italy and Taiwan.

NST: Interesting meetings.

TPL: I presume next time you would be around for that.

NST: Yeah, I should find out when it is. Because I thought they were talking about it late last year James McCoy was the one asked to organise it at Wollongong They were asking me for information about past meetings and so on. This was some time ago.

TPL: You should have come here, although right now, summer is very hot, you come in the fall weather is nice and we don’t work you hard and to relax a little bit so you come and visit us.

NST: Excellent.

• Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.